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+{"text":"\n\\section{Introduction}\\label{sec:intro}\n\nIn recent years, modern TTS systems have largely moved to sequence-to-sequence (Seq2Seq) models, e.g. \\cite{WangY2017_Tacotron,ShenJ2018_Tacotron2,PingW2018_DeepVoice3}, where the alignment between the phonetic or orthographic input sequence and the acoustic output sequence is learned during training and inferred during synthesis. One advantage of this approach is that it leads to a more end-to-end system, in the best case avoiding the need for pre-aligned training data, separate phonetic transcription, or a separate duration model at synthesis. For singing synthesis, not requiring pre-aligned training data is particularly attractive, as many existing tools (e.g. forced alignment with a HMM model) do not yield sufficiently accurate results on expressive singing, often requiring manual correction.\n\nA common approach for Seq2Seq models in TTS is to use a content-based attention mechanism, e.g. \\cite{GehringJ2017_ConvAttention}, sometimes additionally using location-based information, e.g. \\cite{GravesA2013_GMMAttention}. As these mechanisms require access to acoustic information at inference, they are normally used in combination with an autoregressive decoder. Recently, some systems have been proposed that use a feed-forward decoder and an alternative attention mechanism that does not rely on access to acoustic information \\cite{PengK2019_ParaNet,RenY2019_FastSpeech}. These notably provide faster, parallelizable inference, and are reported to produce more robust alignments with fewer mispronounced, repeated or skipped phonemes.\n\nIn the case of singing synthesis, this feed-forward approach is interesting as it avoids the exposure bias problem \\cite{RanzatoMA2016_ExposureBias}, caused by the discrepancy between teacher forced training and fully autoregressive inference. This problem can be especially noticeable in long sustained vowels were prediction errors tend to accumulate over time. Additionally, in our experience reaching similar quality results compared to non-Seq2Seq systems can be quite challenging with standard content-based attention mechanisms.\n\nTo facilitate evaluation of different systems, we only model timbre in this work, and assume F0 to be given. Although we use ground truth F0 extracted from recordings, it is feasible to predict F0 from the input score with an external model, or possibly predict it jointly. Related to this, we use WORLD vocoder features \\cite{MoriseM2016_WORLD} as the output of our system, rather than the more commonly used mel-spectrogram features, as this allows exact control over the synthesized F0. For the best quality results, Seq2Seq systems are typically combined with a neural vocoder, e.g. \\cite{TamamoriA2017_WaveNetVocoder,PrengerR2019_WaveGlow,WangX2019_NSFVocoder}, which can work well from both vocoder or mel-spectrogram features. However, in order to get a better idea of the performance of our model on its own, we do not use this approach in the experiments presented here.\n\nThe principal contributions of this paper are:\n\\begin{inparaenum}\n  \\item Propose a singing synthesizer based on the feed-forward Transformer with a practical Seq2Seq mechanism using an external duration model.\n  \\item Evaluate the quality of this feed-forward model compared to a baseline autoregressive model.\n  \\item Evaluate the importance of self-attention.\n  \\item Evaluate the importance of the accuracy of the duration model used.\n\\end{inparaenum}\n\n\n\\section{Proposed system}\\label{sec:proposed_system}\n\nIn singing synthesis, the alignment between the input phonetic sequence and the output acoustic sequence is strongly constrained by the given musical score. This is a notable difference from TTS, which is generally only weakly constrained by the (average) speech rate. Exploiting this fact, we propose to first generate an approximate initial alignment using note timings and a phoneme duration model. Once the input sequence is roughly aligned to the target output timesteps, we assume that the network is able to gradually refine the alignment through a series of transformations, until reaching something close to the target. Note that this approach is quite different from the approach using content-based attention, as here the initial alignment doesn't use any content at all.\n\nAn important point here is that we assume that the accuracy of the phoneme duration model is not critical to the end results. We assume that the decoder is powerful enough to be able to recover from errors in the initial alignment, to a certain degree. At the same time, the initial alignment can never hugely deviate from the true alignment, as it is heavily constrained by the note timings. To see if this assumption is correct, we purposely use a very simplistic duration model, based on average phoneme durations computed on a different dataset whose segmentation was corrected by hand. While language dependent, in this case the phoneme duration model is not singer dependent and the values could simply be copied from a table, without the need for any data with phonetic timings.\n\n\\subsection{Model architecture}\nThe input to our system is a musical score, consisting of a sequence of notes. Each note consists of an onset, duration, pitch, and a sequence of phonemes, typically corresponding to a syllable. In this work, we define the note onset as the vowel onset, and note end as the onset of the following vowel or silence. Additionally, we provide an external F0 to our system, in order to capture the affect of pitch on timbre. The output of our system is sequence of harmonic and aperiodic vocoder features, which in this case are simply concatenated.\n\nThe main components of our proposed system, as depicted in \\cref{fig:model_architecture}, are the encoder, the aligner and the decoder. The encoder takes the input phonetic sequence and computes a sequence of hidden states corresponding to each phoneme and their local context. The aligner provides a hard alignment by repeating these states according to the predicted phoneme durations, obtaining a sequence of the same length as the output acoustic sequence. Next, some additional conditioning signals derived from F0 and position are added. The decoder, based on the Transformer model \\cite{VaswaniA2017_AttentionIsAllYouNeed}, finally transforms the sequence of encoder hidden states to the target output sequence, through a series of self-attention and convolutional layers.\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=17.8cm]{figures\/model_arch.pdf}\n  \\caption{A diagram of the complete model architecture. On the left is the full system, composed of encoder, aligner and decoder, which themselves are composed of different higher level blocks. On the right, these higher level blocks (sub-layer, gated linear unit (GLU) and attention) are shown in detail.}\n  \\label{fig:model_architecture}\n\\end{figure*}\n\n\\subsection{Encoder}\nOur encoder is based on the encoder proposed in \\cite{PingW2018_DeepVoice3}. First, embeddings are computed from each input phoneme. Then, a series of convolutional blocks with gated linear units (GLUs) \\cite{DauphinYN2017_GLU} allows encoding information about the phonetic context of each phoneme, e.g. corresponding to triphones or pentaphones. Finally a residual shortcut connection from the monophone embeddings is added to the local context output of the convolutional blocks.\n\n\\subsection{Duration model}\nAs noted, we purposely choose to use a very simplistic phoneme duration model in this work. It consists of a simple look-up table, populated with average phoneme durations computed from a dataset of a different singer with manually corrected phonetic segmentation. A simple heuristic is used to ensure that the sum of predicted phoneme durations matches the target note duration.\n\nAs we assume the note onset to correspond to the vowel onset, we first shift all onset consonants of each note to the preceding note (or silence). Then, we look up the sequence of average phoneme durations for each note, $[d_1,d_2,\\dotsc,d_N]$, where $N$ is the corresponding number of phonemes, and $d_1$ corresponds to the duration of the vowel. In order to match the target note duration, $d_n$, we use the predicted consonant durations and fill the remaining duration with the vowel. However, we also ensure at least half of the note's duration is occupied by the vowel by fixing $r_v=0.5$. The scaling factor for the consonants, $r_c$, then becomes,\n\\begin{equation}\nr_c = \n  \\begin{cases}\n  1                                                                    & \\text{ for } N=1,\\\\\n  \\min\\left(1, \\cfrac{d_n - \\rint{r_v d_n}}{\\sum_{i=2}^{N} d_i}\\right) & \\text{ otherwise. }\n  \\end{cases}\n\\end{equation}\nAnd, the adjusted phoneme durations, $[\\hat{d}_1,\\hat{d}_2,\\dotsc,\\hat{d}_N]$,\n\\begin{equation}\n\\hat{d}_i =\n  \\begin{cases}\n  d_n - \\sum\\limits_{j=2}^{N} \\max\\left(1, \\rint{r_c d_j}\\right) & \\text{ for } i=1,\\\\\n  \\max\\left(1, \\rint{r_c d_i}\\right)                             & \\text{ for } i=2,3,\\dotsc,N.\n  \\end{cases}\n\\end{equation}\nNote that all durations here are in integer number of frames, and that there are corrections for rounding errors and zero frame durations.\n\n\\subsection{F0 and position conditioning}\nContinuous log F0 is encoded as a low dimensional vector between zero and one by evaluating several triangular basis functions whose centers are placed at frequencies appropriate for the training data's pitch range.\n\nTransformers typically use an additive trigonometric positional encoding to give the self-attention blocks a sense of the position of their inputs, and provide a linear inductive bias along early on in training. However, in our case we found that a simple $K$-dimensional cyclical encoding of the normalized frame position within each note, $p \\in [0,1] \\subset \\mathbb{R}$, gave slightly better results,\n\\begin{equation}\nv = \\frac{1}{2}\\cos\\left(2\\pi p - 2\\pi\\frac{k-1}{K}\\right) + \\frac{1}{2}\\quad\\text{for }k=1 \\ldots K.\n\\end{equation}\n\n\n\\subsection{Decoder}\n\nOur decoder is based on a feed-forward variant of the Transformer model \\cite{VaswaniA2017_AttentionIsAllYouNeed}, similar to \\cite{RenY2019_FastSpeech}. Each layer consists of a self-attention sub-layer block and a convolutional sub-layer block. Both sub-layers blocks have layer normalization \\cite{BaJ2016_LayerNorm}, dropout and a residual shortcut connection.\n\nFollowing \\cite{VaswaniA2017_AttentionIsAllYouNeed}, our self-attention blocks use the scaled dot product as a scoring function. Additionally, similar to \\cite{SperberM2018_GaussianBiasSelfAttention}, we bias the scores with a Gaussian along the diagonal to favor a more localized self-attention,\n\\begin{equation}\n\\Attention(Q,K,V) = \\softmax\\left(\\frac{QK^{\\transpose}}{\\sqrt{d_{\\text{model}}}} + M\\right) V,\n\\end{equation}\n\\begin{equation}\nM_{j,k} = \\frac{-(j-k)^2}{2\\sigma^2},\n\\end{equation}\nwhere $d_{\\text{model}}$ is the dimensionality of the input vectors, $M \\in \\mathbb{R}^{T \\times T}$ for sequence length $T$, and $\\sigma$ is a learned scale parameter. To reduce memory and computational requirements for the self-attention layers, we may use a reduction factor $r \\geq 1$, which means $r$ frames are predicted per output timestep \\cite{WangY2017_Tacotron,PingW2018_DeepVoice3}. While the use of multi-head attention is typical for NLP applications, we did not find this improved results in our case.\n\nFor the convolutional blocks we use GLUs, which for our case outperform the 2-layer convolutional network with central ReLU activation typically used in Transformer architectures.\n\n\n\\section{Experiments}\\label{sec:experiments}\n\nFor the experiments in this work, we train a model on a proprietary dataset of 41 pop songs performed by a professional English male singer. From this dataset 35 songs were used for training (\\durhm{1}{26} total), 4 for validation and 2 for testing.\n\nOur proposed system uses 64-dimensional input features similar to \\cite{BlaauwM2017_NPSS_MDPI}, extracted with a \\SI{10}{\\milli\\second} hop time. A reduction factor, $r=2$, is used. We use 256-dimensional phoneme embeddings, and an encoder with a single 3x1 GLU block with 64 channels. F0 is coarse coded to a 4-dimensional vector, as is the position within the note, albeit with a cyclical encoding. The decoder consists of 6 layers with (single head) self-attention and 3x1 GLU blocks, all with 256 channels.  The final output projection is to $64r$ channels. Dropout probability is set to 0.1 throughout the model. The learned standard deviation of the Gaussian bias of the self-attention blocks is initialized to 30. Initialization of convolutional layers follows \\cite{GehringJ2017_ConvAttention}. We use the Adam optimizer with $\\beta_1=0.9$, $\\beta_2=0.98$, $\\epsilon=\\num{1e-9}$, and a batch size of 32. We follow the learning rate schedule from \\cite{VaswaniA2017_AttentionIsAllYouNeed}, with a 4000 step warm-up, a base learning rate of \\num{1e-3}, and a total of \\SI{50}{\\kilo\\nounit} updates. Additionally, we use Polyak averaging with a decay of 0.995 for validation and testing. The objective that we optimize is a simple L1 loss between output and target features.\n\nWe compare our proposed feed-forward model, which we label \\mlabel{FFT-NPSS}, to an autoregressive baseline model roughly following \\cite{BlaauwM2017_NPSS_MDPI}, labeled \\mlabel{AR-NPSS}. To study the importance of the accuracy of the approximate initial alignment, we train a version of our model, which uses ground truth phonetic durations rather than predictions by the simple averages duration model. Note that the baseline \\mlabel{AR-NPSS} is a non-Seq2Seq model, so it is also trained on ground truth phonetic durations. To study the importance of self-attention in the model we train a version of our model without attention blocks as well.\n\nWe ran a MOS listening test with 18 participants, which each rated a random subset of 12 out of 20 phrases. Per test 6 stimuli were presented; the 4 systems mentioned previously, and visible and hidden references consisting of a WORLD re-synthesis of the target recording. All systems are presented and rated together to encourage a comparison between them.\n\n\\begin{table}\n  \\centering\n  \\caption{Mean Opinion Score (MOS) ratings on a 1--5 scale with their respective 95\\% confidence intervals.}\n  \\label{tab:mos}\n  \\input{mos_table.tex}\n\\end{table}\n\nThe results of our listening test are shown in \\cref{tab:mos}. We can see that the \\mlabel{FFT-NPSS} system using ground truth phoneme durations performs best, but it is closely followed by the proposed Seq2Seq variant using a simple averages duration model. This shows that the initial alignment provided by the duration model has some importance, but it is not critical. Additionally, our proposed system outperforms the baseline autoregressive \\mlabel{AR-NPSS} system, possibly due to avoiding issues related to exposure bias. Finally, the variant of the \\mlabel{FFT-NPSS} system without self-attention layers performed worst overall, showing that self-attention is an important component for this kind of Seq2Seq mechanism, in our observations especially in terms of providing a coherent timbre over time. While all systems are still rated considerably below the reference WORLD re-synthesis, we expect that this gap would be reduced if we combine our system with a neural vocoder. Some sound examples, both with and without neural vocoder, are available online\\footnote{\\url{https:\/\/mtg.github.io\/singing-synthesis-demos\/transformer\/}}.\n\n\n\\section{Relation to prior work}\\label{sec:prior_work}\n\nOur work is most closely related to the recently proposed FastSpeech model for TTS \\cite{RenY2019_FastSpeech}. This model is also based on the feed-forward Transformer and an initial alignment obtained from a duration model. However, in this case the duration model is trained with the help of a teacher model based on an autogressive Transformer \\cite{LiN2019_TransformerTTS}, which is also used for generating the target mel-spectrogram features. We wanted to avoid the need to train an autoregressive teacher model, as we found this generally challenging for the case of singing voice. Additionally, we apply some modifications to the architecture, such as the use of GLU convolutional blocks, alternative positional encoding and a Gaussian bias for the self-attention layers.\n\nThe ParaNet model \\cite{PengK2019_ParaNet} proposes a different approach to feed-forward TTS. Here, standard content-based encoder-decoder attention is used, but the model is trained trained with the help of attention distillation with an autoregressive teacher model based on \\cite{PingW2018_DeepVoice3}. Besides the reasons mentioned above, we found that the hard alignment used in our approach makes it easier to obtain a quality similar to non-Seq2Seq models, compared to the soft alignment of encoder-decoder attention.\n\nIn singing synthesis, the only Seq2Seq system we are aware of is \\cite{LeeJ2019_KoreanSS}. This model is based on the DCTTS model \\cite{TachibanaH2018_DCTTS}, using content-based encoder-decoder attention, with autoregressive decoder. Similar to our approach, there is an initial alignment of the input states to the output timesteps. However, relying on the fact that the Korean syllable structure has at most one onset and one coda consonant, the first and last frame of the note are assigned to each consonant respectively, and the remaining frames are assigned to the vowel. After which, learning the attention alignment can be facilitated by using diagonally guided attention \\cite{TachibanaH2018_DCTTS}.\n\nNon-Seq2Seq singing synthesizers include those based on autoregressive models \\cite{BlaauwM2017_NPSS_MDPI,BousF2019_SpecEnvSingSynth,YiYH2019_DAR_SS}, feed-forward CNN \\cite{NakamuraK2019_TechnoSpeechCNN}, and feed-forward GAN-based approaches \\cite{HonoY2019_GAN_SS,ChandnaP2019_WGANSing}.\n\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nWe presented a singing synthesizer based on the Transformer model, with a practical Seq2Seq mechanism allowing feed-forward operation. This approach allows training models without the need for pre-aligned training data, which can be cumbersome to prepare for singing data. Compared to a baseline autoregressive model, the proposed model allows for faster inference, avoids issues related to exposure bias, and rates as good or slightly better in listening tests. The use of self-attention resulted to be a key factor in obtaining good quality results, especially in terms of producing coherent timbre. As our model relies on an initial alignment provided by a duration model, we compared a very simplistic duration model to ground truth durations, to see the importance of the initial alignment's accuracy. In listening tests, using ground truth durations was rated highest, but the difference was relatively small. While not shown due to lack of space, in our observations the model can recover from errors in the initial alignment, most likely thanks to the decoder's non-causal convolutions and self-attention layers. For example, while the duration of phrase-final consonants tends to be systematically underpredicted by average durations, in the output of the synthesizer these phonemes have durations close to the target.\\thanksanywhere{This work was funded by TROMPA H2020 No 770376.}\n\n\n\n\n\\clearpage\n\n\n\\bibliographystyle{IEEEbib}\n{\\small","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPerturbation theory provides a useful tool in the studies \nof the weak coupling limit of field theories on a lattice \\cite{Capitani}.\nTo simplify perturbative computations it is helpful to derive analytical \nexpressions for the Feynman integrals.\nTypical integrals emerging in the one-loop approximation have the form\n\\begin{equation}\\label{GenOneLoopInt}\n\\int_{BZ} {dk \\over (2\\pi)^4} {{\\cal P}(\\hat k_{\\mu_0},\\widehat {(k-p_1)}_{\\mu_1},...,\\widehat {(k-p_1-...-p_n)}_{\\mu_1} ) \\over D_{latt}(k) D_{latt}(p_1-k) ... D_{latt}(p_1+...+p_n-k) }\n\\end{equation}\nwhere $\\displaystyle \\hat p_\\mu = {2\\over a} \\sin \\left( {p_\\mu a \\over 2}\\right)$,\n$\\ D_{latt}(k)= \\sum_{\\mu=1}^4 \\hat k_\\mu^2 +m^2 $ (in the boson case), symbol ${\\cal P} $ in the numerator\nmeans ``some polynomial of'' and symbol $BZ$ indicates that integration is performed \nover the Brillouin zone, that is the domain $\\displaystyle -\\;{\\pi\\over a}< k_\\mu < {\\pi\\over a}$.\n\nSuch integrals  cannot be calculated analytically at finite values of $a$.\nIn the limit $a\\to 0$ they can be evaluated by the Kawai--Nakayama--Seo method \\cite{Kawai}.\n\nThe above integrand can be represented in the form \n\\begin{equation}\\label{Kawai001}\nI(k,\\tilde p,m^2;a)=I(k,0,0;a)\\ +\\ (I(k,\\tilde p,m^2;a)-I(k,0,0;a)),\n\\end{equation}\nwhere $\\tilde p$ is the set of all relevant external momenta, $m$ is the mass.\n$I(k,\\tilde p,m^2;a)-I(k,0,0;a)$ has a smooth continuum limit ($pa\\to 0$ and $ma\\to 0$)\nand involves no ultraviolet (UV) divergencies.\nIt can be computed with some continuum regularization\nsuch as dimensional (DR) or with fictitious mass\\footnote{The regularization by fictitious mass\nis obtained by adding the term $\\mu_R^2 = 2\\mu_B^2$ to the denominator of each boson or fermion propagator.} (FMP).\n\nBoth $I(k,0,0;a)$ and $(I(k,\\tilde p,m^2;a)-I(k,0,0;a))$\ninvolve infrared (IR) divergencies.\n\nThough these IR divergencies cancel each other, IR regularization is needed:\n$$\nI(k,\\tilde p,m^2;a) = \\lim_{\\mu_R^2 \\to 0} I(k,\\tilde p,m^2;a,\\mu_R^2).\n$$\nIn this work, we use the infrared regularization introduced below in the formulas\n(\\ref{ScalarBosonPropLatt}) for bosons and (\\ref {DenomFermPropLatt}) for fermions.\nThus the sought-for integral \nis represented as the sum of the integral over the Euclidean momentum space\n(which is readily calculated by well-known method)\nand the \"zero-momentum\" integral over the Brillouin zone.\nCalculation of the latter integrals forms the subject of the present study.\n\nIn recent years, considerable study was given to computations with \nrather complicated actions (see, for example \\cite{Holger1} and \\cite{Holger2}). \nIn so doing, one is confronted with an integrand involving products of \n$(\\hat k_\\mu)^n$ at large values of $n$.\nAlgorithms for computation of such integrals with both\nbosonic and fermionic denominators were proposed in \\cite{BCP} and \\cite{Melnikov}.\n\nAn outline of this paper is as follows.\nIn this work, the algorithm proposed in \\cite{BCP} is employed to obtain \na comprehensive set of the integrals needed in computations\nof various matrix elements. For the reader's convenience, a detailed exposition of the Burgio--Caracciolo--Pelissetto\n(BCP) algorithm is given. In Section~1, we deal with the bosonic case.\nMaking use of the BCP algorithm, we derive an explicit form of the recursion relations:\nformulas (\\ref{FMR_BasBosInt_FinPart_min}), (\\ref{FMR_BasBosInt_FinPart_min_vspom}),\nand (\\ref{RRforJfuncBosNeg}). \nIn Section~2 we describe computations in the fermionic case.\nIn this Section, we also begin with the exposition of the BCP algorithm \nand use it to find an explicit form of the recursion relations for the \nfunctions $B(p,q)$ and $J(p,q)$ related to the functions ${\\cal F}_\\delta(p,q)$ used in \\cite{BCP}.\nThese relations are presented in the Appendices. As a matter of fact,\nthey provide a computer program for a calculation of the general fermionic integrals (\\ref{eq:GenFermInt}).\nThe results obtained in this way are discussed in the Conclusions.\n\n\\section{Boson Integrals}\\label{BosonIntegrals}\n\nWe compute the bosonic `zero-momentum' integrals of the type (we set $a=1$)\n\\begin{equation}\\label{BosIntInitDef}\nF(q,n_1,n_2,n_3,n_4) = \\lim_{\\delta\\to 0} \\int dk\\;{(\\cos k_1 )^{n_1} (\\cos k_2 )^{n_2} (\\cos k_3 )^{n_3} (\\cos k_4 )^{n_4} \\over \\Delta_B^{(q+\\delta)}}\n\\end{equation}\nwhere\n\\begin{equation}\\label{ScalarBosonPropLatt}\n\\Delta_B = 4+\\mu_B^2-\\cos(k_1 )-\\cos(k_2 )-\\cos(k_3 )-\\cos(k_4 )\n\\end{equation}\nis the scalar boson propagator,\n$\\mu_B$ is the infrared regulator mass, and $\\delta$ is an infinitesimal\nparameter needed for an additional intermediate regularization.\nAll integrals in one-loop calculations with the boson propagators can be reduced to the integrals of this type.\n\nSince $F(q;n_1,n_2,n_3,n_4) $ is symmetric in the arguments \n$ n_1, n_2, n_3,$ and $ n_4$, we consider only the case\n\\begin{equation} \nn_1 \\geq n_2 \\geq n_3 \\geq n_4.\n\\end{equation}\n\n\\subsection{Formulas of Reduction}\nA computation of the massless\\footnote{Here $\\mu_B$ is the regulator mass, \nthus we consider the massless limit at the end of computations.} boson integrals \nover the Brillouin zone is based on the following algorithm:\n\\begin{eqnarray}\\label{ReducFormulasGT2}\n&\\mbox{ if} &\\qquad n_4\\geq 2\\qquad \\mbox{ then}  \\qquad   F(q,n_1,n_2,n_3,n_4) = F(q,n_1,n_2,n_3,n_4-2)\\\\ \\nonumber\n&& -\\; {1\\over q-1+\\delta}  \\left((n_4-1) F(q-1,n_1,n_2,n_3,n_4-1) -\n(n_4-2) F(q-1,n_1,n_2,n_3,n_4-3)  \\right),\\\\ \\nonumber\n &  \\mbox{else if}& \\quad n_3\\geq 2 \\qquad \\mbox{ then} \\qquad  F(q,n_1,n_2,n_3,0) = F(q,n_1,n_2,n_3-2,0)\\\\ \\nonumber\n && -\\; {1\\over q-1+\\delta}  \\left((n_3-1) F(q-1,n_1,n_2,n_3-1,0) -\n(n_3-2) F(q-1,n_1,n_2,n_3-3,0)  \\right),\\\\ \\nonumber\n& \\mbox{ else if} &\\quad n_2\\geq 2 \\qquad \\mbox{ then} \\qquad F(q,n_1,n_2,0,0) = F(q,n_1,n_2-2,0,0)\\\\ \\nonumber\n&& -\\; {1\\over q-1+\\delta}  \\left((n_2-1) F(q-1,n_1,n_2-1,0,0) -\n(n_2-2) F(q-1,n_1,n_2-3,0,0)  \\right),\\\\ \\nonumber\n& \\mbox{ else if} &\\quad n_1\\geq 2 \\qquad  \\mbox{ then} \\qquad F(q,n_1,0,0,0) = F(q,n_1-2,0,0,0)\\\\ \\nonumber\n&& -\\; {1\\over q-1+\\delta}  \\left((n_1-1) F(q-1,n_1-1,0,0,0) -\n(n_1-2) F(q-1,n_1-3,0,0,0)  \\right). \\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\\label{ReducFormulasOne}\nF(q,n_1,n_2,n_3,1) &=& (4+\\mu_B^2) F(q,n_1,n_2,n_3,0) - F(q-1,n_1,n_2,n_3,0) \\\\ \\nonumber\n &-& F(q,n_1+1,n_2,n_3,0) - F(q,n_1,n_2+1,n_3,0) - F(q,n_1,n_2,n_3+1,0), \\\\ \\nonumber\nF(q,n_1,n_2,1,0) &=& {1\\over 2} \\left( (4+\\mu_B^2) F(q,n_1,n_2,0,0) - F(q-1,n_1,n_2,0,0)\\right. \\\\ \\nonumber\n && \\left. - F(q,n_1+1,n_2,0,0) - F(q,n_1,n_2+1,0,0)\\right), \\\\ \\nonumber\nF(q,n_1,1,0,0) &=& {1\\over 3} \\left( (4+\\mu_B^2) F(q,n_1,0,0,0) - F(q-1,n_1,0,0,0)\\right. \\\\ \\nonumber\n && \\left. - F(q,n_1+1,0,0,0) \\right), \\\\ \\nonumber\nF(q,1,0,0,0) &=& {1\\over 4} \\left( (4+\\mu_B^2) F(q,0,0,0,0) - F(q-1,0,0,0,0) \\right).  \\nonumber\n\\end{eqnarray}\nThe above identities can be obtained using integration by parts \\cite{BCP};\norder $O(\\delta^2)$ terms should be omitted. \n\nThus we obtain an expression for each integral $F(q,n_1,n_2,n_3,n_4)$\nin terms of the functions\\footnote{In what follows, $\\displaystyle G(q, \\mu_B^2) = lim_{\\delta\\to 0} G_\\delta (q,\\mu_B^2)$.}\n\\begin{equation}\nG_\\delta (q,\\mu_B^2)=\\int {dk \\over (2\\pi)^4 } {1\\over (\\Delta_B)^{q+\\delta}},\n\\end{equation}\nit has the form\n\\begin{eqnarray}\\label{BosFtoGgen}\nF(q,n_1,n_2,n_3,n_4) &=& \\sum_{r=q-n_1-n_2-n_3-n_4}^{q} a_{qr}(\\delta,\\mu_B^2,\\tilde n) \nG_\\delta (r,\\mu_B^2) \\\\ \\nonumber\n&=&\n\\sum_{r=q-n_1-n_2-n_3-n_4}^{0} a_{qr}(\\delta,0,\\tilde n) G_\\delta (r,0) \\;+\\;\n\\sum_{r=1}^{q} a_{qr}(0,\\mu_B^2,\\tilde n) G(r,\\mu_B^2)\\;+\\; {\\cal O}(\\mu_B^2), \\nonumber\n\\end{eqnarray}\nwhere $\\tilde n$ is short-hand notation for $n_1,n_2,n_3,n_4$.\nIn this sum, the terms with $r\\leq 0$ and those with $r>0$ should be considered separately:\n\\begin{itemize}\n\\item  The coefficients $a_{qr}(\\delta,\\mu_B^2,\\tilde n)$ at $r\\leq 0$ involve the pole $\\displaystyle {1\\over \\delta}$:\n\\[\na_{qr}(\\delta,\\mu_B^2,\\tilde n) = a_{qr}^{(sing)}(\\mu_B^2,\\tilde n)\\;{1\\over \\delta} + a_{qr}^{(reg)}(\\mu_B^2,\\tilde n) + O(\\delta),\n\\]\nso that $ G_\\delta (r,0)$ must be expanded to the order ${\\cal O}(\\delta)$.\nSince $ G_\\delta (r,\\mu_B^2)$ has no infrared divergencies at $r \\leq 0$\nand the coefficients $a_{qr}(\\delta,\\mu_B^2,\\tilde n)$ are polynomials in $\\mu_B^2$, \nthe values of $a_{qr}^{(sing)}$ and $a_{qr}^{(reg)}$ should be evaluated at $\\mu_B=0$.\n\n\\item At $r > 0$, the coefficients $a_{qr}(\\delta,\\mu_B^2,\\tilde n)$ involve no \npoles in $\\delta$ and, therefore, $\\delta$ can be safely set to zero. However, at $r>0$,\n$G_\\delta (r,\\mu_B^2)$ involves infrared divergencies, so that\nthe $\\mu_B$ dependence of the coefficients $a_{qr}$ should be kept.\n\\end{itemize}\n\nFrom these properties it follows that we should compute the quantities \nthat appear in the right-hand sides of the formulas\n\\begin{eqnarray}\\label{DefJ}\n\\mbox{for} \\quad r\\leq 0, \\qquad && G_\\delta (r,\\mu_B^2) = {\\cal B}_{-r} + J(r)\\delta +O(\\delta^2)\\;+\\; {\\cal O}(\\mu_B^2), \\\\[1mm] \\nonumber\nr=1 \\qquad && G (1,\\mu_B^2) = J(1)\\;+\\; {\\cal O}(\\mu_B^2), \\\\[1mm] \\nonumber\nr=2 \\qquad && G (2,\\mu_B^2) = J(2) + (\\ln \\mu_B^2 + C)\\;+\\; {\\cal O}(\\mu_B^2), \\\\[1mm] \\nonumber\n\\mbox{for} \\quad r > 2, \\qquad && G (r,\\mu_B^2) = J(r) + D_{r0}(\\ln \\mu_B^2 + C) + \\sum_{k=1}^{r-2} D_{rk}\/(\\mu_B)^{2k}\\;+\\; {\\cal O}(\\mu_B^2),\n\\end{eqnarray}\nwhere $C=0.577...$ is the Euler-Mascheroni constant, $D_{rn}$ and $J(r)$ are some constants to be determined and \n${\\cal B}_r$ are given by \n\\begin{equation}\\label{Bdirect}\n{\\cal B}_r=\\lim_{\\mu_B\\to 0}\\ \\int_{BZ} {dk\\over (2\\pi)^4}\\; \\Delta_B^r\n\\end{equation}\n(note that $r>0$), some of them can be found in Appendix~1.\n\n\\subsection{Computation of the Divergent Part \\\\ (Fictitious Mass Regularization)}\n\nWe consider the representation of $G_\\delta(q,\\mu_B^2)$\nin terms of the modified Bessel function:\n\\begin{equation} \\label{GdeltaReprBessel}\nG_\\delta (q,\\mu_B^2)=\\int {dk \\over (2\\pi)^4 } {1\\over (\\Delta_B)^{q+\\delta}}\n={1\\over \\Gamma(q+\\delta)} \\int_0^\\infty t^{q-1+\\delta}\\; dt\\ \\left[ e^{-4t-\\mu_B^2 t} I_0^4(t)\\right]\n\\end{equation}\nand divide the domain of integration into two parts:\n$\\int_0^\\infty = \\int_0^1 + \\int_1^\\infty $. The integral over\nthe segment $[0,1]$ converges. The divergent part arises from the\nlatter integral and can be isolated by subtracting $q-1$ terms of the\nasymptotic expansion at $z\\to \\infty$ of the function \n\\begin{equation}\\label{InfeldAsExp0}\n\\exp(-4z) I_0^4(z) \\simeq {1\\over (2\\pi z)^2}\n\\left(1+ {b_1\\over z} + {b_2\\over z^2} + ... \\right);\n\\end{equation}\n$b_i$ at $i\\leq 20$ are given in Appendix~1.\nWe isolate the divergent part $\\bar G_{div}^{M} (q,\\delta,\\mu_B)$ as follows:\n\\begin{eqnarray} \\label{G_FMRdef1}\nG_\\delta (q,\\mu_B^2)= \\bar G_{div}^{M} (q,\\delta,\\mu_B) + \\bar J_\\delta(q) &=& {1\\over \\Gamma(q+\\delta)} \\left\\{ \n\\int_0^1 t^{q-1+\\delta}\\; dt\\ \\left[ e^{-4t-\\mu_B^2 t} I_0^4(t)\\right]\\right. + \\\\ \\nonumber\n&&  + \\int_1^\\infty t^{q-1+\\delta}\\; dt\\ e^{-\\mu_B^2 t} \\left[ e^{-4t} I_0^4(t)- {1\\over (2\\pi t)^2} \n\\sum_{n=0}^{q-2} {b_n \\over t^n} \\right]  \\\\ \\nonumber\n&& + \\int_1^\\infty t^{q-1+\\delta}\\; dt\\ \\left. {1\\over (2\\pi t)^2}\\; e^{-\\mu_B^2 t} \\\n\\sum_{n=0}^{q-2} {b_n \\over t^n}  \\right\\}, \\nonumber\n\\end{eqnarray}\nwhere the first and second lines are designated by $\\bar J_\\delta(q)$ and\nthe third---by $\\bar G_{div}^{M}(q,\\delta,\\mu_B)$:\n\\begin{eqnarray}\\label{FMR_DivPart_nonmin}\n&& \\bar G_{div}^{M}(q,\\delta,\\mu_B) =\n{1\\over (2\\pi )^2\\; \\Gamma(q+\\delta)} \\; \\sum_{n=0}^{q-2}\\left( \\int_0^\\infty - \\int_0^1 \\right)\\;\ndt \\ b_n  t^{q-3-n+\\delta}\\; e^{-\\mu_B^2 t} \\\\ \\nonumber\n&& = {1\\over (2\\pi)^2 \\Gamma(q)} \n\\left[ \\; - \\; b_{q-2} l_C\n+ \\sum_{k=1}^{q-2}  b_{q-k-2} \\left({\\Gamma (k) \\over (\\mu_B^2)^k} \n- {1\\over k}  \\right)\\right] + \\\\ \\nonumber\n&+& {\\delta \\over (2\\pi)^2 \\Gamma(q)} \\ \\left[ b_{q-2} \\left( {1\\over 2} l_C^2 + {\\pi^2\\over 12}\n+\\psi(q) l_C \\right)\\right.\n\\left. +\\sum_{n=1}^{q-2} b_{q-2-n} \\left( {\\Gamma(n)\\over (\\mu_B^2)^n }\n(\\psi(n) - \\psi(q) - l_C + C) + {1\\over n^2} \\right) \\right], \\nonumber\n\\end{eqnarray}\nwhere $\\displaystyle \\psi(n)=\\sum_{k=1}^{n-1} {1\\over k} \\ - C$ and $l_C=(\\ln \\mu_B^2 +C)$.\n\nIn the case of purely boson integrals, $O(\\delta)$ terms can be omitted;\nhowever, they are needed  for a computation of the divergent part of the integrals\n(\\ref{eq:GenFermInt}) involving fermion denominators.\n\nWe can also isolate the divergent part in the so called ``minimal way''\n\\begin{equation}\\label{FMR_DivPart_min}\nG_{div}^{M}(q,\\delta,\\mu_B) = {1\\over (2\\pi)^2 \\Gamma(q)} \\left\\{\n\\left[-\\; b_{q-2} l_C + \\sum_{k=1}^{q-2}  b_{q-k-2} {\\Gamma (k) \\over (\\mu_B^2)^k} \n\\right] \\ + \\right. \n\\end{equation}\n\\[\n+ \\delta \\left. \\left[ b_{q-2} \\left( {1\\over 2} l_C^2 \n+ \\psi(q) l_C \\right) +\\sum_{n=1}^{q-2} b_{q-2-n} {\\Gamma(n)\\over (\\mu_B^2)^n }\n\\left(\\psi(n) - \\psi(q) - l_C + C \\right) \\right] \\right\\}; \\nonumber\n\\]\n\n\\begin{equation}\\label{DeltaJ}\n\\!\\! \\bar G_{div}^{M}(q,\\delta,\\mu_B) = G_{div}^{M}(q,\\delta,\\mu_B) - {1\\over (2\\pi)^2 \\Gamma(q)} \n \\sum_{k=1}^{q-2}  {b_{q-k-2} \\over k}\n + {\\delta \\over (2\\pi)^2 \\Gamma(q)} \\ \\left[ b_{q-2} {\\pi^2\\over 12}\n+\\sum_{n=1}^{q-2} {b_{q-2-n} \\over n^2}\\right].\n\\end{equation}\n\nNote that $J(q)$ that appears in (\\ref{DefJ}) is connected with the quantity \n$\\bar J(q)=\\lim_{\\delta\\to 0} \\bar J_\\delta(q)$ (see (\\ref{G_FMRdef1})) by the relations\n\\begin{equation}\nG (q,\\mu_B^2) = \\lim_{\\delta\\to 0} G_\\delta (q, \\mu_B^2) = \n\\bar G_{div}^{M}(q,0,\\mu_B^2) + \\bar J(q) =  G_{div}^{M} (q,0,\\mu_B^2) + J(q),\n\\end{equation}\nso that\n\\begin{equation}\n\\bar J(q) - J(q) = {1\\over (2\\pi)^2 \\Gamma(q)} \\sum_{k=1}^{q-2}  {b_{q-k-2} \\over k}\n\\end{equation}\nand the coefficients $D_{rk}$ from the formula (\\ref{DefJ}) are determined from the\nthe equation (\\ref{FMR_DivPart_min}):\n\\begin{equation}\nD_{rk} = {b_{r-k-2} \\Gamma(k)\\over (2\\pi)^2 \\Gamma(r)} \\ \\ \\mbox{at} \\ \\ 1\\leq k\\leq r-2;\n\\qquad D_{r0} = -\\;{ b_{r-2} \\over (2\\pi)^2 \\Gamma(r)}.\n\\end{equation}\n\n\\subsection{Computation of the Finite Parts \\\\ (Fictitious Mass Regularization)}\n\nUsing the reduction formulas (\\ref{ReducFormulasGT2}) and (\\ref{ReducFormulasOne}), \nwe obtain expressions for the integrals\n$F(q;n_1,n_2,n_3,n_4) \\equiv F(q,\\tilde n)$ \nin terms of the quantities $J(r), {\\cal B}_r$, and $D_{rn}$ determined above (see formula (\\ref{BosFtoGgen})).\n\nThe next step is to use recursion relations \nmaking it possible to express $J(r)$ at $r\\geq 4$\nin terms of the basic boson constants $J(1)$, $J(2)$, and $J(3)$\nand at $r\\leq 0$ in terms of $J(1)$, $J(2)$, $J(3)$, and $J(0)$.\nThe recursion relations are obtained by making use of the trivial identity\n\\begin{equation}\n\\Delta_B-4 - \\mu_B^2 +\\sum_{\\mu=1}^{4} \\cos(k_\\mu)=0.\n\\end{equation}\nInserting this identity in the integrals\n\\begin{eqnarray}\nF(q,1,1,1,1) &=& \\int dk\\;{\\cos(k_1) \\cos(k_2) \\cos(k_3) \\cos(k_4) \\over \\Delta_B^{(q+\\delta)}}  \\nonumber\n\\end{eqnarray}\nwe arrive at\n\\begin{equation}\n(4+\\mu_B^2) F(q,1,1,1,1)= 4 F(q,2,1,1,1)+4 F(q-1,1,1,1,1),\n\\end{equation}\nNow we express $F(q,1,1,1,1)$ etc. in terms of \nthe values $J(r)$ and thus obtain the sought for relations between them.\nWith these relations, $J(r)$ at $r\\leq 0$ and $r\\geq 4$ is readily \nexpressed in terms of $J(1)$, $J(2)$, and $J(3)$.\nWe consider FMR with the finite part (\\ref{FMR_DivPart_min}) defined in the ``minimal''  way.\nGiven\n\\begin{equation}\\label{FMR_DivPart_min0}\nG_{div}^{M}(q,\\mu_B^2) = {1\\over (2\\pi)^2 \\Gamma(q)} \n\\left[b_{q-2} l_C + \\sum_{k=1}^{q-2}  b_{q-k-2} {\\Gamma (k) \\over (\\mu_B^2)^k} \\right],\n\\end{equation}\nwe derive the recursion relations for $J(q)\\quad (q>0)$ as follows:\n\n\\begin{eqnarray}\\label{FMR_BasBosInt_FinPart_min}\nJ(q) &=& {1 \\over 384 (q-1) (q-2)^2 (q-3)} \\\\ \\nonumber\n&& \\left\\{ 16 (q-2) (q-3) \\left[12 + 25 (q-2) (q-3) \\right] J(q-1) \\right. \\\\  \\nonumber\n&& +\\; 4(q-3)^2 \\left[ -17 - 35(q-3)^2 \\right]\\; J(q-2) \\\\ \\nonumber\n&& +\\; 4\\;\\left[1+ 5(q-3)^3(q-4)-5(q-3)(q-4)^2 \\right] \\;J(q-3)\\\\ \\nonumber\n&& \\left. -(q-4)^4\\;J(q-4) \\right\\}\\\\ \\nonumber\n&& +\\; {1\\over (q-2)}\\; D(q)\\\\ \\nonumber\n&& -\\; {25\\over 24 (q-1) (q-2)}\\; (2q-5)\\; D(q-1)\\\\ \\nonumber\n&& + \\; {1\\over 96 (q-1)(q-2)^2}\\; \\left[17+105(q-3)^2\\right] \\; D(q-2)\\\\ \\nonumber\n&& +\\; {5\\over 96 (q-1)(q-2)^2(q-3)}\\; \\left[-1-4(q-3)^2(q-4)+2(q-4)^2\\right]\\; D(q-3)\\\\ \\nonumber\n&& + \\; {5\\over 384 (q-1) (q-2)^2 (q-3)}\\; (q-4)^3 \\; D(q-4);\\nonumber\n\\end{eqnarray}\nwhere $\\displaystyle D(q)={b_{q-2}\\over (q-1)\\!}$; this being so, $D(q)$\nsatisfies the recurrent relation\n\\begin{eqnarray}\\label{FMR_BasBosInt_FinPart_min_vspom}\n D(q) &=& {1\\over 384 (q-1) (q-2)^2 (q-3)} \\\\ \\nonumber\n&& \\left\\{  16 (q-2)(q-3) \\left[12+25(q-2)(q-3) \\right]\\; D(q-1) \\right. \\\\ \\nonumber\n&&  +\\; 4(q-3)^2 \\left[ -17 - 35(q-3)^2\\right]\\; D(q-2)\\\\ \\nonumber\n&&  +\\; 4 \\left[1+ 5(q-3)^3(q-4)-5(q-3)(q-4)^2 \\right]\\; D(q-3)\\\\ \\nonumber\n&& \\left. -\\;(q-4)^4\\;D(q-4)\\right\\};\n\\end{eqnarray}\nwith the initial conditions\n\n\\begin{eqnarray}\\label{J0123boson}\nJ(0)&=& J_0; \\\\ \\nonumber\nJ(1)&=& 2 Z_0; \\\\ \\nonumber\nJ(2)&=& {F_0\\over (2\\pi)^2}; \\\\ \\nonumber\nJ(3)&=& {Z_1\\over 32}\\;+\\; {1\\over (2\\pi)^2} {F_0\\over 4}\n\\;-\\; {1\\over (2\\pi)^2}\\,{13 \\over 48} \\;-\\; {1\\over 128}; \\nonumber\n\\end{eqnarray}\nand\n\n\\begin{equation}\\label{IniCondforDIVboson}\nD(1)=0; \\qquad \\qquad\nD(2)= {1\\over (2\\pi)^{2}}; \\qquad \\qquad\nD(3)= {1\\over (2\\pi)^{2}}\\ {1\\over 4}. \n\\end{equation}\n\nRecurrent relations for $J(q)$ at $q<0$ can be\nderived by the same token, they have the form\n\\begin{eqnarray}\\label{RRforJfuncBosNeg}\nJ(q)&=& -\\; {1\\over q^4}\\ \\left[-4 \\left( 1+5(q+1)q+5(q+1)^2 q^2\\right) J(q+1) \\right. \\\\ \\nonumber\n\t&+& 4 (q+1)^2 (17+35(q+1)^2) J(q+2) \\\\ \\nonumber\n\t&-& 16 (q+2) (q+1) (25 (q+2) (q+1)+12) J(q+3) \\\\ \\nonumber\n\t&+& 384 (q+3) (q+2)^2 (q+1) J(q+4) \\\\[1mm] \\nonumber\n\t&+&{2 q^3 (3+5 (q+3)^2 (q+1)) \\over (q+4)(q+3)(q+2)(q+1)}\\; {\\cal B}_{-q} \\\\[1mm] \\nonumber\n\t&+& 4 \\;{  (-40 q^6-330 q^5-985 q^4-1376 q^3 -1015 q^2-410 q-70)\\over (q+4)(q+3)(q+2)(q+1)}{\\cal B}_{-q-1}\\\\[1mm] \\nonumber\n\t&+& 8 \\;{ (q+1) (105 q^4+788 q^3+1998 q^2+2052 q +788)\\over (q+4)(q+3)(q+2)}{\\cal B}_{-q-2}\\\\[1mm] \\nonumber\n\t&+& 32 \\;{ (q+2) (q+1) (31 (q+4)^2 -81 (q+3)^2)\\over (q+4)(q+3)} {\\cal B}_{-q-3} \\\\[1mm] \\nonumber\n\t&+&\\left. 768\\; {(q+3) (q+2) (q+1)\\over(q+4)} {\\cal B}_{-q-4} \\right] \\qquad \\mbox{for}\\quad  q\\leq -3. \\nonumber\n\\end{eqnarray}\nThe values of ${\\cal B}_q$ can be computed either directly by the formula \n(\\ref{Bdirect}) or with the use of the recurrent relations\n\n\\begin{eqnarray}\\label{RecRelForBbosonic}\n{\\cal B}_{q}&=& {1\\over q^4}\\left[ -\\; 384 (q-1) (q-2)^2 (q-3) \\;{\\cal B}_{q-4} \\right. \\\\ \\nonumber\n          &&\\qquad +\\; 16 (q-1) (q-2) \\big(25 (q-1) (q-2) + 12\\big) \\;{\\cal B}_{q-3} \\\\ \\nonumber\n          &&\\qquad -\\; 4 (q-1)^2 \\big( 35 (q-1)^2 + 17\\big) \\;{\\cal B}_{q-2} \\\\ \\nonumber\n          &&\\qquad \\left. +\\; 4 (q^5 - (q-1)^5) \\;  {\\cal B}_{q-1} \\right]  \\nonumber\n\\end{eqnarray}\n\n\n\nwith the initial conditions\n\n\\begin{equation}\\label{IniCondForBbosonic}\n {\\cal B}_{0} = 1; \\qquad\n {\\cal B}_{1} = 4; \\qquad\n {\\cal B}_{2} = 18;\\qquad\n {\\cal B}_{3} = 88.\n\\end{equation}\n\n\nThe values of $J(-1), J(-2)$, and $J(-3)$ can be determined \nin the same way, the respective identities have the form\n\\begin{eqnarray} \n J(-4) &=& -9\/16 (13\/9-781\/36 J(-3)+83 J(-2)-108 J(-1)+ 32 J(0) ); \\\\ \\nonumber\n J(-3) &=& 16\/27 (-11\/2+211\/12 J(-2)-157\/3 J(-1)+124\/3 J(0)+16 J(1));\\\\ \\nonumber\n J(-2) &=& -3 \\left(-3\/2+{8\\over (2\\pi )^2} -31\/12 J(-1)+13\/3 J(0)+ 4 J(1)\\right);\\\\ \\nonumber\n J(-1) &=& 144 \\left(-1\/36-{ 13\\over 9 (2\\pi)^{2}}+1\/36 J(0)+4\/3 J(2)-16\/3 J(3)\\right); \\nonumber\n\\end{eqnarray}\n\nThe integrals (\\ref{BosIntInitDef}) can also be expressed \nin therms of the quantities $l_C$, and\n\\begin{eqnarray}\\label{BasBosConstNum}\nZ_0 &\\approx& 0.154933390231060214084837208 \\\\ \\nonumber\nZ_1 &\\approx& 0.107781313539874001343391550 \\\\ \\nonumber\nF_0 &\\approx& F_0^C -\\ln 2 = 4.369225233874758 -\\ln 2 \\nonumber\n\\end{eqnarray}\ndetermined from the relations\\footnote{In the review \\cite{Capitani}, the constant $F_0^C \\approx 4.369225233874758$ \nis designated by $F_0$.}\n\\begin{eqnarray}\nF(1,0,0,0,0) &=& 2Z_0+O(\\mu_B^2) \\\\ \\nonumber\nF(2,0,0,0,0) &=& - \\; {l_C\\over (2\\pi)^2}\\;+\\;{F_0\\over (2\\pi)^2} \\; +O(\\mu_B^2)\\\\ \\nonumber\nF(3,0,0,0,0) &=& {1\\over (2\\pi)^2}\\left({1\\over 2\\;\\mu_B^2} - \\; {l_C\\over 4} - {13\\over 48} + {F_0 \\over 4} \\right)\n-{1\\over 128}\\;+\\;{Z_1\\over 32} +O(\\mu_B^2). \\nonumber\n\\end{eqnarray}\nThis being so, the initial conditions for the recurrent relations\nare given by the formula (\\ref{J0123boson}).\n\nIt should be noted that $J_0$ does not appear \nin the ultimate expressions for the integrals of the type (\\ref{BosIntInitDef}),\ntherefore, its numerical value is not needed.\n\n\n\\subsection{Dimensional Regularization \\label{sec:DRbos}}\n\nFirst we introduce the quantity $\\bar J(q;\\tilde n)$\nanalogous to $\\bar J_\\delta(q)$ defined in (\\ref{G_FMRdef1}):\n\\begin{eqnarray} \\label{defJbasic1}\n \\bar J(q;\\tilde n) &=& \\lim_{\\mu_B \\to 0} \\lim_{\\delta \\to 0}{1\\over \\Gamma(q+\\delta)} \\left\\{ \n\\int_0^1 t^{q-1+\\delta}\\; dt\\ \\left[ e^{-(4+\\mu_B^2) t} {{\\cal T}(\\tilde n)} \\right]\\right. + \\\\ \\nonumber\n&+& \\left. \\int_1^\\infty t^{q-1+\\delta}\\; dt\\ e^{-\\mu_B^2 t} \\left[ e^{-4t} {{\\cal T}(\\tilde n)} - {1\\over (2\\pi t)^2} \n\\sum_{k=0}^{q-2} {b_k (\\tilde n) \\over t^k} \\right]\\right\\},  \\nonumber\n\\end{eqnarray}\nwhere\n\\begin{equation} \n{\\cal T}(\\tilde n)= \\left[ \\left( {\\partial\\over \\partial t} \\right)^{n_1} I_0(t)\\right]\\ \n\\left[ \\left( {\\partial\\over \\partial t} \\right)^{n_2} I_0(t)\\right]\\ \n\\left[ \\left( {\\partial\\over \\partial t} \\right)^{n_3} I_0(t)\\right]\\ \n\\left[ \\left( {\\partial\\over \\partial t} \\right)^{n_4} I_0(t)\\right].\n\\end{equation}\nIt represents the finite part of the general boson integral (\\ref{BosIntInitDef})\nprovided that the divergent part is defined by the formula \nsimilar to (\\ref{FMR_DivPart_nonmin}).\nWe omit here $O(\\delta)$ terms because they are only needed for the calculation of fermion integrals\nin the fictitious mass regularization. Thus we set $\\delta=0$.\nThen it should be noted that\n\\begin{eqnarray}\n \\bar J(q;\\tilde n) &=& \\lim_{\\epsilon \\to 0} {1\\over \\Gamma(q)} \\left\\{ \n\\int_0^1 t^{q-1}\\; dt\\ \\left[ e^{-(4-2\\epsilon)t} I_0^{-2\\epsilon}(t) {{\\cal T}(\\tilde n)} \\right]\\right. + \\\\ \\nonumber\n&+& \\int_1^\\infty t^{q-1}\\; dt\\, \\left[ e^{-(4-2\\epsilon)t} I_0^{-2\\epsilon}(t) {{\\cal T}(\\tilde n)} - {1\\over (2\\pi t)^{2-\\epsilon}} \n\\sum_{k=0}^{q-2} {\\tilde b_k(\\tilde n) \\over t^k} \\right],  \\nonumber\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\tilde b_k (\\tilde n) = b_k(\\tilde n) -2\\epsilon d_k (\\tilde n),\n\\end{equation}\nwhere \n$b_k(\\tilde n)$ are the coefficients of the asymptotic expansion at $t \\to \\infty$\n\\begin{equation}\n(2\\pi t)^2\\; e^{-4t} {{\\cal T}(\\tilde n)} \\;\\simeq \\sum_{k=1}^\\infty {b_k (\\tilde n) \\over t^k} \n\\end{equation}\nand $d_k(\\tilde n)$ are the coefficients of the asymptotic expansion\n\\begin{equation}\\label{d_coeff_def}\n(2\\pi t)^2 \\; e^{-4t} {{\\cal T}(\\tilde n)}\\;\\ln \\left[ e^{-t} I_0(t) \\sqrt{2\\pi t}\\right] \\simeq \n\\sum_{n=1}^\\infty {d_n (\\tilde n) \\over t^n} .\n\\end{equation}\n\\vskip 1mm\nNow we {\\bf define} the general boson integral (\\ref{BosIntInitDef}) in the \ndimensional regularization by the formula\n\\begin{equation}\nF(q;\\tilde n) = \\bar J(q;\\tilde n) + F^{DR}_{div}(q;\\tilde n),\n\\end{equation}\nwhere\n\\begin{equation} \nF^{DR}_{div}(q;\\tilde n) = \\int_1^\\infty t^{q-1}\\; dt\\  {1\\over (2\\pi t)^{2-\\epsilon}} \\\n\\sum_{k=0}^{q-2} {\\tilde b_k(\\tilde n) \\over t^k}\n\\end{equation}\nand the dimensional regularization implies that\n\\begin{equation}\n\\int_1^\\infty dt\\ t^{n+\\epsilon} =0 \\quad\\mbox{at}\\ \\ n\\neq -1, \\qquad \\qquad  \n\\int_1^\\infty {dt \\over t^{1-\\epsilon}} = -\\ {1\\over \\epsilon}.\n\\end{equation}\nThis being so,\n\\begin{equation} \nF^{DR}_{div}(q;\\tilde n) = {1\\over (2\\pi )^{2}}  {1\\over \\Gamma(q)} \n\\left\\{ -\\ {1\\over \\epsilon}\\; b_{q-2}(\\tilde n) - \\ln(2\\pi)\\, b_{q-2}(\\tilde n) +2d_{q-2}(\\tilde n) \\right\\}.\n\\end{equation}\nNow we isolate the \"canonical\" divergent part in the dimensional regularization; \nthat is, \n\\begin{equation}\n\\bar F^{DR}_{div}(q;\\tilde n, \\mu^2) =-\\; {1\\over (2\\pi )^{2}} \\, {1\\over \\Gamma(q)} \n\\left[ {1\\over \\epsilon}\\;-C+\\ln\\left({4\\pi\\over\\mu^2}\\right)\\right] b_{q-2}(\\tilde n).\n\\end{equation}\nwhere $\\mu$ is the parameter of dimensional regularization\\footnote{In this subsection it is considered that intergation in (\\ref{BosIntInitDef}) is performed over the $4-2\\epsilon$ dimensional space; the integral\nunder consideration should be multiplied by $\\mu^{2\\epsilon}$};\nthe \"canonical\" divergent part is needed to compensate for the \ninfrared divergent part in the respective continuum integral\nwith nonvanishing external momenta. We see that\n\\begin{eqnarray}\\label{GDR-barGDR}\nF^{DR}_{div}(q;\\tilde n,) &=& \\bar F^{DR}_{div}(q;\\tilde n, \\mu^2) \\\\ \\nonumber\n&+& {1\\over (2\\pi )^{2}} \\, {1\\over \\Gamma(q)} \n\\left[\\left(-C+\\ln {2\\over\\mu^2}\\right) b_{q-2}(\\tilde n)+2d_{q-2}(\\tilde n) \\right]. \\nonumber\n\\end{eqnarray}\nThe respective finite parts can be determined by the formula\n\\begin{equation}\\label{2repr-of-FDR}\nF^{DR}(q;\\tilde n) = \\bar J(q;\\tilde n) + F^{DR}_{div}(q;\\tilde n) = \\bar J^{DR}(q;\\tilde n) + \\bar F^{DR}_{div}(q;\\tilde n),\n\\end{equation}\n\\vskip 1mm\nNow we express $\\bar J^{DR}(q;\\tilde n)$ in terms of the quantity $J(q;\\tilde n)$\nwhich can be calculated in the fictitious mass regularization by the method described below.\nFirst we note that $\\bar J(q;\\tilde n)$ is connected with $\\bar J^{DR}(q;\\tilde n)$ by\nthe relation\n\\begin{equation}\n\\bar J^{DR}(q;\\tilde n)= \\bar J(q;\\tilde n) + {1\\over (2\\pi )^{2}} \\, {1\\over \\Gamma(q)} \n\\left[\\left(-C+\\ln {2}\\right) b_{q-2}(\\tilde n)+2d_{q-2}(\\tilde n) \\right].\n\\end{equation}\nThe relation between $\\bar J(q;\\tilde n)$ and $J(q;\\tilde n)$ is\nderived from the formula\n\\begin{equation}\nF^{FMR}(q;\\tilde n) = \\bar J(q;\\tilde n) + \\bar F^M_{div}(q;\\tilde n) = J(q;\\tilde n) + F^{M}_{div}(q;\\tilde n),\n\\end{equation}\n(it is the definition of $J(q;\\tilde n)$). From the formula analogous to (\\ref{DeltaJ}) it follows that\n\\begin{eqnarray}\\label{FFMR-barFFMR}\n\\bar J(q;\\tilde n) &=& J(q;\\tilde n) + (F^M_{div}(q;\\tilde n) - \\bar F^{M}_{div}(q;\\tilde n))\\\\ \\nonumber\n&=& J(q;\\tilde n)\\; + \\;{1\\over (2\\pi)^2 \\Gamma(q)}  \\sum_{k=1}^{q-2}  {b_{q-k-2} (\\tilde n)\\over k},\n\\end{eqnarray}\nwhere $\\bar F^{M}_{div}(q;\\tilde n)$ and $ F^{M}_{div}(q;\\tilde n)$ are natural\nanalogs of the quantities $\\bar G^{M}_{div}(q)$ and $ G^{M}_{div}(q)$ introduced \nabove. Combining formula (\\ref{FFMR-barFFMR}) with (\\ref{GDR-barGDR}) and (\\ref{2repr-of-FDR}), \nwe arrive at\n\\begin{eqnarray}\nF^{DR}(q;\\tilde n) &=& \\bar F^{DR}_{div}(q;\\tilde n;\\mu)\\;+\\; \\bar J^{DR}(q;\\tilde n;\\mu),\\ \\ \\ \\mbox{where} \\\\ \\nonumber\n\\bar F^{DR}_{div}(q;\\tilde n, \\mu) &=& -\\ {1\\over (2\\pi )^{2}} \\, {1\\over \\Gamma(q)} \n\\left[ {1\\over \\epsilon}\\;-C+\\ln\\left({4\\pi\\over\\mu^2}\\right)\\right] b_{q-2}(\\tilde n), \\\\ \\nonumber\n\\bar J^{DR}(q;\\tilde n;\\mu) &=&  J(q;\\tilde n)\\; + \\;{1\\over (2\\pi)^2 \\Gamma(q)}  \\sum_{k=1}^{q-2}  {b_{q-k-2}(\\tilde n) \\over k}  \\\\ \\nonumber\n&& + {1\\over (2\\pi )^{2}} \\, {1\\over \\Gamma(q)} \n\\left[(-C+\\ln 2) b_{q-2}(\\tilde n)+2d_{q-2}(\\tilde n)\\right], \\nonumber\n\\end{eqnarray}\n where $J(q;\\tilde n)$ can be calculated as follows:\nusing the relations (\\ref{ReducFormulasGT2}) and \n(\\ref{ReducFormulasOne}), $F(q,\\tilde n)$ is transformed to \na linear combination of the quantities $G_\\delta(r,\\mu_B^2)$ (\\ref{BosFtoGgen})\nand the substitutions (\\ref{DefJ}) are employed. \nIn the resulting expression, $\\mu_B^{-1}$ and $l_C$\nare formally set equal to zero, all that remains represents\nthe sought-for $J(q;\\tilde n)$.\n\nIt should be noticed that  $\\displaystyle {1\\over \\epsilon}$ appears in $F^{DR}(q;\\tilde n)$\nonly in the combination $\\displaystyle {1\\over \\epsilon} -\\ln 2 - F_0 + \\ln \\left({4\\pi\\over \\mu^2}\\right)$\n(the Euler-Mascheroni constant $C$ cancels in the total expression).\n\n\n\n\n\\section{Fermion Integrals}\n\n\nHere we consider the integrals (remember that $a=1$)\n\\begin{equation}\\label{eq:GenFermInt}\nF(p,q;\\tilde n)=\\lim_{\\delta \\to 0}\\int {d^4k\\over (2\\pi)^4} \n{\\cos^{n_1}(k_1) \\cos^{n_2}(k_2) \\cos^{n_3}(k_3) \\cos^{n_4}(k_4) \\over \\Delta_B^q\n\\Delta_F^{p+\\delta}}\n\\end{equation}\nwhere $p>0$, $\\delta$ is a regularization parameter, and \n\\begin{equation}\\label{DenomFermPropLatt}\n\\Delta_F = 10-4\\;\\sum_{\\mu=1}^4 \\cos(k_\\mu) + \\sum_{1\\leq \\mu < \\nu \\leq 4} \\cos(k_\\mu) \\cos(k_\\nu) + \\mu_B^2\n\\end{equation}\nis the denominator of the fermionic propagator.\nMaking use of the recursion relations \n\\begin{eqnarray}\\label{RecFermnnnn}\n\\hspace*{-5mm} F(p,q,...,l,...) &=&  F(p,q,...,l-2,...) \\\\ \\nonumber\n&&\\hspace*{-22mm} + \\ \\mu_B^2 \\left(F(p,q,...,l-1,...)-F(p,q,...,l-3,...)\\right) \\\\   \\nonumber\n&&\\hspace*{-22mm} -\\ \\left(F(p,q-1,...,l-1,...)-F(p,q-1,...,l-3,...)\\right) \\\\ \\nonumber\n&&\\hspace*{-22mm} -\\ {q\\over p-1+\\delta} \\; \\left(F(p-1,q+1,...,l-1,...) - F(p-1,q+1,...,l-3,...)\\right) \\\\ \\nonumber\n&&\\hspace*{-22mm} -\\ {1\\over p-1+\\delta}\\; \\left( (l-2) F(p-1,q,...,l-2,...) - (l-3)F(p-1,q,...,l-4,...)\\right);\n\\end{eqnarray} \n\\begin{eqnarray}\\label{RecFerm21} \nF(p,q;n1,n2,n3,2)&=&F(p,q-2,n1,n2,n3,0)-2\\,\\mu_B^2\\,F(p,q-1,n1,n2,n3,0)\\\\  \\nonumber\n&&\t-2\\,F(p-1,q,n1,n2,n3,0)+(4+2\\,\\mu_B^2+\\mu_B^4)\\,F(p,q,n1,n2,n3,0)\\\\  \\nonumber\n&&\t-F(p,q,n1+2,n2,n3,0)-F(p,q,n1,n2+2,n3,0) \\\\ \\nonumber\n&& -F(p,q,n1,n2,n3+2,0), \\\\  \\nonumber\nF(p,q,n1,n2,n3,1)&=&(\\mu_B^2+4)\\,F(p,q,n1,n2,n3,0)\\\\  \\nonumber\n&&\t-F(p,q-1,n1,n2,n3,0)-F(p,q,n1+1,n2,n3,0)\\\\ \\nonumber\n&&\t-F(p,q,n1,n2+1,n3,0)-F(p,q,n1,n2,n3+1,0)\\\\ \\nonumber\nF(p,q;n1,n2,2,0) &=& {1\\over 2} \\left( F(p,q-2,n1,n2,0,0) - 2 \\mu_B^2 F(p,q-1;n1,n2,0,0)\\right. \\\\ \\nonumber\n&& - 2 F(p-1,q;n1,n2,0,0) + (4+2 \\mu_B^2+\\mu_B^4) F(p,q,n1,n2,0,0) \\\\ \\nonumber\n&& \\left. - F(p,q;n1+2,n2,0,0)-F(p,q;n1,n2+2,0,0)\\right); \\\\ \\nonumber\nF(p,q;n1,n2,1,0) &=& {1\\over 2} \\left( (\\mu_B^2+4) F(p,q;n1,n2,0,0) - F(p,q-1;n1,n2,0,0)\\right. \\\\ \\nonumber\n&& \\left. - F(p,q;n1+1,n2,0,0) - F(p,q;n1,n2+1,0,0) \\right); \\\\ \\nonumber\nF(p,q;n1,2,0,0) &=& {1\\over 3} \\Big( F(p,q-2;n1,0,0,0) - 2 \\mu_B^2 F(p,q-1;n1,0,0,0) \\\\ \\nonumber\n&& - 2 F(p-1,q;n1,0,0,0) + (4 + 2 \\mu_B^2 + \\mu_B^4) F(p,q;n1,0,0,0)\\\\ \\nonumber\n&& - F(p,q;n1+2,0,0,0) \\Big); \\\\ \\nonumber\nF(p,q;n1,1,0,0) &=& {1\\over 3} \\Big( (\\mu_B^2+4) F(p,q;n1,0,0,0) - F(p,q-1;n1,0,0,0)\\\\ \\nonumber\n&&  - F(p,q;n1+1,0,0,0)\\Big); \\\\ \\nonumber\nF(p,q;2,0,0,0) &=& {1\\over 4} \\Big( F(p,q-2;0,0,0,0) - 2 \\mu_B^2 F(p,q-1;0,0,0,0)\\\\ \\nonumber\n&& - 2 F(p-1,q,0,0,0,0) + ( 4 + 2 \\mu_B^2 + \\mu_B^4) F(p,q;0,0,0,0)); \\\\  \\nonumber\nF(p,q;1,0,0,0) &=& {1\\over 4} \\Big( (\\mu_B^2 + 4) F(p,q;0,0,0,0) -F (p,q-1;0,0,0,0)\\Big);\\nonumber\n\\end{eqnarray} \n\nwe can express the quantities (\\ref{eq:GenFermInt}) in terms of the integrals\n\\begin{equation}\\label{eq:BasFermInt}\nG_\\delta(p,q;\\mu_B^2)=\\int {d^4k\\over (2\\pi)^4} {1 \\over \\Delta_B^q \\Delta_F^{p+\\delta}}\n\\end{equation}\nas follows:\n\\begin{equation}\\label{FtoGfirst}\nF(p,q;n_1, n_2, n_3, n_4)=\\sum_{r=p-n}^{p} \\ \\sum_{s=q+2p-n-2r}^{q+p-r} C_{pq;\\tilde n}^{rs}(\\mu_B^2,\\delta) G_\\delta(r,s;\\mu_B^2),\n\\end{equation}\nwhere $n=n_1+n_2+n_3+n_4$. The coefficients $C_{pq;n_1 n_2 n_3 n_4}^{rs}$ \nare polynomials in $\\mu_B^2$ (however, at $r+s\\leq 2$ $C_{pq;\\tilde n}^{rs}(\\mu_B^2,\\delta)$\ncan be replaced by $C_{pq;\\tilde n}^{rs}(0,\\delta)$); at $r\\leq 0$ they involve\nthe singularity $\\displaystyle {1\\over \\delta}$; that is, they can be represented in the form\n\\begin{equation} \nC_{pq;n_1 n_2 n_3 n_4}^{rs} = {1\\over \\delta} S_{pq;n_1 n_2 n_3 n_4}^{rs}(\\mu_B^2) + R_{pq;n_1 n_2 n_3 n_4}^{rs}(\\mu_B^2)\n+ O(\\delta),\n\\end{equation}\nwhere $S_{pq;n_1 n_2 n_3 n_4}^{rs}=0$ at $r\\leq 0$ or $p\\leq 0$.\nA straightforward calculation of both $S_{pq;n_1 n_2 n_3 n_4}^{rs}(\\mu_B^2)$ \nand $R_{pq;n_1 n_2 n_3 n_4}^{rs}(\\mu_B^2)$ by employing the above relations is rather simple.\n\nTo compute the basic integrals $G_\\delta(p,q;\\mu_B^2)$, we consider the cases\n$p>0$ and $p\\leq 0$ separately. At $p>0$, only zeroth order of the expansion \nof $G_\\delta(p,q)$ in a power series in $\\delta$ gives a nonvanishing contribution, \nwhereas at $p\\leq 0$ one should also keep the term linear in $\\delta$.\n\nIt is convenient\\footnote{It should be noted that the functions $G_\\delta(p,q)$ are related to the functions ${\\cal F}_\\delta(p,q)$ used in \\cite{BCP} by the formulas ${\\cal F}_\\delta(p,q)= 2^{-p-q} G_\\delta(p,q)$.} to represent $G_\\delta(p,q)$ in the form \n\\begin{eqnarray}\\label{GdeltaExpansion}\nG_\\delta(p,q)&=& D(p,q;\\mu_B^2) + B(p,q) + \\delta \\;(L(p,q;\\mu_B^2)+J(p,q)) + O(\\delta^2) , \\qquad p\\leq 0; \\\\ \\nonumber\nG_\\delta(p,q)&=& D(p,q;\\mu_B^2) + J(p,q) + O(\\delta), \\qquad p > 0.  \\nonumber\n\\end{eqnarray}\nwhere the quantities $B(p,q)$, $D(p,q;\\mu_B^2)$, $L(p,q;\\mu_B^2)$, and $J(p,q)$ \nare defined as follows:\\\\\n$D(p,q;\\mu_B^2)+\\delta L(p,q;\\mu_B^2)$ is \nthe divergent part of $G_\\delta(p,q;\\mu_B^2)$ at $p\\leq 0$ (up to terms $O(\\delta^2)$),\\\\\n$D(p,q;\\mu_B^2)$ is \nthe divergent part of $G_\\delta(p,q;\\mu_B^2)$ at $p > 0$ (up to terms $O(\\delta)$),\\\\\n$B(p,q)+ \\delta \\;J(p,q)$ and $J(p,q)$ are the respective finite parts\\footnote{Note\nthat $J(p,q)$ designates the finite part in the order $O(1)$ at $p>0$ and in the order\n$O(\\delta$ at $p\\leq 0$}.\n\nThe finite and divergent parts are unambiguously fixed by the requirement that \n$D(p,q;\\mu_B^2)$ and $L(p,q;\\mu_B^2)$ can be represented in the form\n\\begin{eqnarray}\\label{DPcoeffOne}\nD(p,q;\\mu_B^2) &=& D_{0}(p,q) (\\ln \\mu_B^2 +C) + \\sum_{r=1}^{p+q-2} {D_{r}(p,q) \\over (\\mu_B^2)^r} \\\\ \\nonumber\nL(p,q;\\mu_B^2) &=& {1\\over 2} L_{0}^{(2)}(p,q) (\\ln \\mu_B^2 +C)^2 + L_{0}^{(1)}(p,q) (\\ln \\mu_B^2 +C) \\\\ \\nonumber\n&& + \\sum_{r=1}^{p+q-2} {L_{r}^{(2)}(p,q) (\\ln \\mu_B^2 + C) \\over (\\mu_B^2)^r} \n+\\sum_{r=1}^{p+q-2} {L_{r}^{(1)}(p,q) \\over (\\mu_B^2)^r}. \\nonumber\n\\end{eqnarray}\n\nIn the domain $p\\leq 0$ we also use the quantities\n\\begin{equation}\\label{BBandJJat_pleq0}\n{\\cal B}(p,q;\\mu_B^2)= B(p,q)+ D(p,q;\\mu_B^2) \\qquad \\mbox{and} \\qquad\n{\\cal J}(p,q;\\mu_B^2) = L(p,q;\\mu_B^2)+J(p,q).  \\nonumber\n\\end{equation}\nAt $p> 0$, \n\\begin{equation}\\label{BBandJJat_pgeq0}\n{\\cal B}(p,q;\\mu_B^2)= B(p,q) = 0 \\qquad \\mbox{and} \\qquad\n{\\cal J}(p,q;\\mu_B^2) = D(p,q;\\mu_B^2)+J(p,q).  \\nonumber\n\\end{equation}\nNote that, at $q < 2-p,\\ \\ $ $D(p,q,\\mu_B^2)=L(p,q,\\mu_B^2)=0$, thus\n${\\cal B}(p,q;\\mu_B^2)=B(p,q)$ and ${\\cal J}(p,q;\\mu_B^2)=J(p,q)$ and one can use both\ndesignations.\n\n\n\\subsection{Divergent Part in the Fictitious Mass Regularization \\label{DPFIFMR}}\n\n\nFirst we note that (symbol ${\\cal D\\!P}$ means `divergent part of')\n\\begin{eqnarray}\\label{CalcDPcoeff}\np\\leq 0, q\\geq2-p \\qquad  &&  D (p,q;\\mu_B^2) = {\\cal D\\!P} \\int {dk\\over (2\\pi)^4} \\; {\\Delta_F^{-p} \\over \\Delta_B^{q}}, \\\\ \\nonumber\np = 0, q\\geq2 \\qquad  &&  L (0,q;\\mu_B^2) = {d\\over d\\delta}\\left|_{\\delta=0} {\\cal D\\!P}\n\\int {dk\\over (2\\pi)^4}\\; { 1 \\over \\Delta_B^{q+\\delta}} \\right. \\\\ \\nonumber\n&& +\\sum_{l=1}^{q-2} {(-1)^l\\over l} \\; \n{\\cal D\\!P} \\int {dk\\over (2\\pi)^4}\\; {\\Delta^l \\over \\Delta_B^{l+q}} , \\\\ \\nonumber\np>0, q\\geq2-p \\qquad  &&  D (p,q;\\mu_B^2) = \\sum_{l=0}^{p+q-2} {(-1)^l\\; (p+l-1)!\\over l! (p-1)!} \n{\\cal D\\!P} \\int {dk\\over (2\\pi)^4} \\; {\\Delta^l \\over \\Delta_B^{p+q+l}},  \\nonumber\n\\end{eqnarray}\n\nThe divergent parts of $\\delta$-independent integrals \nin formulas (\\ref{CalcDPcoeff}) can be \ncalculated as follows. First one employs the recursion relations \n(\\ref{ReducFormulasGT2}) and (\\ref{ReducFormulasOne})\nfor boson integrals to transform the integrand to a linear combination \nof the basic boson integrals (\\ref{GdeltaReprBessel}) and then \nevaluates the divergent part of each integral by the formula \n\\begin{equation}\n{\\cal D\\!P} \\int {dk\\over (2\\pi)^4} \\; {1 \\over \\Delta_B^{q}}\n= {1\\over (2\\pi)^2 \\Gamma(q)} \n\\left[-\\; b_{q-2} l_C + \\sum_{k=1}^{q-2}  b_{q-k-2} {\\Gamma (k) \\over (\\mu_B^2)^k} \n\\right] \n\\end{equation}\n(see derivation of the formula (\\ref{FMR_DivPart_min})), according to the MS prescription\nin the FMR.\nThe $\\delta$-dependent divergent parts that appears in formula (\\ref{CalcDPcoeff})\ncan determined by the same token, however, with the use of the formula\n\\begin{eqnarray} \n&& {d\\over d\\delta}\\left|_{\\delta=0} {\\cal D\\!P} \n\\int {dk\\over (2\\pi)^4}\\; { 1 \\over \\Delta_B^{q+\\delta}} \\right. = \\\\ \\nonumber\n&& ={1\\over (2\\pi)^2 \\Gamma(q)}\n  \\left[ b_{q-2} \\left( {1\\over 2} l_C^2 \n+ \\psi(q) l_C \\right) +\\sum_{n=1}^{q-2} b_{q-2-n} {\\Gamma(n)\\over (\\mu_B^2)^n }\n\\left(\\psi(n) - \\psi(q) - l_C + C \\right) \\right];  \\nonumber\n\\end{eqnarray}\n(see  (\\ref{FMR_DivPart_min})). \nThe divergent parts at $p\\leq 0$ can be obtained by the recursion relations, see below.\n\nThe divergent parts $D(p,q;\\mu_B^2)$ and $L(p,q;\\mu_B^2)$ introduced in the\nformula (\\ref{GdeltaExpansion}) are presented in Appendix~2 at $p+q\\leq 8$;\nat other values it can be readily calculated by the above formulas.\n\n\\subsection{Finite Parts}\n\nGiven the divergent part, we use the recurrent relations (\\ref{RecFermnnnn})\nand (\\ref{RecFerm21}) to express any integral of the type (\\ref{eq:GenFermInt})\nin terms of the functions $B(p,q)$ and $J(p,q)$, which, in their turn, can be found\nby making use of the two types of the recursion relations \\cite{BCP}.\n\nThe relations of the first type (the so called $T$-identities) can be obtained \nby inserting the expression $ \\Delta_B-4 - \\mu_B^2 +\\sum_{\\mu=1}^{4} \\cos(k_\\mu)$\nwhich is identically equals zero, in the integrand\n\\begin{eqnarray}\nF(q,1,1,1,1) &=& \\int dk\\;{\\cos(k_1) \\cos(k_2) \\cos(k_3) \\cos(k_4) \\over\\Delta_F^{(p+\\delta)}\\; \\Delta_B^{q}}  \\nonumber\n\\end{eqnarray}\nThe relations of the second type ($S$-identities) can be obtained by inserting \n$$\n10-4\\;\\sum_{\\mu=1}^4 \\cos(k_\\mu) + \\sum_{1\\leq \\mu < \\nu \\leq 4} \\cos(k_\\mu) \\cos(k_\\nu) + \\mu_B^2 - \\Delta_F,\n$$\nwhich is also identically equals zero, in the same integrand.\nIn so doing, we arrive at\n\\begin{eqnarray} \\label {TandSidentities}\n(4+\\mu_B^2) F(p,q,1,1,1,1) - F(p,q-1,1,1,1,1) - 4 F(p,q,2,1,1,1)) &=& 0, \\\\ \\nonumber\n(4+2 \\mu_B^2+\\mu_B^4) F(p+1,q,1,1,1,1) - 2 \\mu_B^2 F(p+1,q-1,1,1,1,1)&& \\\\ \\nonumber\n- 4 F(p+1,q,3,1,1,1) + F(p+1,q-2,1,1,1,1)-2 F(p,q,1,1,1,1) &=& 0.\n\\end{eqnarray}\nThen we express the functions $F(p,q;\\tilde n)$ emerging here in terms of $B(p,q)$\nand $J(p,q)$ and obtain the sought for identities. Due to the terms singular in $\\delta$,\nthe cases $p=1,2,3$ should be considered separately and, \nin the case $p\\leq 0$, one should remember that only the order $O(\\delta)$ part is nontrivial.\n\nIn what follows, we indicate how to use the explicit form of the derived recurrent relations\nfor the ${\\cal B}$ and ${\\cal J}$ functions presented in Appendices 3-11 in order to compute\\footnote{For more detail, \nsee http:\/\/www.lattice.itep.ru\/$\\sim$pbaivid\/lattpt\/ or contact me via e-mai\n}\n$B(p,q)$ and $J(p,q)$ at $p<9$ and any values of $q$.\nNote that some of the relations deal with $B(p,q)$ and $J(p,q)$,\nwhereas the other---with ${\\cal B}(p,q)$ and ${\\cal J}(p,q)$.\nIn the appendices, $B(p,q)$ and $J(p,q)$ are designated by\n{\\tt B(p,q)} and {\\tt J(p,q)}, respectively, whereas ${\\cal B}(p,q)$ and ${\\cal J}(p,q)$\nare designated by {\\tt BB(p,q)} and {\\tt JJ(p,q)}.\n\nBoth the divergent and the finite parts as well as the recurrent relations were \nobtained using the FORM \\cite{Vermaseren} and (partially) REDUCE \\cite{HEARN} packages.\n\n\\subsubsection{${\\cal B}$ functions}\n\nFirst we compute the ${\\cal B}$ functions defined by the formula\n(\\ref{BBandJJat_pleq0}) (see also (\\ref {GdeltaExpansion})). \n\n\\begin{itemize} \n\\item{\\bf ${\\cal B}$ functions at $p=0$, $q>0$}\n\\end{itemize}\n\\begin{equation}\n{\\cal B}(0,q,\\mu_B^2) = G_{div}^{M}(q,\\mu_B^2)+J(q), \n\\end{equation}\nwhere $G_{div}^{M}(q)$ is defined by the equation (\\ref{FMR_DivPart_min0})\n($G_{div}^{M}(q)\\neq 0$ only at $q\\geq 2$)\\\\\nand $J(q)$ at $q\\geq 4$---by the recurrent relations\\footnote{Note that $J(q)$\nhas nothing to do with $J(p,q)$ or $J(0,q)$.} (\\ref{FMR_BasBosInt_FinPart_min}).\nThe initial conditions are provided by $J(q)$ (and, therefore, ${\\cal B}(0,q,\\mu_B^2)$) \nat $1 \\leq q \\leq 3$ (see the formulas (\\ref{J0123boson}), (\\ref{IniCondforDIVboson}), \nand (\\ref{BasBosConstNum})). \n\n\n\\begin{itemize} \n\\item{\\bf ${\\cal B}$ functions at $p=0$, $q\\leq0$}\n\\end{itemize}\n\\begin{equation}\n{\\cal B}(0,q,\\mu_B^2) = {\\cal B}_{-q}, \n\\end{equation}\nwhere ${\\cal B}_{-q}$ are defined in the formula (\\ref{Bdirect})\nand can also be determined from the recurrent relations (\\ref{RecRelForBbosonic})\nwith the initial conditions (\\ref{IniCondForBbosonic}) \n(some values are given in Appendix~1).\n\n\n\\begin{itemize} \n\\item{\\bf ${\\cal B}$ functions at $p\\leq -1$, $q\\leq 0$}\n\\end{itemize}\nIn this domain, ${\\cal B}(p,q;\\mu_B^2)$ functions\ninvolve no divergencies: ${\\cal B}(p,q;\\mu_B^2) = B(p,q) = {\\cal B}(p,q;0)$. Provided that\n$B(0,q) = {\\cal B}_{-q}$ at $q\\leq 0$ are known, $B(p,q)$ at $p<0, q\\leq 0$\ncan be determined by the recurrent relations \nfor the domain $p<0, q \\geq 4$ (see below), however, with $\\mu_B=0$,\nand, therefore, with $B(p,q)$ instead of ${\\cal B}(p,q)$.\n\n\\begin{itemize} \n\\item{\\bf ${\\cal B}$ functions over the strip $p\\leq -1$, $1\\leq q \\leq 3$}\n\\end{itemize}\n\nIn this domain, ${\\cal B}(p,q) = B(p,q)$ unless $(p,q)=(-1,3)$.\n\n\\noindent ${\\cal B}$ functions at $-2\\leq p \\leq -1$, $1\\leq q \\leq 3$ can be \ncalculated by the formula \n\\begin{equation}\n{\\cal B}(p,q)=\\int dk\\ { \\Delta_F^{-p}(k,\\mu_B^2) \\over \\Delta^q_B(k,\\mu_B^2)},\n\\end{equation}\nthe result is as follows:\n\\begin{eqnarray}\\label{Binput}\nB(-1,1) &=& 1 + 12 \\;Z_1;\\\\ \\nonumber\nB(-1,2) &=& 4 \\;Z_0;\\\\ \\nonumber\nB(-1,3) &=& {1 \\over (2\\pi)^2}\\ \\left({1\\over 2} + F_0  \\right) \\; + \\; {1\\over 2} \\;Z_0,\n\\qquad \\qquad \\left[{\\cal B} (-1,3) = B(-1,3) - {l_C \\over (2\\pi)^2}\\right]; \\\\[1mm] \\nonumber\nB(-2,1) &=&   188\/3 - \\;{1 \\over (2\\pi)^2}\\; {736\\over 9}  - {632\\over 3} \\;Z_1 - {224\\over 3} \\;Z_0;\\\\ \\nonumber\nB(-2,2) &=&  - 11 + \\;{24 \\over (2\\pi)^2} + 114 \\;Z_1 + 24 \\;Z_0;\\\\ \\nonumber\nB(-2,3) &=&   {3\\over 4} - 3 \\;Z_1 + 6 \\;Z_0. 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\n\\Text(240,197)[tr]{2} \n\\Text(260,197)[tr]{3} \n\\Text(280,197)[tr]{4} \n\\Text(300,197)[tr]{5} \n\\Text(320,197)[tr]{6} \n\\Text(340,197)[tr]{7} \n\\Text(360,197)[tr]{8}\n\\Text(380,197)[tr]{9} \n\\Text(198,224)[tr]{1} \n\\Text(198,264)[tr]{3} \n\\Text(198,284)[tr]{4} \n\\Text(198,304)[tr]{5} \n\\Text(198,324)[tr]{6} \n\\Text(198,344)[tr]{7} \n\\Text(198,364)[tr]{8} \n\\Text(198,184)[tr]{-1} \n\\Text(198,164)[tr]{-2} \n\\Text(198,144)[tr]{-3} \n\\Text(198,124)[tr]{-4} \n\\Text(198,104)[tr]{-5} \n\\Text(198,84)[tr]{-6} \n\\Text(196,399)[tr]{\\Large $q$} \n\\Text(400,197)[tr]{\\Large $p$} \n\\end{picture}\n\\end{center}\n\\caption{Values of $(p,q)$ at which the \nfunctions $G(p,q;\\mu_B^2)$ are calculated are shown by dots.\nThe 12 fermionic basic constants are determined from\nthe functions associated with red dots; yellow dots \nshow the constants $X_0 \\div X_3$ that appear in the \nexpressions for $G(p,q;\\mu_B^2)$ but cancel in the\nexpressions for Feynman integrals. Functions shown by gray\nare calculated explicitly and given in the formulas (\\ref{Binput}),\n(\\ref{Ydef1}), (\\ref{Ydef2}), (\\ref{eq:JcrossDOWN}), (\\ref{eq:JcrossLEFT}), \nand (\\ref{eq:JcrossUP}), which provide the initial conditions for the recurrent \nrelations. The order of implementation\nof the recurrent relations is as follows:\n1. ${\\cal B}$ functions are calculated at $p=0$, then at $p<0,q\\leq 0$, then \nover the left strip, then at $p<0, q\\geq 4$.\n2. ${\\cal J}$ functions are calculated in the ``down'' strip, then\nlower-left domain (shown by green), then ``left'' strip, then ``up'' strip\nthen up-left domain, then up-right domain, then down-right domain. }\n\\end{figure}\n\nAt $p\\leq 3$, ${\\cal B}(p,q;\\mu_B^2)$\nfunctions involve no divergencies: ${\\cal B}(p,q;\\mu_B^2) = B(p,q) = {\\cal B}(p,q;0)$. \nThey can be determined by the recurrent relations presented in Appendix~3.\nThe initial conditions are provided by the formulas (\\ref{Binput})\nand $B$ functions calculated previously.\n\nTo express $B(r,1)$ or $B(r,2)$ or $B(r,3)$ at $r\\leq 3$ in terms of $F_0, Z_0$, and $Z_1$, \none should apply these relations beginning with $p=r$ and ending with $p=-3$.\nIn so doing, one must know $B(p,q)$ at $q=0,-1,-2$ and $r\\leq p \\leq 0$\n\n\\begin{itemize} \n\\item{\\bf ${\\cal B}$ functions at $p\\leq -1$, $q\\geq 4$}\n\\end{itemize}\n${\\cal B}$ functions in this domain \nare computed by the recurrent relations presented in the Appendix~4.\nThe initial conditions for these relations  are provided by ${\\cal B}(0,q;\\mu_B^2)$ \nas well as ${\\cal B}(p,q;\\mu_B^2)$ at $p\\leq -1, q\\leq 3$ presented above.\n\n\n\\subsubsection{${\\cal J}$ functions.}\n\nHere, the quantities $J(p,q)$ introduced in (\\ref{GdeltaExpansion})\nare expressed in terms of the boson constants $Z_1$, $Z_0$, $F_0$  \nintroduced in the previous Section, and the quantities $Y_0 \\div Y_{11}$ defined by the relations\n\\begin{eqnarray}\\label{Ydef1}\nY_4={J(1,0)\\over 2}, & \\qquad Y_5 = J(1,-1), & \\qquad Y_6 = 2 J(1,-2), \\\\ \\nonumber\nY_7={J(2,-1)\\over 2}, & \\qquad Y_8 = J(2,-2), & \\qquad Y_9 = {J(3,-2)\\over 2}, \\\\ \\nonumber\nY_{10} = J(3,-3), & \\qquad Y_{11}= 2 J(3,-4), & \\qquad Y_0= {J(2,0)\\over 4}\\; - \\; {F_0 \\over 16\\pi^2},  \\nonumber\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{Ydef2}\nY_1 &=& {1 \\over 48} - {1 \\over 4}\\; Z_0 - {1 \\over 24}\\; J(-1,2) + \n{1 \\over 12}\\; J(0,1) + {1 \\over 12}\\; J(1,0); \\\\ \\nonumber\nY_2 &=&  { 1 \\over 6 } - {1\\over \\pi^2} - Z_0 - { 1 \\over 6 }\\; J(-1,2) + \n{1 \\over 3}\\; J(0,1) -{ 1 \\over 24} \\; J(1,-2) - {1 \\over 12} \\; J(1,-1) - \\\\ \\nonumber\n&& - \n{17 \\over 8}\\; J(1,0) + 4\\; J(1,1) - {1 \\over 48}\\; J(2,-2) + {25 \\over 6}\\; J(2,-1)\n - 4\\; J(2,0); \\\\ \\nonumber\nY_3 &=&  - {1\\over 384\\pi^2} - F_0\\; {1\\over 128\\pi^2} + {1 \\over 96}\\; Z_0 - \n{1 \\over 48}\\; J(-1,3) + {1 \\over 192}  \\; J(0,1) + {1 \\over 48}\\; J(0,2) + {1 \\over 48}\\; J(1,1);\n\\end{eqnarray}\nand the quantities \n\\begin{eqnarray}\nX_0= J(-1,1), &&  X_1= J(-1,3), \\\\ \\nonumber\nX_2= J(0,0), &&  X_3=J(0,2).  \\nonumber\n\\end{eqnarray}\nas well. The numerical values of the quantities $X_0, X_1, X_2$, and $X_3$\nare not needed because expressions for the integrals (\\ref{eq:GenFermInt}) \nin terms of $G(p,q)$ etc. do not involve them. As for the other constants,\nthey are well known (the table below is taken from the review \\cite{Capitani},\nnotation ${\\cal F}(p,q)$ is borrowed from there):\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline\n$Y_0$            & $-$             0.01849765846791657356  \\\\\n$Y_1$            & \\hphantom{$-$}  0.00376636333661866811  \\\\\n$Y_2$            & \\hphantom{$-$}  0.00265395729487879354  \\\\\n$Y_3$            & \\hphantom{$-$}  0.00022751540615147107  \\\\\n$Y_4= {\\cal F}(1,0)$  & \\hphantom{$-$}  0.08539036359532067914  \\\\\n$Y_5= {\\cal F}(1,-1)$ & \\hphantom{$-$}  0.46936331002699614475  \\\\\n$Y_6= {\\cal F}(1,-2)$ & \\hphantom{$-$}  3.39456907367713000586  \\\\\n$Y_7= {\\cal F}(2,-1)$ & \\hphantom{$-$}  0.05188019503901136636  \\\\\n$Y_8= {\\cal F}(2,-2)$ & \\hphantom{$-$}  0.23874773756341478520  \\\\\n$Y_9= {\\cal F}(3,-2)$ & \\hphantom{$-$}  0.03447644143803223145  \\\\\n$Y_{10}= {\\cal F}(3,-3)$ & \\hphantom{$-$}  0.13202727122781293085  \\\\\n$Y_{11}= {\\cal F}(3,-4)$ & \\hphantom{$-$}  0.75167199030295682254  \\\\\n\\hline\n\\end{tabular}\n\\caption{New constants appearing in the general fermionic case.}\n\\label{tab:fermionicconstants}\n\\end{center}\n\\end{table}\n\n\n\\begin{itemize} \n{\\item Thus ${\\cal J}$ functions over the domain\n$\n{\\cal A} = \\left\\{ (p,q): 0 \\leq p \\leq 3,\\  -6 \\leq p \\leq 6-p \\right\\} \\cup $\\\\ \n$\\cup \\left\\{ (p,q): - 4\\leq p \\leq -1,\\  1\\leq q \\leq 3 \\right\\}\n$\ncalculated by the procedure indicated in \\cite{BCP} can \nbe represented in terms of the above constants as follows:}\n\\end{itemize}\n\\begin{eqnarray}\\label{eq:JcrossDOWN}\n&&  J(0,0) = \\; X_2; \\\\ \\nonumber && \n  J(0,-1) = 4 \\; X_2 + 315 \\; Y_{10} - 1218 \\; Y_9 - 134 \\; Y_8 + 804 \\; Y_7 - 2 \\; Y_6 \n+ {25 \\over  2} \\; Y_5  - 114 \\; Y_4 + {15 \\over (2\\pi)^2 }; \\\\ \\nonumber && \n  J(0,-2) = - 443\/12 + 18 \\; X_2 + 525\/16 \\; Y_{11} + 5265\/4 \\; Y_{10} - 7661\/2 \\; Y_9- 8173\/12 \\; Y_8 \\\\ \\nonumber && \n\\quad + 2085 \\; Y_7 - 7\/48 \\; Y_6 + 1657\/24 \\; Y_5 - 339\/2 \\; Y_4 + 35\/4 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n J(0,-3) = - 1549\/27 + 88\\; X_2 + 595\/4\\; Y_{11} + 92395\/6 \\; Y_{10} - 487883\/9\\; Y_9 - 197384\/27 \\; Y_8 \\\\ \\nonumber && \n\\quad + 103330\/3 \\; Y_7 - 6569\/108 \\; Y_6 + 57949\/108 \\; Y_5 - 12829\/3 \\; Y_4 + 9029\/18 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(0,-4) = - 1024087\/576 + 917\/2 \\; X_2 + 1919221\/768 \\; Y_{11} + 7343317\/192 \\; Y_{10} \\\\ \\nonumber && \n\\quad -  20889553\/288 \\; Y_9 - 14398667\/576 \\; Y_8 + 3075865\/144 \\; Y_7 + 325123\/2304 \\; Y_6 \\\\ \\nonumber && \n\\quad +  2538989\/1152 \\; Y_5 + 501607\/96 \\; Y_4 - 910433\/576 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(0,-5) =   22816157\/32400 + 2514 \\; X_2 + 14234765\/4032 \\; Y_{11} + 6270202009\/8400 \\; Y_{10} \\\\ \\nonumber && \n\\quad - 171277520509\/63000 \\; Y_9 - 11273402747\/32400 \\; Y_8 + 33415313987\/18900 \\; Y_7 \\\\ \\nonumber && \n\\quad - 402215297\/129600 \\; Y_6 + 52200958189\/2268000 \\; Y_5 - 4825909001\/21000 \\; Y_4 \\\\ \\nonumber && \n\\quad + 3567631667\/126000 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n   J(0,-6) =  - 59060175671\/583200 + 14376 X_2 - 456897556151\/2268000\\; (2pi)^{-2}   \\\\ \\nonumber && \n\\quad + 290748296317\/1814400 Y_{11} - 46846789343\/252000 Y_{10} + 1099242273317\/226800\\; Y_9  \\\\ \\nonumber && \n\\quad  - 1709413667089\/4082400\\; Y_8 - 8051167629571\/1701000\\; Y_7 + 41141268191\/2332800\\; Y_6 \\\\ \\nonumber && \n\\quad  + 1776831395699\/40824000\\; Y_5 + 61514640887\/54000\\ Y_4; \\\\  \\nonumber && \n  J(1,0) = 2 \\; Y_4; \\\\ \\nonumber && \n  J(1,-1) = \\; Y_5; \\\\ \\nonumber && \n  J(1,-2) =  1\/2 \\; Y_6; \\\\ \\nonumber && \n  J(1,-3) =   52\/3 + 683\/2 \\; Y_{10} - 1342 \\; Y_9 - 439\/3 \\; Y_8 + 924 \\; Y_7\\\\ \\nonumber && \n\\quad - 13\/3 \\; Y_6 + 89\/12 \\; Y_5  - 132 \\; Y_4 + 17 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(1,-4) = - 761\/9 + 683\/12 \\; Y_{11} - 10807\/3 \\; Y_{10} + 48538\/3 \\; Y_9 + 12539\/9 \\; Y_8 \\\\ \\nonumber && \n\\quad - 35284\/3 \\; Y_7 + 1481\/36 \\; Y_6 - 887\/18 \\; Y_5 + 1874 \\; Y_4 - 803\/3 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(1,-5) =    34745\/27 - 52147\/63 \\; Y_{11} + 6069519\/140 \\; Y_{10} - 27087257\/140 \\; Y_9 \\\\ \\nonumber && \n\\quad - 450260\/27 \\; Y_8 + 88135577\/630 \\; Y_7 - 42845\/108 \\; Y_6 + 2730857\/3780 \\; Y_5 \\\\ \\nonumber &&\n\\quad - 3125867\/140 \\; Y_4  + 541505\/168 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(1,-6) = - 3470071\/324 + 123774917\/10800 \\; Y_{11} - 15164760473\/31500 \\; Y_{10} \\\\ \\nonumber && \n\\quad + 34075577647\/15750 \\; Y_9 + 10210300231\/56700 \\; Y_8 - 36997574549\/23625 \\; Y_7 \\\\ \\nonumber && \n\\quad + 5205091\/1296 \\; Y_6 - 4977314411\/567000 \\; Y_5 + 1327628299\/5250 \\; Y_4 \\\\ \\nonumber && \n\\quad - 232435501\/6300 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(2,0) =  4 \\; Y_0 + F_0 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(2,-1) = 2 \\; Y_7; \\\\ \\nonumber && \n  J(2,-2) = \\; Y_8; \\\\ \\nonumber && \n  J(2,-3) = - \\; Y_{10} + 90 \\; Y_9 - 2 \\; Y_8 - 84 \\; Y_7 + 5\/2 \\; Y_5 + 18 \\; Y_4 - 3 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(2,-4) =  - 13\/3 - 1\/4 \\; Y_{11} + \\; Y_{10} - 326 \\; Y_9 + 13\/3 \\; Y_8 + 300 \\; Y_7 + 31\/12 \\; Y_6 \\\\ \\nonumber && \n\\quad - 19\/6 \\; Y_5 - 66 \\; Y_4 + 13 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(2,-5) = 1012\/9 + 53\/63 \\; Y_{11} + 519787\/210 \\; Y_{10} - 316251\/35 \\; Y_9 - 3217\/3 \\; Y_8 \\\\ \\nonumber && \n\\quad + 214918\/35 \\; Y_7  - 343\/9 \\; Y_6 + 17621\/420 \\; Y_5 - 29051\/35 \\; Y_4 + 3859\/42 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(2,-6) =  - 48007\/36 + 8466287\/15120 \\; Y_{11} - 272218199\/6300 \\; Y_{10} + 593087111\/3150 \\; Y_9 \\\\ \\nonumber && \n\\quad + 64920421\/3780 \\; Y_8 - 30568297\/225 \\; Y_7 + 70867\/144 \\; Y_6 - 22014731\/37800 \\; Y_5 \\\\ \\nonumber && \n\\quad + 7424029\/350 \\; Y_4 - 3722933\/1260 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(3,0) =  - 35\/256 \\; Y_{11} - 57\/64 \\; Y_{10} - 301\/2304 \\; Y_9 + 85\/96 \\; Y_8 + 1405\/1152 \\; Y_7\\\\ \\nonumber && \n\\quad - 1\/384 \\; Y_5 - 461\/768 \\; Y_4 - 2 \\; Y_0 + (1433\/1536 - 1\/2 F_0) \\; (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(3,-1) = 155\/96 \\; Y_9 - 83\/48 \\; Y_7 + 11\/32 \\; Y_4 + 4 \\; Y_0  + (F_0 - 31\/64 )\\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(3,-2) =  2 \\; Y_9; \\\\ \\nonumber && \n  J(3,-3) = \\; Y_{10}; \\\\ \\nonumber && \n  J(3,-4) = 1\/2 \\; Y_{11}; \\\\ \\nonumber && \n  J(3,-5) = - 85\/42 \\; Y_{11} - 1649\/140 \\; Y_{10} + 15548\/35 \\; Y_9 + 7\/6 \\; Y_8 - 14264\/35 \\; Y_7 \\\\ \\nonumber && \n\\quad + 5609\/840 \\; Y_5 + 2943\/35 \\; Y_4 - 106\/7 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(3,-6) = - 224\/9 + 2812\/945 \\; Y_{11} + 47669\/1575 \\; Y_{10} - 595726\/225 \\; Y_9 + 6854\/945 \\; Y_8 \\\\ \\nonumber && \n\\quad + 3804196\/1575 \\; Y_7 + 101\/9 \\; Y_6 - 173539\/9450 \\; Y_5 - 90298\/175 \\; Y_4 + 33203\/315 \\; (2\\pi)^{-2}; \\nonumber  \n\\end{eqnarray} \n\\begin{eqnarray}\\label{eq:JcrossLEFT}\n&&  J(-1,1) = \\; X_0; \\\\ \\nonumber && \n  J(-2,1) = - 1135\/9 - 158\/9 \\; X_0 + 448\/3 \\; X_3 + 722\/9 \\; X_2 - 448\/3 \\; X_1 + 1295\/12 \\; Y_{11} \\\\ \\nonumber && \n\\quad + 56237\/9 \\; Y_{10} - 61066\/3 \\; Y_9 - 83036\/27 \\; Y_8 + 108068\/9 \\; Y_7 - 283\/18 \\; Y_6 \\\\ \\nonumber && \n\\quad + 15217\/54 \\; Y_5 - 1274 \\; Y_4 - 19456\/3 \\; Y_3 - 32\/3 \\; Y_2 + 1472\/3 \\; Y_1 + 896\/3 \\; Y_0 \\\\ \\nonumber && \n\\quad + 865\/9 \\;  (2\\pi)^{-2} + 64\/9 F_0 \\;  (2\\pi)^{-2} + 536\/3 \\; Z_1 + 5488\/27 \\; Z_0; \\\\ \\nonumber && \n J(-3,1) = 11845003\/15000 + 105068\/375 \\; X_0 - 371584\/125 \\; X_3 - 200897\/375 \\; X_2 \\\\ \\nonumber && \n\\quad + 371584\/125 \\; X_1 + 204127\/160 \\; Y_{11} + 19090453\/600 \\; Y_{10} - 133042333\/1500 \\; Y_9  \\\\ \\nonumber && \n\\quad - 175667117\/9000 \\; Y_8+ 39223253\/750 \\; Y_7 - 214589\/12000 \\; Y_6 + 8996881\/18000 \\; Y_5 \\\\ \\nonumber && \n\\quad - 207851\/60 \\; Y_4  + 15388672\/125 \\; Y_3 + 27568\/125 \\; Y_2 - 1398848\/125 \\; Y_1  \\\\ \\nonumber && \n\\quad - 640512\/125 \\; Y_0 + 6221759\/15000 \\;  (2\\pi)^{-2} - 153472\/375 F_0 \\;  (2\\pi)^{-2}  \\\\ \\nonumber && \n\\quad - 2696248\/625 \\; Z_1 - 24764192\/5625 \\; Z_0; \\\\ \\nonumber && \n  J(-4,1) = - 3329832752387\/64827000 - 117549296\/25725 \\; X_0 + 446453248\/8575 \\; X_3 \\\\ \\nonumber && \n\\quad + 566923184\/25725 \\; X_2 - 446453248\/8575 \\; X_1 + 737380289\/14112 \\; Y_{11} \\\\ \\nonumber && \n\\quad + 242892361651\/123480 \\; Y_{10} - 5552574235627\/926100 \\; Y_9 \\\\ \\nonumber && \n\\quad - 640505149891\/617400 \\; Y_8 + 1563368068291\/463050 \\; Y_7 - 23776321823\/7408800 \\; Y_6 \\\\ \\nonumber && \n\\quad + 321309701383\/3704400 \\; Y_5 - 89421147379\/308700 \\; Y_4 - 17930338304\/8575 \\; Y_3 \\\\ \\nonumber && \n\\quad - 92940608\/25725 \\; Y_2 + 1782287616\/8575 \\; Y_1 + 2138417152\/25725 \\; Y_0 \\\\ \\nonumber && \n\\quad + 991072905679\/64827000 \\;  (2\\pi)^{-2} + 4988416\/525 F_0 \\;  (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad + 25248697568\/300125 \\; Z_1 + 73082913664\/900375 \\; Z_0; \\\\ \\nonumber && \n J(-1,2) = 2\/3 - 8 \\; X_3 + 8\\; X_1 - 1\/24\\; Y_8 + 50\/3 \\; Y_7 - 1\/24\\; Y_6 - 1\/6\\; Y_5 - 35\/6 \\; Y_4 \\\\ \\nonumber && \n\\quad + 384 \\; Y_3 - 2 \\; Y_2 - 16 \\; Y_1 - 32 \\; Y_0  + 4 (F_0 - 1) \\;  (2\\pi)^{-2} - 10 \\; Z_0; \\\\ \\nonumber && \n J(-2,2) = 135\/4 + 19\/2 \\; X_0 - 48 \\; X_3 - 41\/2 \\; X_2 + 48 \\; X_1 - 105\/16\\; Y_{11} - 1851\/4 \\; Y_{10} \\\\ \\nonumber && \n\\quad + 3075\/2 \\; Y_9 + 827\/4 \\; Y_8 - 841 \\; Y_7 + 11\/8 \\; Y_6 - 269\/8 \\; Y_5 + 175\/2 \\; Y_4 + 2304 \\; Y_3 \\\\ \\nonumber && \n\\quad + 12 \\; Y_2 - 96 \\; Y_1 - 96 \\; Y_0 - 45\/4 \\;  (2\\pi)^{-2} - 114 \\; Z_1 - 76 \\; Z_0; \\\\ \\nonumber && \n J(-3,2) = - 29723\/16 - 2365\/9 \\; X_0 + 7712\/3 \\; X_3 + 32881\/36 \\; X_2 - 7712\/3 \\; X_1 \\\\ \\nonumber && \n\\quad + 56805\/64 \\; Y_{11} + 6355039\/144 \\; Y_{10} - 5035139\/36 \\; Y_9 - 4682939\/216 \\; Y_8  \\\\ \\nonumber && \n\\quad + 1411847\/18 \\; Y_7 - 4543\/48 \\; Y_6 + 2051831\/864 \\; Y_5 - 68044\/9 \\; Y_4 - 330752\/3 \\; Y_3\\\\ \\nonumber && \n\\quad  - 252 \\; Y_2 + 26992\/3 \\; Y_1 + 13952\/3 \\; Y_0 + (54773\/72 + 2336\/9 F_0) \\;  (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad + 13706\/3 \\; Z_1 + 111704\/27 \\; Z_0; \\\\ \\nonumber && \n J(-4,2) = 492009959\/12000 + 87748\/15 \\; X_0 - 324224\/5 \\; X_3 - 275752\/15 \\; X_2 + 324224\/5 \\; X_1 \\\\ \\nonumber && \n\\quad - 3344999\/384 \\; Y_{11} - 240354739\/480 \\; Y_{10} + 5720706571\/3600 \\; Y_9 \\\\ \\nonumber && \n\\quad + 329440181\/1440 \\; Y_8 - 302115271\/360 \\; Y_7 + 2092337\/1920 \\; Y_6  - 518442119\/14400 \\; Y_5 \\\\ \\nonumber && \n\\quad + 285938617\/3600 \\; Y_4 + 2650112 \\; Y_3 + 75952\/15 \\; Y_2 - 1259712\/5 \\; Y_1 - 318976\/3 \\; Y_0 \\\\ \\nonumber && \n\\quad - (423392089\/36000 + 159872\/15 F_0) \\; (2\\pi)^{-2} - 44526248\/375 \\; Z_1 - 40322528\/375 \\; Z_0; \\\\ \\nonumber && \n  J(-1,3) = \\; X_1; \\\\ \\nonumber && \n  J(-2,3) = 1\/12 - 1\/4 \\; X_0 - 12 \\; X_3 + \\; X_2 + 12 \\; X_1 + 105\/2 \\; Y_{10} - 203 \\; Y_9 - 1075\/48 \\; Y_8 \\\\ \\nonumber && \n + 159 \\; Y_7 - 19\/48 \\; Y_6 + 23\/12 \\; Y_5 - 103\/4 \\; Y_4 + 576 \\; Y_3 - 7 \\; Y_2 - 60 \\; Y_1 \\\\ \\nonumber && \n - 48 \\; Y_0 - 7\/2 \\;  (2\\pi)^{-2} + 6 F_0 \\;  (2\\pi)^{-2} + 12 \\; Z_1 - 19 \\; Z_0; \\\\ \\nonumber && \n  J(-3,3) =  871\/2 + 629\/9 \\; X_0 - 1696\/3 \\; X_3 - 1778\/9 \\; X_2 + 1696\/3 \\; X_1 - 175\/2 \\; Y_{11} \\\\ \\nonumber && \n - 50840\/9 \\; Y_{10} + 167119\/9 \\; Y_9 + 279089\/108 \\; Y_8 - 91946\/9 \\; Y_7 + 187\/12 \\; Y_6 \\\\ \\nonumber && \n - 10154\/27 \\; Y_5 + 9446\/9 \\; Y_4 + 77824\/3 \\; Y_3 + 116 \\; Y_2 - 4784\/3 \\; Y_1 - 3136\/3 \\; Y_0 \\\\ \\nonumber && \n - 1739\/6 \\;  (2\\pi)^{-2} - 304\/9 F_0 \\;  (2\\pi)^{-2} - 1502 \\; Z_1 - 28456\/27 \\; Z_0; \\\\ \\nonumber && \n  J(-4,3) =  - 40263778\/1875 - 2997542\/1125 \\; X_0 + 10588096\/375 \\; X_3 + 20165461\/2250 \\; X_2 \\\\ \\nonumber && \n - 10588096\/375 \\; X_1 + 602637\/80 \\; Y_{11} + 161298737\/450 \\; Y_{10} \\\\ \\nonumber && \n - 5056485013\/4500 \\; Y_9 - 4717685677\/27000 \\; Y_8 + 1381579093\/2250 \\; Y_7 \\\\ \\nonumber && \n - 8668453\/12000 \\; Y_6 + 284825729\/13500 \\; Y_5 - 50984761\/900 \\; Y_4 \\\\ \\nonumber && \n - 444755968\/375 \\; Y_3 - 347064\/125 \\; Y_2 + 39247712\/375 \\; Y_1 + 17916928\/375 \\; Y_0 \\\\ \\nonumber && \n + 182925853\/15000 \\;  (2\\pi)^{-2} + 4427968\/1125 F_0 \\;  (2\\pi)^{-2} + 38504584\/625 \\; Z_1 \\\\ \\nonumber && \n + 860640608\/16875 \\; Z_0;  \\nonumber \n\\end{eqnarray}\n\\begin{eqnarray}\\label{eq:JcrossUP}\n&&  J(0,1) =  1\/12 - 4 \\; X_3 + 4 \\; X_1 - 1\/48 \\; Y_8 + 25\/3 \\; Y_7 - 1\/48 \\; Y_6 - 1\/12 \\; Y_5 - 59\/12 \\; Y_4 \\\\ \\nonumber && \n\\quad + 192 \\; Y_3 - \\; Y_2 + 4 \\; Y_1 - 16 \\; Y_0  + (2 F_0 - 2) \\;  (2\\pi)^{-2} - 2 \\; Z_0; \\\\ \\nonumber && \n J(0,2) =  \\; X_3; \\\\ \\nonumber && \n J(0,3) = 91\/1024 + 1\/384 \\; X_0 + 1\/4 \\; X_3 - 1\/96 \\; X_2 - 35\/768 \\; Y_{11} - 27\/32 \\; Y_{10} + 109\/72 \\; Y_9 \\\\ \\nonumber && \n\\quad + 19229\/36864 \\; Y_8 + 383\/2304 \\; Y_7 - 11\/4096 \\; Y_6 - 443\/9216 \\; Y_5 - 1243\/3072 \\; Y_4 \\\\ \\nonumber && \n\\quad - 3\/2 \\; Y_3 - 33\/256 \\; Y_2 + 17\/64 \\; Y_1 - 33\/16 \\; Y_0 - 13\/72 \\;  (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad - 5\/48 F_0 \\;  (2\\pi)^{-2} - 1\/4 \\; Z_1 - 49\/192 \\; Z_0; \\\\ \\nonumber && \n J(0,4) = 80441\/2211840 + 31\/27648 \\; X_0 + 17\/288 \\; X_3 - 31\/6912 \\; X_2 + 1\/288 \\; X_1 \\\\ \\nonumber && \n\\quad + 161\/24576 \\; Y_{11} - 17767\/92160 \\; Y_{10} + 23\/32 \\; Y_9 + 161077\/2949120 \\; Y_8 \\\\ \\nonumber && \n\\quad - 10667\/552960 \\; Y_7 - 14369\/8847360 \\; Y_6 - 16271\/737280 \\; Y_5 - 343291\/2211840 \\; Y_4 \\\\ \\nonumber && \n\\quad - 1379\/1440 \\; Y_3 - 14369\/184320 \\; Y_2 + 9409\/46080 \\; Y_1 - 8609\/11520 \\; Y_0 \\\\ \\nonumber && \n\\quad +( 539\/17280 F_0 - 18031\/86400 ) \\;  (2\\pi)^{-2} - 237\/2560 \\; Z_1 - 965\/9216 \\; Z_0; \\\\ \\nonumber && \n J(0,5) = 7632781\/594542592 + 523\/1327104 \\; X_0 + 209\/13824 \\; X_3 - 523\/331776 \\; X_2   \\\\ \\nonumber && \n\\quad + 25\/13824 \\; X_1 + 77651\/42467328 \\; Y_{11} - 1253633\/17694720 \\; Y_{10} + 56974703\/ \\\\ \\nonumber && \n\\quad      159252480 \\; Y_9 + 782803057\/35672555520 \\; Y_8 + 34653559\/2229534720 \\; Y_7 -  \\\\ \\nonumber && \n\\quad      10633541\/11890851840 \\; Y_6 - 80793859\/8918138880 \\; Y_5 - 291120383\/2972712960 \\; Y_4  \\\\ \\nonumber && \n\\quad - 262597\/645120 \\; Y_3 - 10633541\/247726080 \\; Y_2 + 8290501\/61931520 \\; Y_1 - 5150021\/15482880 \\; Y_0\\\\ \\nonumber && \\quad + (43529\/5806080 F_0 - 8543128133\/78033715200 )\\;  (2\\pi)^{-2} \\\\ \\nonumber &&\n\\quad  - 569479\/20643840 \\; Z_1 - 2292133\/61931520 \\; Z_0; \\\\ \\nonumber && \n  J(0,6) =  1133309347\/237817036800 + 1429\/10616832 \\; X_0 + 2371\/552960 \\; X_3 - 1429\/2654208 \\; X_2 \\\\ \\nonumber && \n\\quad + 401\/552960 \\; X_1 + 10571051\/10192158720 \\; Y_{11} - 639402031\/29727129600 \\; Y_{10} \\\\ \\nonumber && \n\\quad + 48006407249\/267544166400 \\; Y_9 + 5666277343\/1223059046400 \\; Y_8 + 2239163267\/107017666560 \\; Y_7 \\\\ \\nonumber && \\quad - 470661551\/951268147200 \\; Y_6 - 1645375871\/428070666240 \\; Y_5 - 40565004503\/713451110400 \\; Y_4 \\\\ \\nonumber && \\quad  - 23258381\/154828800 \\; Y_3 - 470661551\/19818086400 \\; Y_2 + 406642351\/4954521600 \\; Y_1 \\\\ \\nonumber && \n \\quad - 153477551\/1238630400 \\; Y_0 \n+( 1169977\/232243200 F_0 - 1234692078509\/18728091648000)\\; (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad - 117647549\/14863564800 \\; Z_1 - 202376477\/14863564800 \\; Z_0; \\\\ \\nonumber &&\n  J(1,1) = - 1\/48 + 1\/192 \\; Y_8 - 25\/12 \\; Y_7 + 1\/192 \\; Y_6 + 1\/48 \\; Y_5 + 59\/48 \\; Y_4\\\\ \\nonumber && \n\\quad + 1\/4 \\; Y_2 - \\; Y_1 + 4 \\; Y_0 + (1 + F_0) \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n J(1,2) = - 19\/1536 - 35\/768 \\; Y_{11} - 19\/64 \\; Y_{10} - 173\/288 \\; Y_9 + 5497\/18432 \\; Y_8 \\\\ \\nonumber && \n\\quad - 293\/1152 \\; Y_7 + 19\/6144 \\; Y_6 + 53\/4608 \\; Y_5 + 187\/512 \\; Y_4 + \\; Y_3 + 19\/128 \\; Y_2  \\\\ \\nonumber && \n\\quad - 19\/32 \\; Y_1 + 3\/8 \\; Y_0 + 307\/576 \\;  (2\\pi)^{-2} + 1\/96 \\; Z_0; \\\\ \\nonumber && \n  J(1,3) = - 5491\/737280 + 721\/73728 \\; Y_{11} + 1957\/30720 \\; Y_{10} - 21361\/55296 \\; Y_9 \\\\ \\nonumber && \n\\quad - 542443\/8847360 \\; Y_8 - 266629\/552960 \\; Y_7 + 5959\/2949120 \\; Y_6 + 18289\/2211840 \\; Y_5 \\\\ \\nonumber && \n\\quad + 274901\/737280 \\; Y_4 + 149\/480 \\; Y_3 + 5959\/61440 \\; Y_2 - 5959\/15360 \\; Y_1 \\\\ \\nonumber && \n\\quad + 1799\/3840 \\; Y_0 + (438551\/2764800 + 11\/160 F_0) \\;  (2\\pi)^{-2} - 13\/5120 \\; Z_1 + 35\/3072 \\; Z_0; \\\\ \\nonumber && J(1,4) = - 135181\/27525120 - 1519\/3538944 \\; Y_{11} - 4123\/1474560 \\; Y_{10}- 2532143\/13271040 \\; Y_9 \\\\ \\nonumber && \n\\quad + 12187051\/2972712960 \\; Y_8 - 12605351\/37158912 \\; Y_7 + 1307897\/990904320 \\; Y_6 \\\\ \\nonumber && \n\\quad + 783523\/148635648 \\; Y_5 + 18292649\/82575360 \\; Y_4 - 533\/161280 \\; Y_3 \\\\ \\nonumber && \n\\quad + 1307897\/20643840 \\; Y_2 - 1307897\/5160960 \\; Y_1 + 93817\/1290240 \\; Y_0 \\\\ \\nonumber && \n\\quad + (346641047\/2167603200 - 1\/1680 F_0) \\;  (2\\pi)^{-2} - 22817\/15482880 \\; Z_1 + 146059\/15482880 \\; Z_0; \\\\ \\nonumber && \n  J(1,5) = - 32271257\/9512681472 + 9401131\/2038431744 \\; Y_{11} + 178621489\/5945425920 \\; Y_{10} \\\\ \\nonumber && \n\\quad - 1051530899\/53508833280 \\; Y_9 - 49631503063\/1712282664960 \\; Y_8 \\\\ \\nonumber && \n\\quad - 5652301063\/15288238080 \\; Y_7 + 167849953\/190253629440 \\; Y_6 \\\\ \\nonumber && \n\\quad + 1548254101\/428070666240 \\; Y_5 + 26366380697\/142690222080 \\; Y_4 \\\\ \\nonumber && \n\\quad - 4481117\/30965760 \\; Y_3 + 167849953\/3963617280 \\; Y_2 - 167849953\/990904320 \\; Y_1 \\\\ \\nonumber && \n\\quad + 9544673\/247726080 \\; Y_0 + (260638894247\/3745618329600 + 1339\/215040 F_0) \\; (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad - 541139\/990904320 \\; Z_1 + 806521\/110100480 \\; Z_0; \\\\ \\nonumber && \n  J(2,1) = 23\/4608 - 35\/384 \\; Y_{11} - 19\/32 \\; Y_{10} - 173\/144 \\; Y_9 + 3619\/6144 \\; Y_8 \\\\ \\nonumber && \n\\quad + 2827\/1152 \\; Y_7 - 23\/18432 \\; Y_6 - 31\/4608 \\; Y_5 - 3949\/4608 \\; Y_4 + \\; Y_3 - 23\/384 \\; Y_2 \\\\ \\nonumber && \n\\quad + 23\/96 \\; Y_1 - 19\/24 \\; Y_0 + ( 127\/288 - 1\/4 F_0) \\;  (2\\pi)^{-2} - 1\/96 \\; Z_0; \\\\ \\nonumber && \n J(2,2) = 7043\/1105920 + 343\/12288 \\; Y_{11} + 931\/5120 \\; Y_{10} - 3731\/4608 \\; Y_9 \\\\ \\nonumber && \n\\quad - 806147\/4423680 \\; Y_8 + 339959\/276480 \\; Y_7 - 6467\/4423680 \\; Y_6 - 5879\/1105920 \\; Y_5 \\\\ \\nonumber && \n\\quad - 365953\/1105920 \\; Y_4 + 221\/240 \\; Y_3 - 6467\/92160 \\; Y_2 + 6467\/23040 \\; Y_1 \\\\ \\nonumber && \n\\quad + 4373\/5760 \\; Y_0  + (23\/160 F_0 - 25657\/76800) \\;  (2\\pi)^{-2} - 1\/480 \\; Z_1 - 7\/512 \\; Z_0; \\\\ \\nonumber && \n  J(2,3) = 4611371\/743178240 - 73045\/3538944 \\; Y_{11} - 39653\/294912 \\; Y_{10} \\\\ \\nonumber && \n\\quad - 12722017\/13271040 \\; Y_9 + 393115913\/2972712960 \\; Y_8 + 103603669\/61931520 \\; Y_7 \\\\ \\nonumber && \n\\quad - 4248887\/2972712960 \\; Y_6 - 504563\/82575360 \\; Y_5 - 79536305\/148635648 \\; Y_4 \\\\ \\nonumber && \n\\quad + 155081\/161280 \\; Y_3 - 4248887\/61931520 \\; Y_2 + 4248887\/15482880 \\; Y_1 \\\\ \\nonumber && \n\\quad + 66473\/3870720 \\; Y_0 + 493568683\/6502809600 \\;  (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad - 1093\/26880 F_0 \\;  (2\\pi)^{-2} - 10069\/5160960 \\; Z_1 - 74813\/5160960 \\; Z_0; \\\\ \\nonumber && \n J(2,4) =  196602193\/35672555520 + 3693635\/509607936 \\; Y_{11} + 14035813\/297271296 \\; Y_{10} \\\\ \\nonumber && \n\\quad - 12442566511\/13377208320 \\; Y_9 - 20639077343\/428070666240 \\; Y_8 \\\\ \\nonumber && \n\\quad + 36967799143\/26754416640 \\; Y_7 - 181900981\/142690222080 \\; Y_6 \\\\ \\nonumber && \n\\quad - 530928403\/107017666560 \\; Y_5 - 2755058051\/7134511104 \\; Y_4 + 2424601\/2580480 \\; Y_3 \\\\ \\nonumber && \n\\quad - 181900981\/2972712960 \\; Y_2 + 181900981\/743178240 \\; Y_1 + 46110859\/185794560 \\; Y_0 \\\\ \\nonumber && \n\\quad - 79216495853\/936404582400 \\;  (2\\pi)^{-2} + 4141\/215040 F_0 \\;  (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad - 408367\/247726080 \\; Z_1 - 125077\/9175040 \\; Z_0; \\\\ \\nonumber && \n J(3,1) = - 2473\/2211840 + 1687\/24576 \\; Y_{11} + 4579\/10240 \\; Y_{10} + 28153\/27648 \\; Y_9 \\\\ \\nonumber && \n\\quad - 3930323\/8847360 \\; Y_8 - 866969\/552960 \\; Y_7 + 2797\/8847360 \\; Y_6 + 5689\/2211840 \\; Y_5 \\\\ \\nonumber && \n\\quad + 1137263\/2211840 \\; Y_4 - 211\/480 \\; Y_3 + 2797\/184320 \\; Y_2 - 2797\/46080 \\; Y_1 \\\\ \\nonumber && \n\\quad + 12317\/11520 \\; Y_0  + ( 23\/80 F_0 - 323129\/460800) \\; (2\\pi)^{-2} - 3\/5120 \\; Z_1 + 11\/3072 \\; Z_0; \\\\ \\nonumber && \n  J(3,2) =  - 326465\/148635648 - 206605\/3538944 \\; Y_{11} - 112157\/294912 \\; Y_{10} \\\\ \\nonumber && \n\\quad + 2493365\/2654208 \\; Y_9 + 225122965\/594542592 \\; Y_8 - 27027877\/37158912 \\; Y_7 \\\\ \\nonumber && \n\\quad + 336725\/594542592 \\; Y_6 + 19049\/16515072 \\; Y_5 + 6739903\/148635648 \\; Y_4 \\\\ \\nonumber && \n\\quad - 22955\/32256 \\; Y_3 + 336725\/12386304 \\; Y_2 - 336725\/3096576 \\; Y_1 - 483563\/774144 \\; Y_0 \\\\ \\nonumber && \n\\quad + (429664553\/1300561920 - 331\/2688 F_0) \\;  (2\\pi)^{-2} - 95\/344064 \\; Z_1 + 6823\/1032192 \\; Z_0; \\\\ \\nonumber && \n  J(3,3) = - 69355327\/23781703680 + 4168243\/339738624 \\; Y_{11} + 79196617\/990904320 \\; Y_{10} \\\\ \\nonumber && \n\\quad + 3687114839\/2972712960 \\; Y_9 - 22463493247\/285380444160 \\; Y_8 \\\\ \\nonumber && \n\\quad - 1869533095\/1189085184 \\; Y_7 + 70582891\/95126814720 \\; Y_6 \\\\ \\nonumber && \n\\quad + 45684329\/14269022208 \\; Y_5 + 362000483\/880803840 \\; Y_4 - 1586311\/1720320 \\; Y_3 \\\\ \\nonumber && \n\\quad + 70582891\/1981808640 \\; Y_2 - 70582891\/495452160 \\; Y_1 + 7231211\/123863040 \\; Y_0 \\\\ \\nonumber && \n\\quad - 131572132177\/624269721600 \\;  (2\\pi)^{-2} + 12241\/215040 F_0 \\;  (2\\pi)^{-2} \\\\ \\nonumber && \n\\quad - 34099\/165150720 \\; Z_1 + 1460153\/165150720 \\; Z_0; \\nonumber \n\\end{eqnarray}\n\n\\begin{itemize} \n\\item{\\bf ${\\cal J}(p,q;\\mu_B^2)$ functions  at $0\\leq p\\leq 3$, $q\\leq -7$}\n\\end{itemize}\n\nProvided that ${\\cal J}(p,q;\\mu_B^2)$ over the domain ${\\cal A}$ (at $q\\leq 0$)\nand $B(0,q)$ at $q\\leq 0$ are known, ${\\cal J}$ functions over this strip \ncan be found by the recurrent relations in Appendix~5.\nNote that there are no divergent terms here.\n\n\\begin{itemize} \n\\item{\\bf ${\\cal J}(p,q;\\mu_B^2)$ functions at $ p\\leq -1$, $q\\leq 0$}\n\\end{itemize}\n\nGiven ${\\cal J}$ functions over the strip $0\\leq p \\leq 3,\\ q\\leq 0$\nand ${\\cal B}$ functions at $p\\leq 0, q \\leq 0$,\n${\\cal J}$ functions at $ p\\leq -1$, $q\\leq 0$ can be\ndetermined using the recurrent relations\nin Appendix~8, however, with $\\mu_B=0$ (in this domain, ${\\cal J}(p,q;\\mu_B^2)=J(p,q)$).\n\n\\begin{itemize} \n\\item{\\bf ${\\cal J}(p,q;\\mu_B^2)$ functions at $ p\\leq -5$, $1 \\leq q\\leq 3$}.\n\\end{itemize}\n\nThey can be determined using the recurrent relations presented in Appendix~6.\nThe initial conditions are provided by the formulas (\\ref{eq:JcrossLEFT})\nand $B$ functions determined previously.\nIn calculation of $J(p,q)$ one needs $B(r,s)$ at $r\\geq p$ and $-3\\leq q \\leq 3$.\n\n\\begin{itemize}\n\\item{\\bf ${\\cal J}(p,q;\\mu_B^2)$ functions at $ 0\\leq p\\leq 3$ and $q\\geq 6-p$.}\n\\end{itemize}\n\nIn this domain, the divergent part $D(p,q)$ is calculated separately,\nthe results are partially presented in (\\ref{DivPart_PP_min_domain});\nthus we calculate the functions $J(p,q)$ by the formulas\n\n\\begin{equation}\nJ(n,q+1-n) = { 1 \\over (2q-1)(2q-3)} \\sum_{k=0}^3 M(n,k,q) Z_k(q),\n\\end{equation}\n\nwhere $M(n,k,q)$ are given by \n\\begin{eqnarray}\nM(0,0,q) &=&(q + 1)\/32\/(q - 1); \\\\ \\nonumber\nM(0,1,q) &=& -(8q-3)(q-2)\/32\/q\/(q - 1); \\\\ \\nonumber\nM(0,2,q) &=&-(q-2)\/4\/q\/(q - 1); \\\\ \\nonumber\nM(0,3,q) &=&-(q-2)\/2\/q\/(q-1)^2; \\\\ \\nonumber\nM(1,0,q) &=&(q + 1)\/32; \\\\ \\nonumber\nM(1,1,q) &=&3\/32; \\\\ \\nonumber\nM(1,2,q) &=& -(q-2)\/4\/q; \\\\ \\nonumber\nM(1,3,q) &=& -(q-2)\/2\/q\/(q - 1); \\\\ \\nonumber\nM(2,0,q) &=& q(q + 1)\/32; \\\\ \\nonumber\nM(2,1,q) &=& 3q\/32; \\\\ \\nonumber\nM(2,2,q) &=&3\/16\/q; \\\\ \\nonumber\nM(2,3,q) &=& -(q-2)\/2\/(q - 1); \\\\ \\nonumber\nM(3,0,q) &=& q(q+1)^2\/64; \\\\ \\nonumber\nM(3,1,q) &=&3q(q + 1)\/64; \\\\ \\nonumber\nM(3,2,q) &=&3(q + 1)\/32\/q; \\\\ \\nonumber\nM(3,3,q) &=& - (8q^2-19q+3)\/16\/q\/(q - 1); \\\\ \\nonumber\n\\end{eqnarray}\nand $Z_k(q)$ can be determined from the recurrent relations\npresented in Appendix~7. The initial conditions are provided by the formulas \n(\\ref{eq:JcrossDOWN}), (\\ref{eq:JcrossLEFT}), and (\\ref{eq:JcrossUP});\n${\\cal B}(0,q)$ at $q>0$ and $D(p,q,r)$ at $0\\leq p \\leq 3, q>1-p, 0\\leq r\\leq p+q-2$ are also needed.\n\n\\begin{itemize}\n\\item{\\bf ${\\cal J}(p,q;\\mu_B^2)$ functions at $ p\\leq -1$, $q\\geq 4$.}\n\\end{itemize}\n\nThe respective recurrent relations can be found in Appendix~8,\nthe initial conditions are provided by the ${\\cal B}(p,q)$ \nfunctions at $p\\leq 0$ and $q\\geq -2$ and ${\\cal J}(p,q)$ functions\nat $0\\leq p\\leq 3$ and $-2\\leq q \\leq 3$.\n\n\\begin{itemize}\n\\item{\\bf ${\\cal J}(p,q)$ functions in the domain $p \\geq 4, q\\geq 0 $ }\n\\end{itemize}\nThe recurrent relations in Appendix~9 give expressions for the \nfinite part $J(p,q)$, the divergent part should be found by the \nprocedure described in the beginning of this Section,\nsee also (\\ref{DivPart_PP_min_domain}).\n\nIt should be also noted that the formulas in Appendix~9 {\\bf are valid \nonly in the case $p>4$,} in order to use them at $p=4$ the quantities $J(0,q)$\nthat appear in the right-hand part should be replaced by $B(0,q)$.\n\n\\begin{itemize}\n\\item{\\bf ${\\cal J}(p,q)$ functions in the domain $p \\geq 4, q < 0 $ }\n\\end{itemize}\nThe explicit expressions for the finite parts $J(p,q)$ at $q=-1, -2$ and \n$p\\leq 9$ are presented in the Appendix~10, the recurrent relations\nvalid at $p>4, q<-2$ are given in the Appendix~11.\nTo employ these relations at $p=4$, one should replace $J(0,q)$ that\nappears in the right-hand side by the function $B(0,q)$.\n\nThe divergent parts are given by the formula (\\ref{DivPart_PM_up_to_9}).\n\nTherewith, it should be noted that the constants $X_0 \\div X_3$ \nthat appear in some expressions for $J(p,q)$ at $p\\leq 0$ cancel \nin the expressions for the integrals (\\ref{eq:GenFermInt}) at $p>0$\nand thus their numerical values are not needed.\n\n\\vspace*{-1mm}\n\\subsection{Dimensional Regularization}\n\nIn the dimensional regularization, \neach integral $F(p,q;n1,n2,n3,n4)$ (see (\\ref{eq:GenFermInt})) is associated with\nthe respective boson integral $B_F(p,q;n1,n2,n3,n4)$ that has the same divergent part. \n$B_F(p,q;n1,n2,n3,n4)$ is determined by the procedure \nsimilar to that indicated in subsection (\\ref{DPFIFMR}).\nFor example, at $p>0, q\\geq2-p$\n\\begin{equation}\\label{ABIexample1}\nB_F (p,q;n1,n2,n3,n4) = \\sum_{l=0}^{p+q-2} {(-1)^l\\; (p+l-1)!\\over l! (p-1)!} \n\\int {dk\\over (2\\pi)^4} \\; {\\Delta^l  \\cos^{n_1}(k_1)\\ .... \\cos^{n_4}(k_4)\\ \\over \\Delta_B^{p+q+l}}. \\\\ \\nonumber\n\\end{equation}\nThen we compute $B_F(p,q;n1,n2,n3,n4)$ in the dimensional regularization as \nit is described in subsection (\\ref{sec:DRbos}) and \n$F(p,q;n1,n2,n3,n4)-B_F(p,q;n1,n2,n3,n4)$ (which is convergent) in the fictitious mass regularization.\nThe sum of these quantities provides the sought for result.\\\\[1mm]\n\n\\vspace*{-1mm}\n\\section{Conclusions}\n\nThe BCP algorithm described above and the explicit formulas obtained with it and presented\nin the Appendices make it a straightforward matter to express an integral \nof the type (\\ref{eq:GenFermInt}) at $p\\leq 9$ and arbitrary values of \n$n_1, n_2, n_3, n_4$ and $q$ in terms of the constants $F_0, Z_0, Z_1$ and $Y_0 \\div Y_{11}$.\nIn fact, these formulas provide a computer program,\nwhich can easily be realized with various packages. \nSuch program was written in FORM and performed, some of \nthe results are presented on the web: \\ \\ \n{\\tt http:\/\/www.lattice.itep.ru\/$\\sim$pbaivid\/lattpt\/ }\n\nThese are\n\\begin{itemize} \n\\item the values of the functions $J(p,q)$ and $B(p,q)$\nat $-26 \\leq p \\leq 0,\\ \\ -56 - 2p \\leq q \\leq 34 $ \nand the values of $J(p,q)$  at $1\\leq p \\leq 9, \\ \\ -28 \\leq q \\leq 33 - p$;\n\\item the expressions for the integrals of the type (\\ref{eq:GenFermInt})\nat some particular values of $p$ and $q$ and $n_1\\leq 6$;\n\\item the program for the computation of the integrals \n(\\ref{eq:GenFermInt}) at $0\\leq p,q \\leq 9$ and $n_\\mu^{max}\\leq 6$\nthat can be readily used by anyone.\n\\end{itemize} \n\nI hope that this work will facilitate using the BCP algorithm in practical computations.\n\n{\\large \\bf Acknowledgments:} I am grateful to H.Perlt, A.Schiller, and V. Bornyakov for stimulating discussions,\nto P.Buividovich for the help with the presentation on the web,\nand to the Leipzig University, where this study was started, for hospitality.\nThis work was supported in part by the grant for scientific schools no. NSh-679.2008.2\nand by the Russian Foundation for Basic Research (RFBR grant no. 07-02-0237).\n\n\\newpage\n\n{\\Large \\bf Appendix 1.}\\\\[2mm]\n\n\nSome values used in the text are listed below.\n\nThe integrals defined in (\\ref{Bdirect}) at $4\\leq q\\leq 12$\n(see also (\\ref{IniCondForBbosonic}))\n\\begin{eqnarray}\\label{BBosonicTable1}\n&& {\\cal B}_{4} = 917\/2; \\\\ \\nonumber\n&& {\\cal B}_{5} = 2514; \\\\ \\nonumber\n&& {\\cal B}_{6} = 14376; \\\\ \\nonumber\n&& {\\cal B}_{7} = 85152; \\\\ \\nonumber\n&& {\\cal B}_{8} = 16628949\/32; \\\\ \\nonumber\n&& {\\cal B}_{9} = 26026877\/8; \\\\ \\nonumber\n&& {\\cal B}_{10} = 333148183\/16; \\\\ \\nonumber\n&& {\\cal B}_{11} = 543325293\/4; \\\\ \\nonumber\n&& {\\cal B}_{12} = 14415564199\/16;  \\nonumber\n\\end{eqnarray}\n\nThe coefficients introduced in formula (\\ref{InfeldAsExp0}) are\n\\begin{eqnarray}\n&&\tb_{0}=1; \\\\ \\nonumber\n&&\tb_{1}=1\/2; \\\\ \\nonumber\n&&\tb_{2}=3\/8; \\\\ \\nonumber\n&&\tb_{3}=13\/32; \\\\ \\nonumber\n&&\tb_{4}=77\/128; \\\\ \\nonumber\n&&\tb_{5}=297\/256; \\\\ \\nonumber\n&&\tb_{6}=5727\/2048; \\\\ \\nonumber\n&&\tb_{7}=66687\/8192; \\\\ \\nonumber\n&&\tb_{8}=912303\/32768; \\\\ \\nonumber\n&&\tb_{9}=3586545\/32768; \\\\ \\nonumber\n&&\tb_{10}=127448505\/262144; \\\\ \\nonumber\n&&\tb_{11}=2523924765\/1048576; \\\\ \\nonumber\n&&\tb_{12}=110207056005\/8388608; \\\\ \\nonumber\n&&\tb_{13}=657259273755\/8388608; \\\\ \\nonumber\n&&\tb_{14}=68022530602425\/134217728; \\\\ \\nonumber\n&&\tb_{15}=1897008475419225\/536870912; \\\\ \\nonumber\n&&\tb_{16}=56719614296927925\/2147483648; \\\\ \\nonumber\n&&\tb_{17}=226232753142332475\/1073741824; \\\\ \\nonumber\n&&\tb_{18}=15346146376168947675\/8589934592; \\\\ \\nonumber\n&&\tb_{19}=275641831899783381375\/17179869184; \\\\ \\nonumber\n&&\tb_{20}=41819089838429396989125\/274877906944;  \\nonumber\n\\end{eqnarray}\nThe coefficients $c_q(n_1,n_2,n_3,n_4)$ used \nin the dimensional regularization are introduced in (\\ref{d_coeff_def}):\n\\begin{eqnarray}\n   c_0(n_1,n_2,n_3,n_4) &=& 0, \\\\ \\nonumber\n   c_1(n_1,n_2,n_3,n_4) &=& 1\/8, \\nonumber\n\\end{eqnarray}\nthese equations at $q=0,1$ are valid for all $n_\\mu$.\nThe coefficients $c_q=c_q(0,0,0,0)$ at $q\\leq 10$ are\n\\begin{eqnarray}\n   c_2(0,0,0,0) &=& 1\/8; \\\\ \\nonumber\n   c_3(0,0,0,0) &=& 55\/384; \\\\ \\nonumber\n   c_4(0,0,0,0) &=& 5\/24; \\\\ \\nonumber\n   c_5(0,0,0,0) &=& 1973\/5120; \\\\ \\nonumber\n   c_6(0,0,0,0) &=& 54583\/61440; \\\\ \\nonumber\n   c_7(0,0,0,0) &=& 8558131\/3440640; \\\\ \\nonumber\n   c_8(0,0,0,0) &=& 4727509\/573440; \\\\ \\nonumber\n   c_9(0,0,0,0) &=& 652905649\/20643840; \\\\ \\nonumber\n   c_{10}(0,0,0,0) &=& 2276619691\/16515072;  \\nonumber\n\\end{eqnarray}\nAnd the values some other of the coefficients $c_q(n_1,n_2,n_3,n_4)$:\n\\begin{eqnarray}\n   c_2(1,0,0,0) &=& 1\/16; \\\\ \\nonumber\n   c_2(1,1,0,0) &=& 0; \\\\ \\nonumber\n   c_2(1,1,1,0) &=& - 1\/16; \\\\ \\nonumber\n   c_2(1,1,1,1) &=& - 1\/8; \\\\ \\nonumber\n   c_2(2,0,0,0) &=& 0; \\\\ \\nonumber\n   c_2(2,1,0,0) &=& - 1\/16; \\\\ \\nonumber\n   c_2(2,1,1,0) &=& - 1\/8; \\\\ \\nonumber\n   c_2(2,1,1,1) &=& - 3\/16; \\\\ \\nonumber\n   c_3(1,0,0,0) &=&  25\/384; \\\\ \\nonumber\n   c_3(1,1,0,0) &=&  7\/384; \\\\ \\nonumber\n   c_3(1,1,1,0) &=&  1\/384; \\\\ \\nonumber\n   c_4(1,0,0,0) &=&  27\/256; \\\\ \\nonumber\n   c_4(1,1,0,0) &=&  19\/384; \\\\ \\nonumber\n   c_4(1,1,1,0) &=&  19\/768; \\\\ \\nonumber\n   c_4(1,1,1,1) &=&  1\/64; \\\\ \\nonumber\n   c_4(2,0,0,0) &=&  55\/384; \\\\ \\nonumber\n   c_5(1,0,0,0) &=&  1143\/5120; \\\\ \\nonumber\n   c_5(1,1,0,0) &=&  1999\/15360; \\\\ \\nonumber\n   c_5(2,0,0,0) &=&  1433\/5120; \\\\ \\nonumber\n   c_6(1,0,0,0) &=&  7085\/12288; \\\\ \\nonumber\n   c_6(1,1,0,0) &=&  23387\/61440; \\\\ \\nonumber\n   c_6(2,0,0,0) &=&  40867\/61440; \\nonumber\n\\end{eqnarray}\n\\pagebreak \n\n{\\Large \\bf Appendix 2.}\n\n\\begin{eqnarray}\\label{DPpartQ_eq_0_dlt}\n L(0,2) &=& ( \\;l_C + 1\/2 \\;l_C^2  ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n L(0,3) &=& ( - 3\/4 \\mu_B^{-2} - 1\/2 \\; \\mu_B^{-2} \\;l_C + 5\/8 \\;l_C + 1\/8 \\;l_C^2 ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n L(0,4) &=& ( ( - 1\/6 \\; \\mu_B^{-4} - 1\/12 \\; \\mu_B^{-2} + 137\/960 ) \\;l_C \\\\ \\nonumber \n  &&   - 5\/36 \\; \\mu_B^{-4} - 31\/144 \\; \\mu_B^{-2}   + 1\/32  \\;l_C^2  ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n L(0,5) &=& (  ( - 1\/12 \\; \\mu_B^{-6} - 1\/48 \\; \\mu_B^{-4}  - 1\/64 \\; \\mu_B^{-2} + 15527\/322560 ) \\;l_C \\\\ \\nonumber \n  && - 7\/144 \\; \\mu_B^{-6}  - 101\/2880 \\; \\mu_B^{-4} - 151\/3840 \\; \\mu_B^{-2}+ 13\/1536 \\;l_C^2  ) \/ (2\\pi)^{2} ; \\\\ \\nonumber  \n L(0,6) &=& ( ( - 1\/20 \\; \\mu_B^{-8} - 1\/120 \\; \\mu_B^{-6} - 1\/320 \\; \\mu_B^{-4} - 13\/3840 \\; \\mu_B^{-2} + 172241\/12902400 ) \\;l_C  \\\\ \\nonumber \n  && - 9\/400 \\; \\mu_B^{-8} - 77\/7200 \\; \\mu_B^{-6} - 709\/134400  \\; \\mu_B^{-4} - 3953\/403200 \\; \\mu_B^{-2} \\\\ \\nonumber \n  && + 77\/30720 \\;l_C^2 ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n L(0,7) &=& ( (- 1\/30 \\; \\mu_B^{-10} - 1\/240 \\; \\mu_B^{-8} - 1\/960 \\; \\mu_B^{-6} \\\\ \\nonumber \n&& - 13\/23040 \\; \\mu_B^{-4} - 77\/92160 \\; \\mu_B^{-2}  + 457867\/94617600 ) \\;l_C  \\\\ \\nonumber \n&&  - 11\/900 \\; \\mu_B^{-10} - 439\/100800 \\; \\mu_B^{-8} - 53\/38400 \\; \\mu_B^{-6} \\\\ \\nonumber \n&&  - 5371\/4838400 \\; \\mu_B^{-4} - 183101\/77414400 \\; \\mu_B^{-2} + 33\/40960 \\;l_C^2  ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n L(0,8) &=& (  ( - 1\/42 \\; \\mu_B^{-12} - 1\/420 \\; \\mu_B^{-10} - 1\/2240 \\; \\mu_B^{-8} - 13\/80640 \\;\\mu_B^{-6}\\\\ \\nonumber \n   && - 11\/92160 \\; \\mu_B^{-4} - 33\/143360 \\; \\mu_B^{-2} + 51135377\/31791513600 )\\;l_C \\\\ \\nonumber \n  &&  - 13\/1764 \\; \\mu_B^{-12} - 743\/352800 \\; \\mu_B^{-10} - 2789\/5644800 \\; \\mu_B^{-8} \\\\ \\nonumber \n  &&  - 3371\/13547520 \\;\\mu_B^{-6} - 28441\/121651200 \\; \\mu_B^{-4} - 10159\/14450688  \\; \\mu_B^{-2} \\\\ \\nonumber \n  &&   + 1909\/6881280 \\;l_C^2  ) \/ (2\\pi)^{2} ;  \\nonumber \n\\end{eqnarray}\n\\begin{eqnarray}\\label{DivPart_PP_min_domain}\n D(0,2) &=&  - \\;l_C  \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(0,3) &=& (   1\/2 \\; \\mu_B^{-2} - 1\/4 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(0,4) &=& (   1\/6 \\; \\mu_B^{-4} + 1\/12 \\; \\mu_B^{-2} - 1\/16 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(0,5) &=& (   1\/12 \\; \\mu_B^{-6} + 1\/48 \\; \\mu_B^{-4} + 1\/64 \\; \\mu_B^{-2}  - 13\/768 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(0,6) &=& (   1\/20 \\; \\mu_B^{-8} + 1\/120 \\; \\mu_B^{-6} + 1\/320 \\; \\mu_B^{-4} + 13\/3840 \\; \\mu_B^{-2} - 77\/15360 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(0,7) &=& ( 1\/30 \\; \\mu_B^{-10} + 1\/240 \\; \\mu_B^{-8} + 1\/960 \\; \\mu_B^{-6} + 13\/23040 \\; \\mu_B^{-4}\\\\ \\nonumber \n&&    + 77\/92160 \\; \\mu_B^{-2} - 33\/20480 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(0,8) &=& ( 1\/42 \\; \\mu_B^{-12} + 1\/420 \\; \\mu_B^{-10} + 1\/2240 \\; \\mu_B^{-8} + 13\/80640 \\; \\mu_B^{-6}\\\\ \\nonumber \n&&   + 11\/92160 \\; \\mu_B^{-4} + 33\/143360 \\; \\mu_B^{-2} - 1909\/3440640 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(1,1) &=& (    - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(1,2) &=& (   1\/2 \\; \\mu_B^{-2} ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(1,3) &=& ( 1\/6 \\; \\mu_B^{-4} + 1\/48 \\; \\mu_B^{-2} - 11\/160 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(1,4) &=& ( 1\/12 \\;\\mu_B^{-6} + 1\/120 \\; \\mu_B^{-4} + 7\/480 \\; \\mu_B^{-2} + 1\/1680 \\;l_C ) \/ (2\\pi)^{2}; \\\\ \\nonumber \n   D(1,5) &=& (  1\/20 \\; \\mu_B^{-8} + 1\/240 \\; \\mu_B^{-6} + 3\/1120 \\; \\mu_B^{-4} + 19\/21504 \\; \\mu_B^{-2} - 1339\/215040 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(1,6) &=& (  1\/30 \\; \\mu_B^{-10} + 1\/420 \\; \\mu_B^{-8} + 23\/26880 \\; \\mu_B^{-6} + 13\/53760 \\; \\mu_B^{-4} \\\\ \\nonumber \n && + 181\/215040 \\; \\mu_B^{-2} - 67\/630784 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(1,7) &=& ( 1\/42 \\;\\mu_B^{-12} + 1\/672 \\;\\mu_B^{-10} + 29\/80640 \\;\\mu_B^{-8} + 277\/3225600 \\;\\mu_B^{-6} \\\\ \\nonumber \n&& + 15163\/141926400 \\;\\mu_B^{-4} + 46523\/567705600 \\; \\mu_B^{-2} - 52001\/75694080 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(2,1) &=& (  1\/2 \\; \\mu_B^{-2} + 1\/4 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(2,2) &=& ( 1\/6 \\; \\mu_B^{-4} - 1\/24 \\; \\mu_B^{-2} - 23\/160 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(2,3) &=& ( 1\/12 \\;\\mu_B^{-6} - 1\/240 \\; \\mu_B^{-4} + 1\/40 \\;\\mu_B^{-2} + 1093\/26880 \\;l_C ) \/ (2\\pi)^{2}; \\\\ \\nonumber \n D(2,4) &=& ( 1\/20 \\;\\mu_B^{-8} + 13\/3360 \\;\\mu_B^{-4} - 37\/8960 \\;\\mu_B^{-2} - 4141\/215040 \\;l_C )\/(2\\pi)^{2};\\\\ \\nonumber \n D(2,5) &=& ( 1\/30 \\; \\mu_B^{-10} + 1\/1680 \\; \\mu_B^{-8} + 29\/26880 \\; \\mu_B^{-6} - 1\/3072 \\; \\mu_B^{-4} \\\\ \\nonumber \n&& + 31\/15360 \\; \\mu_B^{-2} + 34689\/6307840 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(2,6) &=& ( 1\/42 \\;\\mu_B^{-12} + 1\/1680 \\; \\mu_B^{-10} + 11\/26880 \\; \\mu_B^{-8} - 17\/537600 \\;\\mu_B^{-6} \\\\ \\nonumber\n&&  + 4747\/23654400 \\; \\mu_B^{-4} - 36473\/94617600 \\; \\mu_B^{-2} - 128103\/50462720 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(3,1) &=& (  1\/6 \\; \\mu_B^{-4} - 5\/48 \\; \\mu_B^{-2} - 23\/80 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(3,2) &=& (  1\/12 \\; \\mu_B^{-6} - 1\/60 \\; \\mu_B^{-4} + 3\/64 \\; \\mu_B^{-2}  + 331\/2688 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(3,3) &=& ( 1\/20 \\; \\mu_B^{-8} - 1\/240 \\; \\mu_B^{-6} + 3\/448 \\;\\mu_B^{-4} - 19\/1344 \\; \\mu_B^{-2} - 12241\/215040 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(3,4) &=& ( 1\/30 \\; \\mu_B^{-10} - 1\/840 \\; \\mu_B^{-8} + 23\/13440 \\;\\mu_B^{-6} - 19\/13440 \\; \\mu_B^{-4} \\\\ \\nonumber \n&& + 481\/86016 \\; \\mu_B^{-2} +  35879\/1576960 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(3,5) &=& ( 1\/42 \\; \\mu_B^{-12} - 1\/3360 \\; \\mu_B^{-10} + 1\/1680 \\;\\mu_B^{-8} - 53\/215040 \\;\\mu_B^{-6} \\\\ \\nonumber \n&& + 4801\/9461760 \\; \\mu_B^{-4} - 33797\/18923520 \\; \\mu_B^{-2} - 492689\/50462720 \\;l_C ) \/ (2\\pi)^{2} ;  \\nonumber \n\\end{eqnarray}\n\\begin{eqnarray}\n   D(4,1) &=& ( 1\/12 \\; \\mu_B^{-6} - 7\/240 \\; \\mu_B^{-4} + 77\/960 \\;\\mu_B^{-2} + 7201\/26880 \\;l_C ) \/ (2\\pi)^{2}; \\\\ \\nonumber \n   D(4,2) &=& ( 1\/20 \\; \\mu_B^{-8} - 1\/120 \\; \\mu_B^{-6} + 5\/448 \\;\\mu_B^{-4} - 85\/2688 \\; \\mu_B^{-2} - 29671\/215040 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(4,3) &=& ( 1\/30 \\; \\mu_B^{-10} - 1\/336 \\; \\mu_B^{-8} + 37\/13440 \\;\\mu_B^{-6} - 19\/5760 \\; \\mu_B^{-4} \\\\ \\nonumber \n&& + 17267\/1290240 \\; \\mu_B^{-2} + 51223\/788480 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(4,4) &=& ( 1\/42 \\; \\mu_B^{-12} - 1\/840 \\; \\mu_B^{-10} + 37\/40320 \\; \\mu_B^{-8} - 11\/17920 \\; \\mu_B^{-6} \\\\ \\nonumber \n&& + 16853\/14192640 \\; \\mu_B^{-4}       - 923\/177408 \\; \\mu_B^{-2} - 2284033\/75694080 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(5,1) &=& ( 1\/20 \\; \\mu_B^{-8} - 1\/80 \\; \\mu_B^{-6} + 29\/1680 \\; \\mu_B^{-4}  \\\\ \\nonumber \n&& - 2117\/35840 \\; \\mu_B^{-2} - 30869\/107520 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(5,2) &=& ( 1\/30 \\;\\mu_B^{-10} - 1\/210 \\;\\mu_B^{-8} + 113\/26880 \\;\\mu_B^{-6} - 1009\/161280 \\;\\mu_B^{-4} \\\\ \\nonumber \n&& + 35873\/1290240  \\; \\mu_B^{-2} + 482633\/3153920 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(5,3) &=& ( 1\/42 \\;\\mu_B^{-12} - 1\/480 \\; \\mu_B^{-10} + 37\/26880 \\; \\mu_B^{-8} - 767\/645120 \\; \\mu_B^{-6} \\\\ \\nonumber\n&& + 69689\/28385280 \\;\\mu_B^{-4} - 116749\/9461760 \\; \\mu_B^{-2} - 5912603\/75694080 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(6,1) &=& ( 1\/30 \\; \\mu_B^{-10} - 11\/1680 \\; \\mu_B^{-8} + 163\/26880 \\; \\mu_B^{-6} \\\\ \\nonumber \n&& - 3407\/322560 \\; \\mu_B^{-4} + 66911\/1290240 \\; \\mu_B^{-2} + 400193\/1261568 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n D(6,2) &=& ( 1\/42 \\; \\mu_B^{-12} - 1\/336 \\; \\mu_B^{-10} + 53\/26880 \\; \\mu_B^{-8} - 467\/230400 \\;\\mu_B^{-6} \\\\ \\nonumber \n && + 162457\/35481600 \\; \\mu_B^{-4} - 1214401\/47308800 \\; \\mu_B^{-2} - 26852377\/151388160 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(7,1) &=& (  1\/42 \\; \\mu_B^{-12} - 13\/3360 \\; \\mu_B^{-10} + 109\/40320 \\; \\mu_B^{-8} - 10267\/3225600 \\; \\mu_B^{-6}\\\\ \\nonumber \n&&  + 53257\/6758400  \\; \\mu_B^{-4} - 27585673\/567705600 \\; \\mu_B^{-2} - 18404583\/50462720 \\;l_C ) \/ (2\\pi)^{2};\\nonumber \n\\end{eqnarray}\n\\begin{eqnarray}\\label{DivPart_PM_up_to_9}\n   D(2,0) &=& ( - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(3,0) &=& ( 1\/2 \\; \\mu_B^{-2} + 1\/2 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(3,-1) &=& (  - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(4,0) &=& ( 1\/6 \\; \\mu_B^{-4} - 1\/6 \\; \\mu_B^{-2} - 1\/2 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(4,-1) &=& ( 1\/2 \\; \\mu_B^{-2} + 3\/4 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(4,-2) &=& ( - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(5,0) &=& ( 1\/12 \\; \\mu_B^{-6} - 1\/24 \\; \\mu_B^{-4} + 1\/8 \\; \\mu_B^{-2} + 95\/192 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(5,-1) &=& ( 1\/6 \\; \\mu_B^{-4} - 11\/48 \\; \\mu_B^{-2} - 25\/32 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(5,-2) &=& (  1\/2 \\; \\mu_B^{-2} + \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(5,-3) &=& ( - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(6,0) &=& ( 1\/20 \\; \\mu_B^{-8} - 1\/60 \\; \\mu_B^{-6} + 1\/40 \\; \\mu_B^{-4} - 19\/192 \\; \\mu_B^{-2} - 1027\/1920 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(6,-1) &=& ( 1\/12 \\;\\mu_B^{-6} - 13\/240 \\;\\mu_B^{-4} + 29\/160 \\;\\mu_B^{-2} + 3163\/3840 \\;l_C )\/(2\\pi)^{2}; \\\\ \\nonumber \n  D(6,-2) &=& ( 1\/6 \\; \\mu_B^{-4} - 7\/24 \\; \\mu_B^{-2} - 181\/160 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(6,-3) &=& ( 1\/2 \\; \\mu_B^{-2} + 5\/4 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(6,-4) &=& (  - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(7,0) &=& ( 1\/30 \\; \\mu_B^{-10} - 1\/120 \\; \\mu_B^{-8} + 1\/120 \\; \\mu_B^{-6} - 19\/1152 \\; \\mu_B^{-4} \\\\ \\nonumber \n&& + 1027\/11520 \\; \\mu_B^{-2} + 3067\/5120 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(7,-1) &=& ( 1\/20 \\; \\mu_B^{-8} - 1\/48 \\; \\mu_B^{-6} + 11\/320 \\; \\mu_B^{-4} - 1181\/7680 \\; \\mu_B^{-2} - 14099\/15360 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(7,-2) &=& ( 1\/12 \\; \\mu_B^{-6} - 1\/15 \\; \\mu_B^{-4} + 239\/960 \\;\\mu_B^{-2} + 2447\/1920 \\;l_C ) \/ (2\\pi)^{2}; \\\\ \\nonumber \n  D(7,-3) &=& ( 1\/6 \\; \\mu_B^{-4} - 17\/48 \\; \\mu_B^{-2} - 31\/20 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(7,-4) &=& ( 1\/2 \\; \\mu_B^{-2} + 3\/2 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(7,-5) &=& (  - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(8,0) &=& ( 1\/42 \\; \\mu_B^{-12} - 1\/210 \\; \\mu_B^{-10} + 1\/280 \\; \\mu_B^{-8} - 19\/4032 \\; \\mu_B^{-6}  \\\\ \\nonumber \n&& + 1027\/80640 \\; \\mu_B^{-4} - 3067\/35840 \\; \\mu_B^{-2} - 74609\/107520 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(8,-1) &=& ( 1\/30 \\; \\mu_B^{-10} - 17\/1680 \\; \\mu_B^{-8} + 37\/3360  \\; \\mu_B^{-6} \\\\ \\nonumber \n&&  - 3923\/161280 \\; \\mu_B^{-4} + 9281\/64512 \\; \\mu_B^{-2} + 150949\/143360 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(8,-2) &=& (1\/20 \\; \\mu_B^{-8} - 1\/40 \\; \\mu_B^{-6} + 61\/1344 \\; \\mu_B^{-4} - 405\/1792 \\; \\mu_B^{-2} - 79489\/53760 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(8,-3) &=& ( 1\/12 \\; \\mu_B^{-6} - 19\/240 \\; \\mu_B^{-4} + 21\/64 \\;\\mu_B^{-2} + 10037\/5376 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n  D(8,-4) &=& ( 1\/6 \\; \\mu_B^{-4} - 5\/12 \\; \\mu_B^{-2} - 163\/80 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(8,-5) &=& ( 1\/2 \\; \\mu_B^{-2} + 7\/4 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(8,-6) &=& (    - \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(9,-1) &=& ( 1\/42 \\; \\mu_B^{-12} - 19\/3360 \\; \\mu_B^{-10} + 41\/8960 \\;\\mu_B^{-8} - 4303\/645120 \\; \\mu_B^{-6}  \\\\ \\nonumber \n&& + 50513\/2580480 \\; \\mu_B^{-4} - 163217\/1146880 \\; \\mu_B^{-2} - 8535263\/6881280 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(9,-2) &=& ( 1\/30 \\; \\mu_B^{-10} - 1\/84 \\; \\mu_B^{-8} + 379\/26880 \\; \\mu_B^{-6} \\\\ \\nonumber \n&& - 791\/23040 \\; \\mu_B^{-4} + 5087\/23040 \\; \\mu_B^{-2} + 501267\/286720 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(9,-3) &=& (       1\/20 \\; \\mu_B^{-8} - 7\/240 \\; \\mu_B^{-6} + 13\/224 \\; \\mu_B^{-4} - 6841\/21504 \\; \\mu_B^{-2} - 487141\/215040 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(9,-4) &=& (   1\/12 \\; \\mu_B^{-6} - 11\/120 \\; \\mu_B^{-4} + 67\/160 \\;\\mu_B^{-2} + 8807\/3360 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(9,-5) &=& (    1\/6 \\; \\mu_B^{-4} - 23\/48 \\; \\mu_B^{-2} - 83\/32 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(9,-6) &=& (   1\/2 \\; \\mu_B^{-2} + 2 \\;l_C ) \/ (2\\pi)^{2} ; \\\\ \\nonumber \n   D(9,-7) &=& (    - \\;l_C ) \/ (2\\pi)^{2} ;  \\nonumber \n\\end{eqnarray}\n\n\\pagebreak[1]\n{\\Large \\bf Appendix 3.}\n\\begin{verbatim}\nB(p,1) = ( 1536*B(p+2,2)*(80*p^4 + 596*p^3 + 1628*p^2 + 1935*p + 846)\n + 64*B(p+1,2)*(520*p^4 + 3404*p^3 + 8106*p^2+ 8413*p+3245)\n + 48*B( p,2)*(40*p^4 + 216*p^3 + 394*p^2 + 290*p + 75)\n + 73728*B(p+4,1)*(8*p^5 + 92*p^4 + 410*p^3 + 885*p^2 + 927*p + 378)\n + 1536*B(p+3,1)*(320*p^5 + 3220*p^4 + 12772*p^3 + 24943*p^2 + 23993*p + 9110)\n + 64*B(p+2,1)*(1080*p^5 + 8260*p^4 + 24470*p^3 + 35247*p^2 + 24823*p + 6920)\n + 48*B(p+1,1)*( - 40*p^5 - 480*p^4 - 1946*p^3 - 3508*p^2 - 2911*p - 915)\n + 1152*B(p+5,-3)*(4*p^5 + 48*p^4 + 221*p^3 + 489*p^2 + 522*p + 216)\n + 12*B(p+4,-3)*( - 724*p^5 - 6504*p^4 - 22567*p^3 - 37962*p^2 - 31033*p - 9858)\n + 9*B(p+3,-3)*( - 128*p^5 - 828*p^4 - 1924*p^3 - 1961*p^2 - 867*p - 130)\n + 6*B(p+2,-3)*(32*p^5 + 180*p^4 + 296*p^3 + 15*p^2 - 328*p - 195)\n + 7296*B(p+5,-2)*(4*p^5 + 48*p^4 + 221*p^3 + 489*p^2 + 522*p + 216)\n + 4*B(p+4,-2)*( - 13756*p^5 - 133560*p^4 - 508645*p^3 - 953406*p^2\n    - 881659*p - 322086)\n + 2*B(p+3,-2)*( - 3588*p^5 - 25448*p^4 - 67995*p^3 - 85874*p^2\n    - 52149*p - 12410)\n + 4*B(p+2,-2)*(244*p^5 + 1600*p^4 + 3635*p^3 + 3080*p^2 + 216*p - 585)\n + 3072*B(p+5,-1)*(4*p^5 + 48*p^4 + 221*p^3 + 489*p^2 + 522*p + 216)\n + 32*B(p+4,-1)*( - 5908*p^5 - 62136*p^4 - 256375*p^3 - 518814*p^2\n    - 515197*p - 201018)\n + 8*B(p+3,-1)*( - 1508*p^5 - 9252*p^4 - 19927*p^3 - 16451*p^2 - 2040*p + 2084)\n + 2*B(p+2,-1)*(1712*p^5 + 13228*p^4 + 37256*p^3 + 46893*p^2 + 26223*p + 5070)\n + 3*B(p+1,-1)*( - 64*p^5 - 472*p^4 - 1072*p^3 - 422*p^2 + 1160*p + 975)\n + 73728*B(p+5,0)*( - 4*p^5 - 48*p^4 - 221*p^3 - 489*p^2 - 522*p - 216)\n + 768*B(p+4,0)*( - 400*p^5 - 4432*p^4 - 19118*p^3 - 40113*p^2 - 41003*p - 16386)\n + 8*B(p+3,0)*(10100*p^5 + 96976*p^4 + 370679*p^3 + 704416*p^2\n    + 663875*p + 247102)\n + 4*B(p+2,0)*(3220*p^5 + 22680*p^4 + 62143*p^3 + 84494*p^2 + 57851*p + 16040)\n + 4*B(p+1,0)*( - 492*p^5 - 3268*p^4 - 7609*p^3 - 7163*p^2 - 2243*p - 120)\n)\/12\/(32*p^5 + 176*p^4 + 336*p^3 + 280*p^2 + 106*p + 15);\n\nB(p,2) = (  2304*B(p+2,2)*(2*p^2 + 7*p + 6)\n + 32*B(p+1,2)*(4*p^2 + 12*p + 11)\n + 768*B(p+3,1)*( - 10*p^3 - 59*p^2 - 110*p - 64)\n + 128*B(p+2,1)*( - 60*p^3 - 299*p^2 - 495*p - 271)\n + 24*B(p+1,1)*( - 40*p^3 - 144*p^2 - 172*p - 69)\n + 6*B(p+4,-3)*( - 10*p^3 - 61*p^2 - 115*p - 66)\n + 12*B(p+3,-3)*p*(10*p^2 + 31*p + 22)\n + 38*B(p+4,-2)*( - 10*p^3 - 61*p^2 - 115*p - 66)\n + B(p+3,-2)*(742*p^3 + 2795*p^2 + 3167*p + 1090)\n + 18*B(p+2,-2)*p*(2*p^2 + 5*p + 3)\n + 16*B(p+4,-1)*( - 10*p^3 - 61*p^2 - 115*p - 66)\n + 4*B(p+3,-1)*(590*p^3 + 2719*p^2 + 4057*p + 1958)\n + 6*B(p+2,-1)*(20*p^3 + 50*p^2 + 84*p + 59)\n + 384*B(p+4,0)*(10*p^3 + 61*p^2 + 115*p + 66)\n + 4*B(p+3,0)*(1178*p^3 + 6705*p^2 + 12285*p + 7174)\n + 2*B(p+2,0)*( - 302*p^3 - 591*p^2 + 69*p + 416)\n + 6*B(p+1,0)*( - 10*p^3 - 23*p^2 - 10*p + 3)\n)\/48\/(2*p + 3)\/(p+1);\n\nB(p,3) = ( 4*B(p+1,2)*( - 20*p^2 - 46*p - 23) \n + 192*B(p+3,1)*( - 2*p^3 - 11*p^2 - 19*p - 10) \n + 2*B(p+2,1)*( - 138*p^3 - 589*p^2 - 827*p - 376)\n + 3*B(p+1,1)*( - 4*p^3 - 14*p^2 - 14*p - 3) \n + 3*B(p+4,-3)*( - p^3 - 6*p^2 - 11*p - 6) \n + 6*B(p+3,-3)*p*(p^2 + 3*p + 2) \n + 19*B(p+4,-2)*( - p^3 - 6*p^2 - 11*p - 6) \n + 2*B(p+3,-2)*(19*p^3 + 70*p^2 + 77*p + 26) \n + 8*B(p+4,-1)*( - p^3 - 6*p^2 - 11*p - 6)\n + B(p+3,-1)*(124*p^3 + 559*p^2 + 809*p + 374)\n + 3*B(p+2,-1)*( - 2*p^3 - 8*p^2 - 5*p + 1)\n + 192*B(p+4,0)*(p^3 + 6*p^2 + 11*p + 6)\n + 2*B(p+3,0)*(89*p^3 + 472*p^2 + 793*p + 410)\n + 2*B(p+2,0)*( - 37*p^3 - 119*p^2 - 122*p - 40)\n)\/12\/(2*p + 3);\n\\end{verbatim}\n\n\n{\\Large \\bf Appendix 4.}\n\nHere and below, the symbol {\\tt BB(p,q)} in any Appendix\ndesignates ${\\cal B}(p,q)$, whereas {\\tt B(p,q)} designates $B(p,q)$.\n\n\\begin{verbatim}\nBB(p,q)= ((  24*BB(p+1,q-1)*(5*p+3)*(q-2)*(q-3) \n + 2*BB(p+1,q-2)*(q-3)*(3+p*(20*p +21*q-28)-3*(q-5)*(2*q-5))\n + 24*BB(p+2,q-2)*(p+1)*(q-3)*(11*p+14) \n + 4*BB(p+2,q-3)*(p+1)*(10*p*(3*p+2*q+2) -15*q^2+121*q-175)\n + BB(p+2,q-4)*(p+1)*(2*p*(5*p-4*q+26) -3*(7*q-20)*(q-4))\n + 24*BB(p+3,q-3)*(p+2)*(p+1)*(15*p+31) \n - 2*BB(p+3,q-4)*(p+2)*(p+1)*(73*p+ 66*q - 85) \n - 4*BB(p+3,q-5)*(p+2)*(p+1)*(10*p + 10*q - 23)\n - 6*BB(p+3,q-6)*(p+2)*(p+1)*(p + q - 3) \n + (-192*BB(p+4,q-4)+20*BB(p+4,q-5) +3*BB(p+4,q-6))*(p+3)*(p+2)*(p+1))\/12 \n+muB^2*( - 24*BB(p+1,q)*(q-1)*(q-2)*(q-3) \n + 6*BB(p+1,q-1)*(-7*p+4*q-13)*(q-2)*(q-3)\n - 120*BB(p+2,q-1)*(p+1)*(q-2)*(q-3)\n + 2*BB(p+2,q-2)*(p+1)*(-80*p+39*q-204)*(q-3) \n + BB(p+2,q-3)*(p+1)*(2*p*(-5*p+8*q-38)+3*(21*q^2 -131*q+203))\n + 264*BB(p+3,q-2)*(p+2)*(p+1)*(-q+3)\n + 2*BB(p+3,q-3)*(p+2)*(p+1)*( - 47*p + 92*q - 383)\n + 2*BB(p+3,q-4)*(p+2)*(p+1)*(30*p + 64*q - 183)\n + 6*BB(p+3,q-5)*(p+2)*(p+1)*(3*p + 4*(q-3))\n + (- 168*BB(p+4,q-3) + 106*BB(p+4,q-4) + 31*BB(p+4,q-5) \n     + 6*BB(p+4,q-6))*(p+3)*(p+2)*(p+1))\/12 \n+muB^4*( - 12*BB(p+1,q)*(q-1)*(q-2)*(q-3) \n - 18*BB(p+2,q-1)*(p+1)*(q-2)*(q-3) \n + BB(p+2,q-2)*(p+1)*(- 8*p - 63*q + 165)*(q-3) \n - 12*BB(p+3,q-2)*(p+2)*(p+1)*(q-3) \n + 2*BB(p+3,q-3)*(p+2)*(p+1)*( - 10*p - 68*q + 209) \n - 18*BB(p+3,q-4)*(p+2)*(p+1)*(p + 2*(q-3)) \n - (6*BB(p+4,q-3)+ 61*BB(p+4,q-4)\n + 18*BB(p+4,q-5))*(p+3)*(p+2)*(p+1))\/12 \n+muB^6*( 7*BB(p+2,q-1)*(p+1)*(q-2)*(q-3) \n + 16*BB(p+3,q-2)*(p+2)*(p+1)*(q-3) \n + 2*BB(p+3,q-3)*(p+2)*(p+1)*(p + 4*(q-3)) \n + (9*BB(p+4,q-3)+6*BB(p+4,q-4))*(p+3)*(p+2)*(p+1))\/4 \n+ muB^8*( - BB(p+3,q-2)*(p+2)*(p+1)*(q-3) \n - BB(p+4,q-3)*(p+3)*(p+2)*(p+1))\/2)\n\/(-2)\/(q-1)\/(q-2)\/(q-3);\n\\end{verbatim}\n\n{\\Large \\bf Appendix 5.}\n\n\\begin{verbatim}\nJ(0,p) = ( 6144*J(3,3+p)*( - 4428972*p^9 - 96411618*p^8\n - 738214452*p^7 - 2743289766*p^6 - 5415151530*p^5\n - 5585181043*p^4 - 2539249652*p^3 - 67504115*p^2 + 233419148*p + 30299280)\n + 256*J(3,2+p)*(139751460*p^9 + 3070483974*p^8\n + 23818829328*p^7 + 89626132062*p^6 + 178818682872*p^5\n + 185987859019*p^4 + 85026655094*p^3 + 2165966651*p^2\n - 7934451500*p - 1033526160)\n + 32*J(3,1+p)*( - 53064072*p^9 - 1189331640*p^8\n - 9510540174*p^7 - 36782676126*p^6 - 74957974878*p^5\n - 78555090042*p^4 - 34288910369*p^3 + 1761870029*p^2\n + 4642726412*p + 613660240)\n + 72*J(3,p)*( - 682164*p^9 - 21229552*p^8 - 194740397*p^7\n - 812923675*p^6 - 1742215447*p^5 - 1924011565*p^4\n - 966215776*p^3 - 94303632*p^2 + 58837408*p + 8169600)\n + 18*J(3,-1+p)*( - 69084*p^9 + 722493*p^8 + 15902908*p^7\n + 89008228*p^6 + 226205991*p^5 + 276568093*p^4\n + 134439291*p^3 - 7926512*p^2 - 20882848*p - 2804160)\n + 3072*J(2,4+p)*(4428972*p^10 + 122985450*p^9\n + 1316684160*p^8 + 7172576478*p^7 + 21874890126*p^6\n + 38076090223*p^5 + 36050335910*p^4 + 15303002027*p^3\n + 171605542*p^2 - 1430814168*p - 181795680)\n + 128*J(2,3+p)*( - 113259276*p^10 - 3205411506*p^9\n - 35040435228*p^8 - 194508285894*p^7 - 602752373652*p^6\n - 1062870299215*p^5 - 1016468592718*p^4 - 434436290355*p^3\n - 4442421788*p^2 + 41237682352*p + 5258528640)\n + 32*J(2,2+p)*(130442076*p^10 + 3413592018*p^9\n + 34321240080*p^8 + 177812525358*p^7 + 522305372877*p^6\n + 884026310037*p^5 + 817641485963*p^4 + 335486857709*p^3\n - 4288900710*p^2 - 34908660968*p - 4428920480)\n + 8*J(2,1+p)*(93154860*p^10 + 2376526194*p^9\n + 22985820378*p^8 + 114170267016*p^7 + 322016440995*p^6\n + 526074477498*p^5 + 474729514373*p^4 + 196318386058*p^3\n + 4895262676*p^2 - 16149703888*p - 2017957440)\n + 36*J(2,p)*(2556162*p^10 + 61577829*p^9\n + 565688878*p^8 + 2669754107*p^7 + 7157734171*p^6\n + 11142549507*p^5 + 9630402307*p^4 + 3844859343*p^3\n + 101897232*p^2 - 294481376*p - 35482560)\n + 18*J(2,-1+p)*( - 115758*p^10 - 3068601*p^9\n - 28796323*p^8 - 133726220*p^7 - 342937513*p^6\n - 494864596*p^5 - 374444902*p^4 - 102361975*p^3\n + 32325936*p^2 + 22856992*p + 2804160)\n + 192*J(1,4+p)*(8878356*p^10 + 259596072*p^9\n + 2924912646*p^8 + 16568071920*p^7 + 51995667138*p^6\n + 92411299412*p^5 + 88851864277*p^4 + 38204834002*p^3\n + 531726629*p^2 - 3565245012*p - 454489200)\n + 16*J(1,3+p)*( - 9079560*p^11 - 430232670*p^10\n - 7372801566*p^9 - 61656609141*p^8 - 287230567893*p^7\n - 786492878195*p^6 - 1266898022654*p^5 - 1131366942039*p^4\n - 454714067077*p^3 + 2641691135*p^2 + 44246230980*p + 5521098480)\n + 4*J(1,2+p)*(59152572*p^11 + 1593555408*p^10\n + 16912911906*p^9 + 95861230578*p^8 + 324080159613*p^7\n + 679347181198*p^6 + 874596683962*p^5 + 642957167296*p^4\n + 207762312811*p^3 - 18032380112*p^2 - 24531833312*p - 2885392320)\n + 3*J(1,1+p)*( - 1971756*p^11 - 49219953*p^10\n - 469272444*p^9 - 2325010713*p^8 - 6687694463*p^7\n - 11583569599*p^6 - 11836532108*p^5 - 6263432419*p^4\n - 618308369*p^3 + 977759600*p^2 + 412312224*p + 42062400)\n + 36*J(1,p)*p*(1593*p^10 + 20446*p^9 + 98415*p^8\n + 200300*p^7 + 49659*p^6 - 470802*p^5 - 694935*p^4\n - 123400*p^3 + 456948*p^2 + 373456*p + 88320)\n + 54*J(1,-1+p)*p^2*(378*p^9 + 4783*p^8 + 23383*p^7\n + 54276*p^6 + 53436*p^5 - 8397*p^4 - 62973*p^3\n - 49606*p^2 - 14224*p - 1056)\n + 559872*J(0,5+p)*p*( - 7*p^10 - 151*p^9 - 1386*p^8\n - 7002*p^7 - 20859*p^6 - 35475*p^5 - 26276*p^4\n + 13252*p^3 + 41616*p^2 + 29376*p + 6912)\n + 23328*J(0,4+p)*p*( - 16*p^11 - 145*p^10 + 657*p^9\n + 15080*p^8 + 91206*p^7 + 283311*p^6 + 476125*p^5\n + 331666*p^4 - 199140*p^3 - 545816*p^2 - 368832*p - 84096)\n + 1944*J(0,3+p)*p*(200*p^11 + 2865*p^10 + 14981*p^9\n + 25008*p^8 - 73738*p^7 - 437223*p^6 - 855551*p^5\n - 635826*p^4 + 311548*p^3 + 911928*p^2 + 602560*p + 133248)\n + 1944*J(0,2+p)*p*( - 70*p^11 - 1015*p^10 - 5959*p^9\n - 17811*p^8 - 26156*p^7 - 6696*p^6 + 36010*p^5\n + 49732*p^4 + 13471*p^3 - 20274*p^2 - 17296*p - 3936)\n + 486*J(0,1+p)*p*(40*p^11 + 539*p^10 + 2915*p^9\n + 8077*p^8 + 11771*p^7 + 6819*p^6 - 4591*p^5\n - 11321*p^4 - 9191*p^3 - 4018*p^2 - 944*p - 96)\n )\/972\/p^5\/(p^7 + 12*p^6 + 54*p^5 + 108*p^4\n + 69*p^3 - 72*p^2 - 124*p - 48)\n +(\n 16*B(0,p+5)*(9569448*p^11 + 263172276*p^10\n + 2796333210*p^9 + 14979144042*p^8 + 45346896150*p^7\n + 81451195130*p^6 + 86698023173*p^5 + 50610254573*p^4\n + 11013851543*p^3 - 3191145997*p^2 - 1994622412*p - 245554896)\n + 4*B(0,p+4)*( - 61193772*p^11 - 1412170434*p^10\n - 12221193447*p^9 - 55077683067*p^8 - 145501643868*p^7\n - 235023215404*p^6 - 229902193451*p^5 - 124704441431*p^4\n - 24368641382*p^3 + 8307539200*p^2 + 4697326336*p + 557574080)\n + 3*B(0,p+3)*(2924316*p^11 + 54602601*p^10\n + 399544548*p^9 + 1560379173*p^8 + 3620624261*p^7\n + 5121679611*p^6 + 4177323684*p^5 + 1389082017*p^4\n - 624082689*p^3 - 825833082*p^2 - 322097816*p - 46392864)\n + 18*B(0,p+2)*( - 25866*p^11 - 271739*p^10\n - 1238954*p^9 - 3192049*p^8 - 4973923*p^7\n - 4312062*p^6 - 536457*p^5 + 3801002*p^4 + 5378048*p^3\n + 3758272*p^2 + 1396000*p + 217728)\n)\/486\/(p+2)^2\/(p+1)^4\/p^5\/(p-1);\n\nJ(3, - 6 + q) = (  8*B(0,q-1)*(2*q^3 - 7*q^2 - 3*q + 18)\n + 2*B(0,q-2)*( - 13*q^3 + 109*q^2 - 298*q + 264)\n + 1536*J(3,q-3)*( - q + 3)\n + 64*J(3,q-4)*(31*q - 87)\n + 8*J(3,q-5)*( - 6*q + 13)\n + 768*J(2,q-2)*q*(q - 3)\n + 32*J(2,q-3)*( - 25*q^2 + 61*q + 30)\n + 8*J(2,q-4)*(27*q^2 - 127*q + 148)\n + 4*J(2,q-5)*(12*q^2 - 65*q + 84)\n + 6*J(2,q-6)*(q^2 - 6*q + 8)\n + 48*J(1,q-2)*(2*q^2 - 3*q - 9)\n + 4*J(1,q-3)*( - 2*q^3 - 22*q^2 + 150*q - 201)\n + J(1,q-4)*(13*q^3 - 130*q^2 + 418*q - 420)\n)\/6\/(q - 2);\n\nJ(2, - 6 + q) = ( 16*B(0,q-1)*( - 2034*q^4 + 13237*q^3\n    - 18362*q^2 - 27483*q + 55062)\n + 4*B(0,q-2)*(13197*q^4 - 150272*q^3 + 634577*q^2 - 1175522*q + 803760)\n + 12*B(0,q-3)*(q^4 - 46*q^2 + 165*q - 162)\n + 9*B(0,q-4)*( - 7*q^4 + 95*q^3 - 470*q^2 + 1000*q - 768)\n + 3072*J(3,q-3)*(1017*q^2 - 6110*q + 9177)\n + 128*J(3,q-4)*( - 31698*q^2 + 184307*q - 267447)\n + 16*J(3,q-5)*(8463*q^2 - 44580*q + 56333)\n + 1536*J(2,q-2)*q*( - 1017*q^2 + 6110*q - 9177)\n + 64*J(2,q-3)*(25677*q^3 - 140330*q^2 + 160103*q + 88998)\n + 16*J(2,q-4)*( - 28941*q^3 + 222222*q^2 - 561953*q + 468596)\n + 16*J(2,q-5)*( - 5571*q^3 + 47436*q^2 - 132356*q + 120561)\n + 96*J(1,q-2)*( - 54*q^4 - 1494*q^3 + 7171*q^2 + 3216*q - 29475)\n + 8*J(1,q-3)*(2709*q^4 + 9326*q^3 - 193019*q^2 + 616413*q - 583827)\n + 2*J(1,q-4)*( - 14142*q^4 + 181553*q^3 - 859787*q^2 + 1768658*q - 1324320)\n + 24*J(1,q-5)*(11*q^4 - 141*q^3 + 664*q^2 - 1356*q+1008)\n + 18*J(1,q-6)*(q^4 - 14*q^3 + 71*q^2 - 154*q + 120)\n)\/12294\/(q-2)\/(q-3)\/(q-4);\n\nJ(1, - 6 + q) = (  8392704*J(3,q-1)*(738162*q^7 - 13826958*q^6 + 107438526*q^5\n  - 446532922*q^4 + 1064669687*q^3 - 1443484631*q^2\n  + 1020689473*q - 289691337)\n + 349696*J(3,q-2)*( - 23305518*q^7 + 431941470*q^6\n    - 3310148214*q^5 + 13505408608*q^4 - 31391328545*q^3\n    + 41045263193*q^2 - 27524008819*q + 7242283425)\n + 64*J(3,q-3)*(5997560976*q^7 - 107967926286*q^6\n    + 794611952772*q^5 - 3056894462088*q^4\n    + 6480621271814*q^3 - 7204980963707*q^2\n    + 3391870943690*q - 197859183171)\n + 288*J(3,q-4)*q*(71372070*q^6 - 1276658510*q^5\n    + 9289274641*q^4 - 35007711040*q^3 + 71462127680*q^2\n    - 73822280850*q + 29283876009)\n + 36*J(3,q-5)*q*( - 24179544*q^6 + 434455572*q^5\n    - 3159380738*q^4 + 11827334899*q^3 - 23816366035*q^2\n    + 24112882577*q - 9374746731)\n + 4196352*J(2,q)*(-738162*q^8 + 12350634*q^7 - 79784610*q^6 + 231655870*q^5\n   - 171603843*q^4 - 685854743*q^3 + 1866279789*q^2\n   -1751687609*q + 579382674)\n + 174848*J(2,q-1)*(18876546*q^8 - 305743830*q^7\n    + 1860247722*q^6 - 4615222660*q^5 - 558487079*q^4\n    + 27753054687*q^3 - 58619476249*q^2 + 49578489187*q - 15063949524)\n + 64*J(2,q-2)*( - 14810167224*q^8 + 270992616750*q^7 - 2041549790688*q^6\n  + 8136869079315*q^5 - 18296821447304*q^4 + 22800932361744*q^3\n  - 14321555406258*q^2 + 3748781983207*q - 395718366342)\n + 16*J(2,q-3)*( - 10824070743*q^8 + 207455511240*q^7 - 1660383188328*q^6\n  + 7176976971729*q^5 - 18064650946958*q^4 + 26521521351577*q^3\n  - 21365258254495*q^2 + 7986599358662*q - 791436732684)\n + 36*J(2,q-4)*q*( - 541141251*q^7 + 10676099500*q^6\n    - 88342419106*q^5 + 396420066920*q^4 - 1038067530681*q^3\n    + 1577944291122*q^2 - 1278897041426*q + 420807674922)\n + 36*J(2,q-5)*q*(15986898*q^7 - 324636501*q^6\n    + 2767299563*q^5-12801558868*q^4 + 34573795993*q^3\n    - 54197750750*q^2 + 45244769538*q - 15277905873)\n + 262272*J(1,q)*( - 1476324*q^8 + 22486782*q^7\n    - 118088346*q^6+140996162*q^5 + 996391080*q^4\n    - 4565718547*q^3 + 8063013471*q^2 - 6565443637*q + 2027839359)\n + 21856*J(1,q-1)*(1479240*q^9 - 718824*q^8 - 290505888*q^7\n    + 3040697422*q^6 - 14276034470*q^5 + 36530607805*q^4\n    - 52418121878*q^3 + 40019451544*q^2 - 13989602036*q + 1448456685)\n + 8*J(1,q-2)*q*( - 6672113604*q^8 + 143144560506*q^7 - 1317848051532*q^6\n    + 6794557026194*q^5 - 21434683657627*q^4 + 42303073328518*q^3\n    - 50886066609614*q^2 + 33974478049038*q - 9569982531879)\n + 6*J(1,q-3)*q*(221313528*q^8 - 5165620026*q^7  + 51969200474*q^6\n    - 293881218901*q^5 + 1019347369493*q^4 - 2213379274564*q^3\n    + 2923819741196*q^2 - 2132296331629*q + 649364820429)\n + 27*J(1,q-4)*q*( - 91046*q^8 + 2191855*q^7 - 22772339*q^6\n    + 133314169*q^5 - 480172079*q^4 + 1085700520*q^3\n    -1495620096*q^2 + 1136307096*q-358858080)\n + 324*J(1,q-5)*q*( - 3304*q^8 + 86533*q^7 - 972784*q^6\n    + 6115624*q^5 - 23438146*q^4 + 55819747*q^3\n    - 80159286*q^2 + 62865696*q - 20314080)\n + 63732096*J(0,q)*q*( - q^9 + 23*q^8 - 228*q^7\n    + 1278*q^6 - 4461*q^5 + 10047*q^4 - 14582*q^3\n    + 13132*q^2 - 6648*q + 1440)\n + 2655504*J(0,q-1)*q*(25*q^9 - 625*q^8 + 6812*q^7\n    - 42490*q^6 + 167117*q^5 - 429505*q^4 + 720358*q^3\n    - 758580*q^2 + 453528*q - 116640)\n + 663876*J(0,q-2)*q*( - 35*q^9 + 945*q^8 - 11182*q^7\n    + 76062*q^6 - 327403*q^5  + 923145*q^4 - 1700348*q^3 + 1964808*q^2\n    - 1284552*q + 358560)\n + 663876*J(0,q-3)*q*(5*q^9 - 145*q^8 + 1845*q^7 - 13500*q^6\n    + 62476*q^5 - 189120*q^4 + 373040*q^3 - 459875*q^2 + 318994*q - 93720)\n + 165969*J(0,q-4)*q*( - q^9 + 31*q^8 - 421*q^7 + 3281*q^6 - 16130*q^5\n    + 51704*q^4 - 107584*q^3 + 139264*q^2 - 100864*q + 30720)\n)\/25515\/q\/(q-1)\/(q-2)\/(q-3)\/(q-4)\/(q-4)\/(q-5)\/(q-5)\/(7*q-24)\n +(87424*B(0,q+1)*( - 1476324*q^6 + 16581486*q^5 - 47333430*q^4\n    - 98082016*q^3 + 746063306*q^2 - 1287219275*q + 675946453)\n + 43712*B(0,q)*(4862637*q^6 - 78129165*q^5 + 502175106*q^4\n    - 1641458936*q^3 + 2853167747*q^2 - 2474965235*q + 830365446)\n + 24*B(0,q-1)*( - 187645068*q^6 + 3303988746*q^5 - 23822963042*q^4\n    + 90075021407*q^3 - 188498675784*q^2 + 207202955323*q - 93562325614)\n + 108*B(0,q-2)*( - 2349349*q^6 + 47038978*q^5 - 386433111*q^4\n    + 1668058574*q^3 - 3990763652*q^2 + 5015869656*q - 2585112480)\n + 324*B(0,q-3)*(125620*q^6 - 2697750*q^5 + 23873044*q^4\n    - 111347697*q^3 + 288317245*q^2 - 392099154*q + 218030520)\n + 1701*B(0,q-4)*( - 631*q^6 + 14513*q^5 - 137798*q^4\n    + 690128*q^3 - 1917728*q^2 + 2792192*q - 1653760)\n)\/51030\/(q-2)\/(q-4)\/(q-4)\/(q-5)\/(q-5)\/(7*q-24);\n\\end{verbatim}\n\n\n\n{\\Large \\bf Appendix 6.}\n\\begin{verbatim}\nJ(p,1)= -( J(p+1,0)*(123*p^3 + 325*p^2 + 141*p + 8)\n\/(3*(8*p^3 + 12*p^2 + 6*p + 1))+\nJ(p+2,0)*( - 1610*p^4 - 7315*p^3 - 12784*p^2 - 10287*p - 3208)\n\/(3*(16*p^4 + 48*p^3 + 48*p^2 + 20*p + 3))+\nJ(p+3,0)*(2*( - 10100*p^5 - 96976*p^4 - 370679*p^3 - 704416*p^2 - 663875*p\n - 247102))\/(3*(32*p^5 + 176*p^4 + 336*p^3 + 280*p^2 + 106*p + 15))+\nJ(p+4,0)*(64*(400*p^5 + 4432*p^4 + 19118*p^3 + 40113*p^2\n + 41003*p + 16386))\/(32*p^5 + 176*p^4 + 336*p^3 + 280*p^2 + 106*p + 15)+\nJ(p+5,0)*(6144*(2*p^4 + 21*p^3 + 79*p^2 + 126*p + 72))\n\/(16*p^4 + 64*p^3 + 72*p^2 + 32*p + 5)+\nJ(p+1,-1)*(16*p^3 + 54*p^2 - 8*p - 65)\/(4*(8*p^3 + 12*p^2 + 6*p + 1))+\nJ(p+2,-1)*( - 856*p^4 - 4474*p^3 - 7443*p^2 - 4839*p\n - 1014)\/(6*(16*p^4 + 48*p^3 + 48*p^2 + 20*p + 3))+\nJ(p+3,-1)*(2*(1508*p^5 + 9252*p^4 + 19927*p^3 + 16451*\np^2 + 2040*p - 2084))\/(3*(32*p^5 + 176*p^4 + 336*p^3\n + 280*p^2 + 106*p + 15))+\nJ(p+4,-1)*(8*(5908*p^5 + 62136*p^4 + 256375*p^3 + \n518814*p^2 + 515197*p + 201018))\/(3*(32*p^5\n + 176*p^4 + 336*p^3 + 280*p^2 + 106*p + 15))+\nJ(p+5,-1)*(256*( - 2*p^4 - 21*p^3 - 79*p^2 - 126*p - \n72))\/(16*p^4 + 64*p^3 + 72*p^2 + 32*p + 5)+\nJ(p+2,-2)*( - 61*p^3 - 156*p^2 - 56*p + 39)\n\/(3*(8*p^3 + 12*p^2 + 6*p + 1))+\nJ(p+3,-2)*(1794*p^4 + 8239*p^3 + 13400*p^2 + 9437*p + \n2482)\/(6*(16*p^4 + 48*p^3 + 48*p^2 + 20*p + 3))+\nJ(p+4,-2)*(13756*p^5 + 133560*p^4 + 508645*p^3 +\n953406*p^2 + 881659*p + 322086)\/(3*(32*p^5 +\n 176*p^4 + 336*p^3 + 280*p^2 + 106*p + 15))+\nJ(p+5,-2)*(608*( - 2*p^4 - 21*p^3 - 79*p^2 - 126*p -\n72))\/(16*p^4 + 64*p^3 + 72*p^2 + 32*p + 5)+\nJ(p+2,-3)*( - 8*p^3 - 13*p^2 + 8*p + 13)\/(2*(8*p^3 + 12*p^2 + 6*p + 1))+\nJ(p+3,-3)*(3*(64*p^4 + 254*p^3 + 327*p^2 + 163*p + 26)\n)\/(4*(16*p^4 + 48*p^3 + 48*p^2 + 20*p + 3))+\nJ(p+4,-3)*(724*p^5 + 6504*p^4 + 22567*p^3 + 37962*p^2\n + 31033*p + 9858)\/(32*p^5 + 176*p^4 + 336*p^3 + 280*p^2 + 106*p + 15)+\nJ(p+5,-3)*(96*( - 2*p^4 - 21*p^3 - 79*p^2 - 126*p - 72\n))\/(16*p^4 + 64*p^3 + 72*p^2 + 32*p + 5)+\nJ(p+1,1)*(4*(10*p^3 + 80*p^2 + 129*p + 61))\/(8*p^3 + 12*p^2 + 6*p + 1)+\nJ(p+2,1)*(16*( - 540*p^4 - 2780*p^3 - 5285*p^2 - \n4411*p - 1384))\/(3*(16*p^4 + 48*p^3 + 48*p^2 + 20*p + 3))+\nJ(p+3,1)*(128*( - 320*p^5 - 3220*p^4 - 12772*p^3 - \n24943*p^2 - 23993*p - 9110))\/(32*p^5 +\n 176*p^4 + 336*p^3 + 280*p^2 + 106*p + 15)+\nJ(p+4,1)*(6144*( - 4*p^4 - 40*p^3 - 145*p^2 - 225*p\n- 126))\/(16*p^4 + 64*p^3 + 72*p^2 + 32*p + 5)+\nJ(p,2)*(4*( - 10*p^2 - 14*p - 5))\/(8*p^3 + 12*p^2 + 6*p + 1)+\nJ(p+1,2)*(16*( - 260*p^3 - 1052*p^2 - 1423*p - 649))\/\n(3*(16*p^4 + 48*p^3 + 48*p^2 + 20*p + 3))+\nJ(p+2,2)*(128*( - 40*p^3 - 238*p^2 - 457*p - 282))\/(\n16*p^4 + 64*p^3 + 72*p^2 + 32*p + 5)+\nB(p+1,0)*( - 100*p^4 + 579*p^3 + 2723*p^2 + 2859*p + \n677)\/(6*(8*p^5 + 36*p^4 + 58*p^3 + 43*p^2 + 15*p + 2))+\nB(p+2,0)*(2920*p^5 + 14282*p^4 + 21517*p^3 + 7182*p^2\n - 8141*p - 5168)\/(6*(16*p^6 + 128*p^5 + 384*\np^4 + 548*p^3 + 391*p^2 + 135*p + 18))+\nB(p+3,0)*(4*(960*p^6 - 5788*p^5 - 128500*p^4 - 629377*\np^3 - 1383907*p^2 - 1447282*p - 585016))\n\/(3*(32*p^7 + 400*p^6 + 1952*p^5 + 4744*p^4\n + 6098*p^3 + 4117*p^2 + 1377*p + 180))+\nB(p+4,0)*(3968*(2*p^3 + 13*p^2 + 27*p + 18))\/(16*p^5 +\n 128*p^4 + 328*p^3 + 320*p^2 + 133*p + 20)+\nB(p+1,-1)*(44*p^3 + 244*p^2 + 440*p + 259)\/(4*(8*p^5 +\n 36*p^4 + 58*p^3 + 43*p^2 + 15*p + 2))+\nB(p+2,-1)*(1600*p^5 + 8748*p^4 + 7904*p^3 - 25623*p^2\n - 49512*p - 23133)\/(12*(16*p^6 + 128*p^5 + \n384*p^4 + 548*p^3 + 391*p^2 + 135*p + 18))+\nB(p+3,-1)*( - 400*p^6 - 1728*p^5 - 6008*p^4 - 71868*p^3\n - 306657*p^2 - 490625*p - 262964)\/(3*(32*p^7\n + 400*p^6 + 1952*p^5 + 4744*p^4 + 6098*p^3 + 4117*p^2 + 1377*p + 180))+\nB(p+4,-1)*(5920*(2*p^3 + 13*p^2 + 27*p + 18))\/(16*p^5\n+ 128*p^4 + 328*p^3 + 320*p^2 + 133*p + 20)+\nB(p+2,-2)*(60*p^3 - 163*p^2 - 777*p - 554)\n\/(6*(8*p^4 + 28*p^3 + 30*p^2 + 13*p + 2))+\nB(p+3,-2)*( - 120*p^4 + 6946*p^3 + 27377*p^2 + 31658*p\n + 11347)\/(12*(16*p^5 + 96*p^4 + 192*p^3 + 164*p^2 + 63*p + 9))+\nB(p+4,-2)*(2816*(2*p^3 + 13*p^2 + 27*p + 18))\/(16*p^5 \n+ 128*p^4 + 328*p^3 + 320*p^2 + 133*p + 20)+\nB(p+2,-3)*(3*( - 8*p^2 - 21*p - 13))\n\/(2*(8*p^4 + 28*p^3 + 30*p^2 + 13*p + 2))+\nB(p+3,-3)*(6*(22*p^3 + 83*p^2 + 95*p + 34))\n\/(16*p^5 + 96*p^4 + 192*p^3 + 164*p^2 + 63*p + 9)+\nB(p+4,-3)*(576*(2*p^3 + 13*p^2 + 27*p + 18))\n\/(16*p^5 + 128*p^4 + 328*p^3 + 320*p^2 + 133*p + 20)+\nB(p,1)*(30*p^3 + 90*p^2 + 85*p + 29)\n\/(4*p*(4*p^4 + 16*p^3 + 21*p^2 + 11*p + 2))+\nB(p+1,1)*( - 2400*p^6 - 17780*p^5 - 43716*p^4 -\n32887*p^3 + 23271*p^2 + 43533*p + 15615)\n\/(6*(16*p^7 + 144*p^6 + 512*p^5 + 932*p^4 + 939*p^3\n + 526*p^2 + 153*p + 18))+\nB(p+2,1)*(2*( - 3600*p^7 - 67792*p^6 - 501064*p^5 - \n1910808*p^4 - 4105397*p^3 - 5021161*p^2\n - 3276852*p - 894496))\/(3*(32*p^8 + 464*p^7\n + 2752*p^6 + 8648*p^5 + 15586*p^4 + 16313*p^3\n + 9611*p^2 + 2934*p + 360))+\nB(p+3,1)*(64*( - 1000*p^5 - 10760*p^4 - 45112*p^3 - \n92251*p^2 - 92195*p - 36122))\/(32*p^7 + 400*p^6 + 1952*p^5 + 4744*p^4\n + 6098*p^3 + 4117*p^2 + 1377*p + 180)+\nB(p+4,1)*(6144*( - 2*p^3 - 13*p^2 - 27*p - 18))\n\/(16*p^5 + 128*p^4 + 328*p^3 + 320*p^2 + 133*p + 20)+\nB(p,2)*( - 440*p^4 - 1476*p^3 - 1390*p^2 - 255*p +\n 87)\/(3*p*(8*p^5 + 36*p^4 + 58*p^3 + 43*p^2+15*p+2))+\nB(p+1,2)*(4*( - 160*p^5 - 2736*p^4 - 12072*p^3 - \n21842*p^2 - 17255*p - 4765))\/(3*(16*p^7 \n+ 144*p^6 + 512*p^5 + 932*p^4 + 939*p^3 + 526*p^2 + 153*p + 18))+\nB(p+2,2)*(128*(40*p^5 + 404*p^4 + 1558*p^3\n + 2847*p^2 + 2426*p + 750))\/(16*p^7 + 208*p^6\n + 1064*p^5 + 2728*p^4 + 3701*p^3 + 2605*p^2 + 898*p + 120)+\nB(p-1,3)*( - 16*p - 29)\/(p*(4*p^3 + 12*p^2 + 9*p + 2))+\nB(p,3)*(4*(88*p^3 + 452*p^2 + 762*p + 447))\n\/(16*p^6 + 128*p^5 + 384*p^4 + 548*p^3 + 391*p^2 + 135*p + 18) +\nB(p+1,3)*(768*(4*p^5 + 42*p^4 + 167*p^3 + 308*p^2 +\n 251*p + 63))\/(16*p^8 + 224*p^7 + 1272*p^6\n + 3792*p^5 + 6429*p^4 + 6306*p^3 + 3503*p^2 + 1018*p + 120)           );\n\nJ(p,2)= - (J(p+1,0)*(5*p - 1)\/8 + \nJ(p+2,0)*(302*p^3 + 591*p^2 - 69*p - 416)\n \/(24*(2*p^2 + 5*p + 3))+ \nJ(p+3,0)*( - 1178*p^3 - 6705*p^2 - 12285*p - 7174)\n \/(12*(2*p^2 + 5*p + 3))+ \nJ(p+4,0)*(8*( - 10*p^3 - 61*p^2 - 115*p - 66))\/(2*p^2 + 5*p + 3)+ \nJ(p+2,-1)*( - 20*p^3 - 50*p^2 - 84*p - 59)\/(8*(2*p^2 + 5*p + 3))+ \nJ(p+3,-1)*( - 590*p^3 - 2719*p^2 - 4057*p - 1958)\n \/(12*(2*p^2 + 5*p + 3))+ \nJ(p+4,-1)*(10*p^3 + 61*p^2 + 115*p + 66)\/(3*(2*p^2 + 5*p + 3))+ \nJ(p+2,-2)*( - 3*p)\/8+ \nJ(p+3,-2)*( - 742*p^3 - 2795*p^2 - 3167*p - 1090)\n \/(48*(2*p^2 + 5*p + 3))+ \nJ(p+4,-2)*(19*(10*p^3 + 61*p^2 + 115*p + 66))\/(24*(2*p^2 + 5*p + 3))+ \nJ(p+3,-3)*(p*( - 10*p^2 - 31*p - 22))\/(4*(2*p^2 + 5*p + 3))+ \nJ(p+4,-3)*(10*p^3 + 61*p^2 + 115*p + 66)\/(8*(2*p^2 + 5*p + 3))+ \nJ(p+1,1)*(20*p^2 + 42*p + 23)\/(2*(p + 1))+ \nJ(p+2,1)*(8*(60*p^3 + 299*p^2 + 495*p + 271))\n \/(3*(2*p^2 + 5*p + 3))+ \nJ(p+3,1)*(16*(10*p^3 + 59*p^2 + 110*p + 64))\/(2*p^2 + 5*p + 3)+ \nJ(p+1,2)*(2*( - 4*p^2 - 12*p - 11))\/(3*(2*p^2 + 5*p + 3))+ \nJ(p+2,2)*(48*( - p - 2))\/(p + 1)+ \nB(p+1,0)*(5*p^2 + 20*p + 9)\/(8*(p^2 + 3*p + 2))+ \nB(p+2,0)*( - 438*p^4 - 1663*p^3 - 1857*p^2 - 1002*p - 542)\n \/(24*(2*p^4 + 15*p^3 + 40*p^2 + 45*p + 18))+ \nB(p+3,0)*( - 144*p^3 - 1246*p^2 - 2905*p - 1978)\n \/(6*(2*p^3 + 11*p^2 + 18*p + 9))+ \nB(p+2,-1)*( - 40*p^4 - 300*p^3 - 708*p^2 - 528*p - 53)\n \/(8*(2*p^4 + 15*p^3 + 40*p^2 + 45*p + 18))+ \nB(p+3,-1)*(5*(12*p^3 - 292*p^2 - 985*p - 706))\n \/(24*(2*p^3 + 11*p^2 + 18*p + 9))+ \nB(p+2,-2)*(3*( - p^2 - 4*p - 2))\/(8*(p^2 + 3*p + 2))+ \nB(p+3,-2)*(18*p^3 - 1643*p^2 - 5213*p - 3710)\n \/(48*(2*p^3 + 11*p^2 + 18*p + 9))+ \nB(p+3,-3)*(3*( - 10*p^2 - 31*p - 22))\n \/(4*(2*p^3 + 11*p^2 + 18*p + 9))+ \nB(p+1,1)*(120*p^5 + 1240*p^4 + 4674*p^3  + 8128*p^2 + 6639*p + 2082)\n \/(8*(2*p^5 + 17*p^4 + 55*p^3 + 85*p^2 + 63*p + 18))+ \nB(p+2,1)*(540*p^4 + 6980*p^3 + 26255*p^2 + 38886*p + 20048)\n \/(12*(2*p^4 + 15*p^3 + 40*p^2 + 45*p + 18))+ \nB(p+3,1)*(8*(10*p^2 + 31*p + 22))\/(2*p^3 + 11*p^2 + 18*p + 9)+ \nB(p,2)*1\/(p^2 + 3*p + 2)+ \nB(p+1,2)*( - 200*p^4 - 1168*p^3 - 2588*p^2 - 2525*p - 854)\n \/(6*(2*p^5 + 17*p^4 + 55*p^3 + 85*p^2 + 63*p + 18))+ \nB(p,3)*( - 20*p^2 - 60*p - 57)\n \/(2*(2*p^4 + 13*p^3 + 29*p^2 + 27*p + 9))        );\n\nJ(p,3)= (   12*B(p,3)*(4*p^3 + 21*p^2 + 36*p + 21)\n + 4*B(p+1, 2)*(20*p^4 + 92*p^3 + 125*p^2 + 36*p - 23)\n + 192*B(p+3,1)*( - p^4 - 6*p^3 - 13*p^2 - 12*p - 4)\n + 2*B(p+2,1)*( - 239*p^4 - 1382*p^3 - 2873*p^2 - 2556*p - 826)\n + 3*B(p+1,1)*( - 10*p^4 - 60*p^3 - 133*p^2 - 132*p - 51)\n + 18*B(p+3,-3)*(p^4 + 6*p^3 + 13*p^2 + 12*p + 4)\n + 88*B(p+3,-2)*(p^4 + 6*p^3 + 13*p^2 + 12*p + 4)\n + 185*B(p+3,-1)*(p^4 + 6*p^3 + 13*p^2 + 12*p + 4)\n + 3*B(p+2,-1)*( - 4*p^4 - 34*p^3 - 97*p^2 - 108*p - 41)\n + 124*B(p+3,0)*(p^4 + 6*p^3 + 13*p^2 + 12*p + 4)\n + 2*B(p+2,0)*( - 103*p^4 - 570*p^3 - 1123*p^2 -948*p -292)\n + 4*J(p+1,2)*( - 20*p^5 - 166*p^4 - 519*p^3 - 764*p^2 - 529*p - 138)\n + 192*J(p+3,1)*( - 2*p^6 - 23*p^5 - 107*p^4 - 257*p^3\n\t - 335*p^2 - 224*p - 60)\n + 2*J(p+2,1)*( - 138*p^6 - 1417*p^5 - 5879*p^4 - 12645*p^3\n\t - 14887*p^2 - 9098*p - 2256)\n + 3*J(p+1,1)*( - 4*p^6 - 38*p^5 - 142*p^4 - 265*p^3\n\t - 256*p^2 - 117*p - 18)\n + 3*J(p+4,-3)*( - p^6 - 12*p^5 - 58*p^4 - 144*p^3 - 193*p^2 - 132*p - 36)\n + 6*J(p+3,-3)*p*(p^5 + 9*p^4 + 31*p^3\t + 51*p^2 + 40*p + 12)\n + 19*J(p+4,-2)*( - p^6 - 12*p^5 - 58*p^4 - 144*p^3\n\t - 193*p^2 - 132*p - 36)\n + 2*J(p+3,-2)*(19*p^6 + 184*p^5 + 706*p^4 + 1372*p^3\n\t + 1423*p^2 + 748*p + 156)\n + 8*J(p+4,-1)*( - p^6 - 12*p^5 - 58*p^4 - 144*p^3\n\t - 193*p^2 - 132*p - 36)\n + J(p+3,-1)*(124*p^6 + 1303*p^5 + 5527*p^4 + 12121*p^3\n\t + 14497*p^2 + 8968*p + 2244)\n + 3*J(p+2,-1)*( - 2*p^6 - 20*p^5 - 75*p^4 - 129*p^3 - 97*p^2 - 19*p + 6)\n + 192*J(p+4,0)*(p^6 + 12*p^5 + 58*p^4 + 144*p^3\n\t + 193*p^2 + 132*p + 36)\n + 2*J(p+3,0)*(89*p^6 + 1006*p^5 + 4604*p^4 + 10894*p^3\n\t + 14015*p^2 + 9268*p + 2460)\n + 2*J(p+2,0)*( - 37*p^6 - 341*p^5 - 1243*p^4 - 2303*p^3\n \t- 2296*p^2 - 1172*p - 240)\n)\/12\/(p+1)\/(p+2)\/(p+3)\/(2*p+3);\n\\end{verbatim}\n\n{\\Large \\bf Appendix 7.}\n\nHere $B(0,q,n)$ is the coefficient of the expansion \n\\[\n{\\cal B}(0,q;\\mu_B^2)=\\sum_{n=0}^{q-2} {B(0,q,n)\\over (\\mu_B^2)^n } + D_0(0,q)(\\ln \\mu_B^2 + C),\n\\]\n(see (\\ref{DPcoeffOne}) and (\\ref{BBandJJat_pleq0}))\nand $D(p,q,r)$ is nothing but $D_{r}(p,q)$ defined in formula (\\ref{DPcoeffOne}).\n\n\\begin{verbatim}\nZ_0(q) = (( - 414720*q^2 + 5184000*q + 105460920\/(q-1) + 22917120\/(q-2)\n   + 1462140\/(q-3) - 337161300\/(q+1) + 5257440)*B(0,q - 4,0)\n + (2903040*q^2 - 30274560*q - 350758440\/(q-1) - 49733280\/(q-2)\n   - 1462140\/(q-3) + 1236495780\/(q+1) - 26287200)*B(0,q - 3,1)\n + (8294400*q^2 - 84602880*q + 426474720\/(q-1) - 12792960\/(q-2)\n   - 4267440\/(q-3) - 1713368880\/(q+1) + 420923520)*B(0,q - 3,0)\n + ( - 8709120*q^2 + 72783360*q + 388058400\/(q-1) + 26816160\/(q-2)\n   - 1679831040\/(q+1) + 52574400)*B(0,q - 2,2)\n + ( - 50595840*q^2 + 413061120*q - 681805440\/(q-1) + 32651520\/(q-2)\n   + 4409120640\/(q+1) - 1598987520)*B(0,q - 2,1)\n + ( - 58060800*q^2 + 443750400*q - 1814227200\/(q-1) - 29468160\/(q-2)\n   + 8956742400\/(q+1) - 2136844800)*B(0,q - 2,0)\n + (14515200*q^2 - 91238400*q - 142760880\/(q-1) + 1003590000\/(q+1)\n   - 52574400)*B(0,q - 1,3)\n + (128563200*q^2 - 787968000*q + 234077760\/(q-1) - 4178364480\/(q+1)\n   + 2271421440)*B(0,q - 1,2)\n + (306892800*q^2 - 1758412800*q + 1767352320\/(q-1) - 17987028480\/(q+1)\n   + 6567989760)*B(0,q - 1,1)\n + (165888000*q^2 - 729907200*q + 388177920\/(q-1) - 2986813440\/(q+1)\n   + 1305262080)*B(0,q - 1,0)\n + (414720*q^2 + 829440*q)*B(0,q + 3,7)\n + (9123840*q^2 + 18247680*q)*B(0,q + 3,6)\n + (74649600*q^2 + 149299200*q)*B(0,q + 3,5)\n + (282009600*q^2 + 564019200*q)*B(0,q + 3,4)\n + (491028480*q^2 + 982056960*q)*B(0,q + 3,3)\n + (318504960*q^2 + 637009920*q)*B(0,q + 3,2)\n + ( - 2903040*q^2 + 207360*q)*B(0,q + 2,6)\n + ( - 53913600*q^2 + 1658880*q)*B(0,q + 2,5)\n + ( - 356659200*q^2 - 20736000*q)*B(0,q + 2,4)\n + ( - 1011916800*q^2 - 265420800*q)*B(0,q + 2,3)\n + ( - 1141309440*q^2 - 915701760*q)*B(0,q + 2,2)\n + ( - 318504960*q^2 - 955514880*q)*B(0,q + 2,1)\n + (8709120*q^2 - 18662400*q + 5257440\/(q+1) - 5257440)*B(0,q + 1,5)\n + (132710400*q^2 - 273715200*q - 336216960\/(q+1) + 336216960)*B(0,q + 1,4)\n + (680140800*q^2 - 1277337600*q - 2232368640\/(q+1) + 2232368640)*B(0,q + 1,3)\n + (1343692800*q^2 - 1891123200*q - 2815395840\/(q+1)\n   + 2815395840)*B(0,q + 1,2)\n + (809533440*q^2 + 212336640*q + 3530096640\/(q+1) - 3530096640)*B(0,q + 1,1)\n + (955514880*q + 4140564480\/(q+1) - 4140564480)*B(0,q + 1,0)\n + ( - 14515200*q^2 + 61171200*q - 228350880\/(q+1) + 26287200)*B(0,q,4)\n + ( - 174182400*q^2 + 713318400*q + 1824491520\/(q+1) - 1429574400)*B(0,q,3)\n + ( - 646963200*q^2 + 2463436800*q + 11243888640\/(q+1) - 6663513600)*B(0,q,2)\n + ( - 779673600*q^2 + 2322432000*q + 6705192960\/(q+1) - 4209131520)*B(0,q,1)\n + ( - 159252480*q^2 - 278691840*q - 11515944960\/(q+1) + 5381406720)*B(0,q,0)\n + (3034260\/(q-1) + 706560\/(q-2) + 32940\/(q-3) + 604860\/(q-5)\n   - 3963900\/(q+1))*D(3,q - 8,1)\n + ( - 22299660\/(q-1) - 4327200\/(q-2) - 164700\/(q-3) - 2903328\/(q-4)\n   - 1814580\/(q-5) + 28191708\/(q+1))*D(3,q - 7,2)\n + (2175210\/(q-1) + 974320\/(q-2) + 1169730\/(q-3) - 1752336\/(q-4)\n   + 3125110\/(q-5) - 12880514\/(q+1))*D(3,q - 7,1)\n + (70144380\/(q-1) + 11037600\/(q-2) + 5773140\/(q-3) + 8709984\/(q-4)\n   + 1814580\/(q-5) - 85867524\/(q+1))*D(3,q - 6,3)\n + ( - 36941910\/(q-1) - 11226160\/(q-2) + 1505790\/(q-3) - 14098512\/(q-4)\n   - 6149410\/(q-5) + 112114682\/(q+1))*D(3,q - 6,2)\n + ( - 59868660\/(q-1) + 3139840\/(q-2) + 3407220\/(q-3) - 11231232\/(q-4)\n   + 10685860\/(q-5) + 86076892\/(q+1))*D(3,q - 6,1)\n + ( - 122426100\/(q-1) - 19848480\/(q-2) - 16660620\/(q-3) - 8709984\/(q-4)\n   - 604860\/(q-5) + 145025724\/(q+1))*D(3,q - 5,4)\n + (98060730\/(q-1) + 17506320\/(q-2) + 24591150\/(q-3) + 28615152\/(q-4)\n   + 2721870\/(q-5) - 303376182\/(q+1))*D(3,q - 5,3)\n + (252260220\/(q-1) - 25022880\/(q-2) + 30365100\/(q-3) - 48235296\/(q-4)\n   - 604860\/(q-5) - 349490604\/(q+1))*D(3,q - 5,2)\n + (239320080\/(q-1) + 52675200\/(q-2) + 9501840\/(q-3) + 87594624\/(q-4)\n   - 16936080\/(q-5) - 306353424\/(q+1))*D(3,q - 5,1)\n + (129869280\/(q-1) + 25993440\/(q-2) + 16495920\/(q-3) + 2903328\/(q-4)\n   - 146231568\/(q+1))*D(3,q - 4,5)\n + ( - 112768560\/(q-1) - 32463600\/(q-2) - 49388040\/(q-3) - 12764304\/(q-4)\n   + 425803704\/(q+1))*D(3,q - 4,4)\n + ( - 543929760\/(q-1) - 33583200\/(q-2) + 64962720\/(q-3) + 1550304\/(q-4)\n   + 828260736\/(q+1))*D(3,q - 4,3)\n + (350288640\/(q-1) - 32133120\/(q-2) - 93375360\/(q-3) + 81593856\/(q-4)\n   - 526175616\/(q+1))*D(3,q - 4,2)\n + ( - 841824000\/(q-1) + 77160960\/(q-2) - 173082240\/(q-3) + 8418816\/(q-4)\n   + 13071744\/(q+1))*D(3,q - 4,1)\n + ( - 85719600\/(q-1) - 19195200\/(q-2) - 5476680\/(q-3)\n   + 87167160\/(q+1))*D(3,q - 3,6)\n + (66969720\/(q-1) + 42256800\/(q-2) + 22104900\/(q-3)\n   - 350303580\/(q+1))*D(3,q - 3,5)\n + (662632560\/(q-1) - 9288000\/(q-2) + 9513720\/(q-3)\n   - 1072048680\/(q+1))*D(3,q - 3,4)\n + ( - 897347520\/(q-1) - 51736320\/(q-2) - 171905760\/(q-3)\n   + 1579946400\/(q+1))*D(3,q - 3,3)\n + ( - 347765760\/(q-1) + 318689280\/(q-2) - 67564800\/(q-3)\n   + 886268160\/(q+1))*D(3,q - 3,2)\n + (1177804800\/(q-1) - 349470720\/(q-2) + 99671040\/(q-3)\n   + 1938539520\/(q+1))*D(3,q - 3,1)\n + (33372000\/(q-1) + 5633280\/(q-2) - 27393120\/(q+1))*D(3,q - 2,7)\n + ( - 23814000\/(q-1) - 17400960\/(q-2) + 173510640\/(q+1))*D(3,q - 2,6)\n + ( - 389590560\/(q-1) - 33592320\/(q-2) + 729384480\/(q+1))*D(3,q - 2,5)\n + (592565760\/(q-1) + 186209280\/(q-2) - 1316528640\/(q+1))*D(3,q - 2,4)\n + (1726202880\/(q-1) + 196715520\/(q-2) - 2554398720\/(q+1))*D(3,q - 2,3)\n + ( - 2683514880\/(q-1) - 91791360\/(q-2) + 5111009280\/(q+1))*D(3,q - 2,2)\n + ( - 1648926720\/(q-1) + 139345920\/(q-2) - 2179768320\/(q+1))*D(3,q - 2,1)\n + ( - 5974560\/(q-1) + 2656800\/(q+1))*D(3,q - 1,8)\n + (4801680\/(q-1) - 49314960\/(q+1))*D(3,q - 1,7)\n + (108125280\/(q-1) - 232264800\/(q+1))*D(3,q - 1,6)\n + ( - 136183680\/(q-1) + 456347520\/(q+1))*D(3,q - 1,5)\n + ( - 1061475840\/(q-1) + 1740510720\/(q+1))*D(3,q - 1,4)\n + (173352960\/(q-1) - 4119275520\/(q+1))*D(3,q - 1,3)\n + (3423928320\/(q-1) - 10377953280\/(q+1))*D(3,q - 1,2)\n + (2229534720\/(q-1) - 1592524800\/(q+1))*D(3,q - 1,1)\n + 414720\/(q+1)*D(3,q,9) + 6428160\/(q+1)*D(3,q,8)\n + 21150720\/(q+1)*D(3,q,7) - 75479040\/(q+1)*D(3,q,6)\n - 296939520\/(q+1)*D(3,q,5) + 1074954240\/(q+1)*D(3,q,4)\n + 4804116480\/(q+1)*D(3,q,3) + 4459069440\/(q+1)*D(3,q,2)\n + ( - 55219230\/(q-1) - 8610480\/(q-2) - 247050\/(q-3) - 1451664\/(q-4)\n   - 907290\/(q-5) + 102412674\/(q+1) + 1866240)*D(2,q - 7,1)\n + (172115550\/(q-1) + 21855600\/(q-2) + 3215970\/(q-3) + 4354992\/(q-4)\n   + 907290\/(q-5) - 327435642\/(q+1) - 7464960)*D(2,q - 6,2)\n + ( - 183874230\/(q-1) - 31308000\/(q-2) - 3285090\/(q-3) - 2452320\/(q-4)\n   - 2117010\/(q-5) + 549853290\/(q+1) - 21841920)*D(2,q - 6,1)\n + ( - 297787050\/(q-1) - 31999440\/(q-2) - 8659710\/(q-3) - 4354992\/(q-4)\n   - 302430\/(q-5) + 590587782\/(q+1) + 17418240)*D(2,q - 5,3)\n + (477897210\/(q-1) + 66902640\/(q-2) + 7588890\/(q-3) + 9710640\/(q-4)\n   + 1512150\/(q-5) - 1517830410\/(q+1) + 75479040)*D(2,q - 5,2)\n + (246864120\/(q-1) - 42446400\/(q-2) - 7535160\/(q-3) + 20407872\/(q-4)\n   - 6048600\/(q-5) - 286075752\/(q+1) + 33315840)*D(2,q - 5,1)\n + (309786840\/(q-1) + 29772720\/(q-2) + 8412660\/(q-3) + 1451664\/(q-4)\n   - 654424524\/(q+1) - 26127360)*D(2,q - 4,4)\n + ( - 663575040\/(q-1) - 71833200\/(q-2) - 18628200\/(q-3) - 7107984\/(q-4)\n   + 2314426344\/(q+1) - 149022720)*D(2,q - 4,3)\n + ( - 710805600\/(q-1) + 40844160\/(q-2) - 36370800\/(q-3) + 28281600\/(q-4)\n   + 1218275280\/(q+1) - 73681920)*D(2,q - 4,2)\n + (1596136320\/(q-1) - 11470080\/(q-2) - 28428480\/(q-3) - 89899776\/(q-4)\n   - 4522732224\/(q+1) + 201000960)*D(2,q - 4,1)\n + ( - 194803920\/(q-1) - 16395840\/(q-2) - 2771280\/(q-3)\n   + 452746080\/(q+1) + 26127360)*D(2,q - 3,5)\n + (518633640\/(q-1) + 48381600\/(q-2) + 13039380\/(q-3)\n   - 2107865820\/(q+1) + 183859200)*D(2,q - 3,4)\n + (1012815360\/(q-1) + 40938240\/(q-2) - 50103360\/(q-3)\n   - 2232649920\/(q+1) + 76723200)*D(2,q - 3,3)\n + ( - 2834017920\/(q-1) + 23185920\/(q-2) + 140374080\/(q-3)\n   + 10110626880\/(q+1) - 742348800)*D(2,q - 3,2)\n + (731934720\/(q-1) - 143769600\/(q-2) + 135060480\/(q-3)\n   - 1296829440\/(q+1) + 511488000)*D(2,q - 3,1)\n + (69033600\/(q-1) + 3964320\/(q-2) - 188030880\/(q+1) - 17418240)*D(2,q - 2,6)\n + ( - 219587760\/(q-1) - 17556480\/(q-2) + 1152483120\/(q+1)\n   - 145152000)*D(2,q - 2,5)\n + ( - 728297280\/(q-1) + 29093760\/(q-2) + 2024331840\/(q+1)\n   - 28339200)*D(2,q - 2,4)\n + (2019525120\/(q-1) - 57300480\/(q-2) - 10609781760\/(q+1)\n   + 1346457600)*D(2,q - 2,3)\n + (1086750720\/(q-1) - 235653120\/(q-2) - 4727715840\/(q+1)\n   - 199065600)*D(2,q - 2,2)\n + ( - 2711162880\/(q-1) + 273162240\/(q-2) + 19434885120\/(q+1)\n   - 5240954880)*D(2,q - 2,1)\n + ( - 10711440\/(q-1) + 41089680\/(q+1) + 7464960)*D(2,q - 1,7)\n + (40821840\/(q-1) - 357097680\/(q+1) + 71608320)*D(2,q - 1,6)\n + (183746880\/(q-1) - 876052800\/(q+1) - 13409280)*D(2,q - 1,5)\n + ( - 500601600\/(q-1) + 5522446080\/(q+1) - 1304985600)*D(2,q - 1,4)\n + ( - 908236800\/(q-1) + 7611770880\/(q+1) - 818380800)*D(2,q - 1,3)\n + (1542205440\/(q-1) - 20560711680\/(q+1) + 7406346240)*D(2,q - 1,2)\n + ( - 145981440\/(q-1) - 16137584640\/(q+1) + 7843184640)*D(2,q - 1,1)\n + ( - 207360\/(q+1) + 207360)*D(2,q + 1,9)\n + ( - 2488320\/(q+1) + 2488320)*D(2,q + 1,8)\n + (3732480\/(q+1) - 3732480)*D(2,q + 1,7)\n + (131880960\/(q+1) - 131880960)*D(2,q + 1,6)\n + (343388160\/(q+1) - 343388160)*D(2,q + 1,5)\n + ( - 806215680\/(q+1) + 806215680)*D(2,q + 1,4)\n + ( - 3808788480\/(q+1) + 3808788480)*D(2,q + 1,3)\n + ( - 3503554560\/(q+1) + 3503554560)*D(2,q + 1,2)\n + ( - 2864160\/(q+1) - 1866240)*D(2,q,8)\n + (53412480\/(q+1) - 20183040)*D(2,q,7)\n + (136045440\/(q+1) + 15068160)*D(2,q,6)\n + ( - 1351987200\/(q+1) + 651110400)*D(2,q,5)\n + ( - 3259975680\/(q+1) + 1014681600)*D(2,q,4)\n + (7019274240\/(q+1) - 4265533440)*D(2,q,3)\n + (18685624320\/(q+1) - 9900195840)*D(2,q,2)\n + (5414584320\/(q+1) - 3503554560)*D(2,q,1)\n + (5806080*q + 525346245\/(q-1) + 60022440\/(q-2) + 271485\/(q-3)\n   + 3629160\/(q-4) + 756075\/(q-5) - 1441292205\/(q+1) + 56987280)*D(1,q -5,1)\n + ( - 20321280*q - 1095709320\/(q-1) - 90395640\/(q-2) - 7076160\/(q-3)\n   - 3629160\/(q-4) + 3353470920\/(q+1) - 170961840)*D(1,q - 4,2)\n + ( - 88542720*q + 128255400\/(q-1) + 20111040\/(q-2) + 4887540\/(q-3)\n   - 3279168\/(q-4) - 2442027132\/(q+1) + 755926560)*D(1,q - 4,1)\n + (40642560*q + 1140907140\/(q-1) + 66553680\/(q-2) + 6895170\/(q-3)\n   - 4146890790\/(q+1) + 284936400)*D(1,q - 3,3)\n + (269982720*q + 169401240\/(q-1) + 995040\/(q-2) + 5354100\/(q-3)\n   + 3574083060\/(q+1) - 1710309600)*D(1,q - 3,2)\n + (385689600*q - 2864138400\/(q-1) - 61000320\/(q-2) + 5040720\/(q-3)\n   + 16590738000\/(q+1) - 3805142400)*D(1,q - 3,1)\n + ( - 50803200*q - 595411200\/(q-1) - 21235200\/(q-2) + 2887281600\/(q+1)\n   - 284936400)*D(1,q - 2,4)\n + ( - 457228800*q - 378488160\/(q-1) - 15764640\/(q-2) - 2653613760\/(q+1)\n   + 2041070400)*D(1,q - 2,3)\n + ( - 1026432000*q + 2902803840\/(q-1) + 28934400\/(q-2) - 25745139840\/(q+1)\n   + 7837274880)*D(1,q - 2,2)\n + ( - 120268800*q + 479473920\/(q-1) + 80524800\/(q-2) - 2808011520\/(q+1)\n   + 539619840)*D(1,q - 2,1)\n + (40642560*q + 125452800\/(q-1) - 1090694160\/(q+1) + 170961840)*D(1,q - 1,5)\n + (464486400*q + 170800560\/(q-1) + 1167529680\/(q+1)\n   - 1351296000)*D(1,q - 1,4)\n + (1451520000*q - 940357440\/(q-1) + 18635400000\/(q+1)\n   - 7927856640)*D(1,q - 1,3)\n + (481075200*q - 833817600\/(q-1) + 9558397440\/(q+1)\n   - 3023516160)*D(1,q - 1,2)\n + ( - 2202992640*q + 830822400\/(q-1) - 27323412480\/(q+1)\n   + 13092710400)*D(1,q - 1,1)\n - 725760*q*D(1,q + 2,8)\n - 13893120*q*D(1,q + 2,7)\n - 85017600*q*D(1,q + 2,6)\n - 120268800*q*D(1,q + 2,5)\n + 550748160*q*D(1,q + 2,4)\n + 1897758720*q*D(1,q + 2,3)\n + 1592524800*q*D(1,q + 2,2)\n + (5806080*q - 8141040\/(q+1) + 8141040)*D(1,q + 1,7)\n + (95800320*q + 66152160\/(q+1) - 66152160)*D(1,q + 1,6)\n + (485222400*q + 769979520\/(q+1) - 769979520)*D(1,q + 1,5)\n + (481075200*q + 1112209920\/(q+1) - 1112209920)*D(1,q + 1,4)\n + ( - 2202992640*q - 3838924800\/(q+1) + 3838924800)*D(1,q + 1,3)\n + ( - 4591779840*q - 9196830720\/(q+1) + 9196830720)*D(1,q + 1,2)\n + ( - 1592524800*q - 3822059520\/(q+1) + 3822059520)*D(1,q + 1,1)\n + ( - 20321280*q + 189367200\/(q+1) - 56987280)*D(1,q,6)\n + ( - 283046400*q - 362849760\/(q+1) + 468715680)*D(1,q,5)\n + ( - 1150848000*q - 6244715520\/(q+1) + 3941015040)*D(1,q,4)\n + ( - 721612800*q - 6432307200\/(q+1) + 3339394560)*D(1,q,3)\n + (3304488960*q + 18636963840\/(q+1) - 12536709120)*D(1,q,2)\n + (3490283520*q + 19292774400\/(q+1) - 13052067840)*D(1,q,1)\n + (2903040*q^2 - 27786240*q - 41109120\/(q-1) - 3663360\/(q-2) + 285120\/(q-3)\n   - 8389440\/(q+1) + 81907200)*D(0,q - 3,1)\n + ( - 8709120*q^2 + 66562560*q + 43856640\/(q-1) + 1451520\/(q-2)\n   + 74960640\/(q+1) - 163814400)*D(0,q - 2,2)\n + ( - 50595840*q^2 + 396472320*q + 249246720\/(q-1) + 14929920\/(q-2)\n   + 498908160\/(q+1) - 999475200)*D(0,q - 2,1)\n + (14515200*q^2 - 82944000*q - 15344640\/(q-1) - 119439360\/(q+1)\n   + 163814400)*D(0,q - 1,3)\n + (128563200*q^2 - 754790400*q - 135613440\/(q-1) - 1170754560\/(q+1)\n   + 1530316800)*D(0,q - 1,2)\n + (306892800*q^2 - 1924300800*q - 282839040\/(q-1) - 3881779200\/(q+1)\n   + 4227655680)*D(0,q - 1,1)\n + (414720*q^2 + 829440*q)*D(0,q + 3,7)\n + (9123840*q^2 + 18247680*q)*D(0,q + 3,6)\n + (74649600*q^2 + 149299200*q)*D(0,q + 3,5)\n + (282009600*q^2 + 564019200*q)*D(0,q + 3,4)\n + (491028480*q^2 + 982056960*q)*D(0,q + 3,3)\n + (318504960*q^2 + 637009920*q)*D(0,q + 3,2)\n + ( - 2903040*q^2 - 207360*q)*D(0,q + 2,6)\n + ( - 53913600*q^2 - 1658880*q)*D(0,q + 2,5)\n + ( - 356659200*q^2 + 20736000*q)*D(0,q + 2,4)\n + ( - 1011916800*q^2 + 265420800*q)*D(0,q + 2,3)\n + ( - 1141309440*q^2 + 915701760*q)*D(0,q + 2,2)\n + ( - 318504960*q^2 + 955514880*q)*D(0,q + 2,1)\n + (8709120*q^2 - 16174080*q - 16381440\/(q+1) + 16381440)*D(0,q + 1,5)\n + (132710400*q^2 - 257126400*q - 265420800\/(q+1) + 265420800)*D(0,q + 1,4)\n + (680140800*q^2 - 1443225600*q - 1575106560\/(q+1) + 1575106560)*D(0,q + 1,3)\n + (1343692800*q^2 - 3483648000*q - 4283228160\/(q+1)\n   + 4283228160)*D(0,q + 1,2)\n + (809533440*q^2 - 3450470400*q - 5202247680\/(q+1) + 5202247680)*D(0,q + 1,1)\n + ( - 14515200*q^2 + 54950400*q + 74649600\/(q+1) - 81907200)*D(0,q,4)\n + ( - 174182400*q^2 + 680140800*q + 970444800\/(q+1) - 1040947200)*D(0,q,3)\n + ( - 646963200*q^2 + 2712268800*q + 4494735360\/(q+1) - 4476487680)*D(0,q,2)\n + ( - 779673600*q^2 + 3914956800*q + 8944680960\/(q+1) - 7697203200)*D(0,q,1)\n + (1517130\/(q-1) + 353280\/(q-2) + 16470\/(q-3) + 302430\/(q-5)\n   - 1981950\/(q+1))*J(3,q - 8)\n + (6662520\/(q-1) + 1568960\/(q-2) + 626040\/(q-3) - 150336\/(q-4)\n   + 2016200\/(q-5) - 13488184\/(q+1))*J(3,q - 7)\n + ( - 144679680\/(q-1) - 25155840\/(q-2) + 607680\/(q-3) - 1002240\/(q-4)\n   - 19355520\/(q-5) + 177696960\/(q+1))*J(3,q - 6)\n + (129669120\/(q-1) + 65802240\/(q-2) - 44904960\/(q-3)\n   + 9621504\/(q-4) + 215824896\/(q+1))*J(3,q - 5)\n + (2016645120\/(q-1) - 189112320\/(q-2) + 113909760\/(q-3)\n   - 4272721920\/(q+1))*J(3,q - 4)\n + ( - 4910284800\/(q-1) + 159252480\/(q-2) + 10590289920\/(q+1))*J(3,q - 3)\n + (7585650\/(q-1) + 1413120\/(q-2) + 49410\/(q-3) + 302430\/(q-5)\n   - 13873650\/(q+1) - 207360)*J(2,q - 8)\n + (29684340\/(q-1) + 5413440\/(q-2) + 1285020\/(q-3) - 150336\/(q-4)\n   + 604860\/(q-5) - 84893004\/(q+1) + 2764800)*J(2,q - 7)\n + ( - 34566480\/(q-1) + 12218240\/(q-2) + 3280320\/(q-3) - 300672\/(q-4)\n   - 4032400\/(q-5) + 14409632\/(q+1) - 5944320)*J(2,q - 6)\n + ( - 311880960\/(q-1) + 21319680\/(q-2) + 2508480\/(q-3) + 2004480\/(q-4)\n   + 19355520\/(q-5) + 728736960\/(q+1) - 19353600)*J(2,q - 5)\n + ( - 546186240\/(q-1) - 48936960\/(q-2) + 51586560\/(q-3) - 9621504\/(q-4)\n   + 1238552064\/(q+1) - 165335040)*J(2,q - 4)\n + ( - 498216960\/(q-1) + 199065600\/(q-2) - 113909760\/(q-3)\n   - 3636817920\/(q+1) + 1293926400)*J(2,q - 3)\n + (4817387520\/(q-1) - 159252480\/(q-2) - 5613649920\/(q+1)\n   - 1751777280)*J(2,q - 2)\n + ( - 725760*q - 100585665\/(q-1) - 14945280\/(q-2) - 90495\/(q-3)\n   - 756075\/(q-5) + 256898475\/(q+1) - 8141040)*J(1,q - 6)\n + (12441600*q - 88154460\/(q-1) - 10180320\/(q-2) - 4797900\/(q-3)\n   + 375840\/(q-4) + 604860\/(q-5) + 650604780\/(q+1) - 137954880)*J(1,q - 5)\n + ( - 60134400*q + 913299840\/(q-1) + 20880000\/(q-2) - 2079360\/(q-3)\n   - 300672\/(q-4) - 4009344768\/(q+1) + 724688640)*J(1,q - 4)\n + (280108800\/(q-1) - 11750400\/(q-2) - 9106560\/(q-3) - 1342730880\/(q+1)\n   + 256711680)*J(1,q - 3)\n + (550748160*q - 391219200\/(q-1) - 36495360\/(q-2) + 11039016960\/(q+1)\n   - 4306452480)*J(1,q - 2)\n + ( - 796262400*q - 905748480\/(q-1) - 1403412480\/(q+1)\n   + 2826731520)*J(1,q - 1)\n + ( - 414720*q^2 + 4769280*q + 12597120\/(q-1) + 2211840\/(q-2) - 285120\/(q-3)\n   - 5400000\/(q+1) - 16381440)*J(0,q - 4)\n + (8294400*q^2 - 81285120*q - 111352320\/(q-1) - 17971200\/(q-2)\n   + 1140480\/(q-3) - 33557760\/(q+1) + 244684800)*J(0,q - 3)\n + ( - 58060800*q^2 + 485222400*q + 236390400\/(q-1) + 32071680\/(q-2)\n   + 976527360\/(q+1) - 1326274560)*J(0,q - 2)\n + (165888000*q^2 - 1260748800*q - 49766400\/(q-1) - 4439162880\/(q+1)\n   + 3413975040)*J(0,q - 1)\n + ( - 159252480*q^2 + 1552711680*q + 6210846720\/(q+1) - 4379443200)*J(0,q)\n  ) \/ 19906560;\n\nZ_1(q) = (( - 41148*q - 194304\/(q-1) - 26292\/(q-2) + 246888)*B(0,q - 3,0)\n + (164592*q + 414900\/(q-1) + 26292\/(q-2) - 740664)*B(0,q - 2,1)\n + ( - 72000*q - 318960\/(q-1) + 34320\/(q-2) + 324000)*B(0,q - 2,0)\n + ( - 246888*q - 220596\/(q-1) + 740664)*B(0,q - 1,2)\n + (98208*q + 178656\/(q-1) - 294624)*B(0,q - 1,1)\n + (1127232*q + 1319616\/(q-1) - 3381696)*B(0,q - 1,0)\n - 41148*q*B(0,q + 1,4)\n - 45792*q*B(0,q + 1,3)\n + 848448*q*B(0,q + 1,2)\n + 2377728*q*B(0,q + 1,1)\n + 1327104*q*B(0,q + 1,0)\n + (164592*q - 246888)*B(0,q,3)\n + (19584*q - 29376)*B(0,q,2)\n + ( - 2003328*q + 3004992)*B(0,q,1)\n + ( - 2529792*q + 3794688)*B(0,q,0)\n + ( - 8832\/(q-1) - 1476\/(q-2) - 5892\/(q-4))*D(3,q - 7,1)\n + (54468\/(q-1) + 7380\/(q-2) + 17676\/(q-3) + 17676\/(q-4))*D(3,q - 6,2)\n + (2482\/(q-1) - 11622\/(q-2) + 12150\/(q-3) - 30442\/(q-4))*D(3,q - 6,1)\n + ( - 139860\/(q-1) - 32436\/(q-2) - 53028\/(q-3) - 17676\/(q-4))*D(3,q - 5,3)\n + (40598\/(q-1) + 20394\/(q-2) + 81390\/(q-3) + 59902\/(q-4))*D(3,q - 5,2)\n + (37216\/(q-1) - 35580\/(q-2) + 76032\/(q-3) - 104092\/(q-4))*D(3,q - 5,1)\n + (197292\/(q-1) + 67788\/(q-2) + 53028\/(q-3) + 5892\/(q-4))*D(3,q - 4,4)\n + ( - 74622\/(q-1) - 101910\/(q-2) - 169770\/(q-3) - 26514\/(q-4))*D(3,q - 4,3)\n + ( - 137916\/(q-1) - 92724\/(q-2) + 278604\/(q-3) + 5892\/(q-4))*D(3,q - 4,2)\n + ( - 469008\/(q-1) - 116208\/(q-2) - 507120\/(q-3) + 164976\/(q-4))*D(3,q - 4,1)\n + ( - 164916\/(q-1) - 60408\/(q-2) - 17676\/(q-3))*D(3,q - 3,5)\n + (60978\/(q-1) + 163356\/(q-2) + 76230\/(q-3))*D(3,q - 3,4)\n + (475212\/(q-1) - 144264\/(q-2) - 2772\/(q-3))*D(3,q - 3,3)\n + ( - 351744\/(q-1) + 64896\/(q-2) - 498240\/(q-3))*D(3,q - 3,2)\n + (1008576\/(q-1) + 993024\/(q-2) - 92736\/(q-3))*D(3,q - 3,1)\n + (78048\/(q-1) + 19152\/(q-2))*D(3,q - 2,6)\n + ( - 36576\/(q-1) - 69480\/(q-2))*D(3,q - 2,5)\n + ( - 448848\/(q-1) - 73296\/(q-2))*D(3,q - 2,4)\n + (733248\/(q-1) + 630720\/(q-2))*D(3,q - 2,3)\n + (1196928\/(q-1) + 450432\/(q-2))*D(3,q - 2,2)\n + ( - 705024\/(q-1) - 483840\/(q-2))*D(3,q - 2,1)\n - 16200\/(q-1)*D(3,q - 1,7)\n + 11556\/(q-1)*D(3,q - 1,6)\n + 204552\/(q-1)*D(3,q - 1,5)\n - 266112\/(q-1)*D(3,q - 1,4)\n - 1772928\/(q-1)*D(3,q - 1,3)\n - 1548288\/(q-1)*D(3,q - 1,2)\n + (108198\/(q-1) + 11070\/(q-2) + 8838\/(q-3) + 8838\/(q-4) - 56700)*D(2,q - 6,1)\n + ( - 276030\/(q-1) - 30978\/(q-2) - 26514\/(q-3) - 8838\/(q-4)\n   + 170100)*D(2,q - 5,2)\n + (256764\/(q-1) + 58086\/(q-2) + 12708\/(q-3) + 20622\/(q-4)\n   - 218592)*D(2,q - 5,1)\n + (378366\/(q-1) + 48654\/(q-2) + 26514\/(q-3) + 2946\/(q-4)\n   - 283500)*D(2,q - 4,3)\n + ( - 560046\/(q-1) - 91206\/(q-2) - 56898\/(q-3) - 14730\/(q-4)\n   + 522720)*D(2,q - 4,2)\n + ( - 110904\/(q-1) + 96552\/(q-2) - 129432\/(q-3) + 58920\/(q-4)\n   - 180144)*D(2,q - 4,1)\n + ( - 295938\/(q-1) - 37584\/(q-2) - 8838\/(q-3) + 283500)*D(2,q - 3,4)\n + (605046\/(q-1) + 101820\/(q-2) + 42534\/(q-3) - 665280)*D(2,q - 3,3)\n + (530352\/(q-1) + 80592\/(q-2) - 168480\/(q-3) - 181440)*D(2,q - 3,2)\n + ( - 825312\/(q-1) + 180864\/(q-2) + 532512\/(q-3) + 1518336)*D(2,q - 3,1)\n + (125892\/(q-1) + 11052\/(q-2) - 170100)*D(2,q - 2,5)\n + ( - 336048\/(q-1) - 51912\/(q-2) + 475200)*D(2,q - 2,4)\n + ( - 752208\/(q-1) + 156240\/(q-2) + 777024)*D(2,q - 2,3)\n + (1178112\/(q-1) - 340992\/(q-2) - 2460672)*D(2,q - 2,2)\n + ( - 177408\/(q-1) - 767232\/(q-2) - 1442304)*D(2,q - 2,1)\n + ( - 22824\/(q-1) + 56700)*D(2,q - 1,6)\n + (80244\/(q-1) - 180576)*D(2,q - 1,5)\n + (246960\/(q-1) - 713232)*D(2,q - 1,4)\n + ( - 468288\/(q-1) + 1347840)*D(2,q - 1,3)\n + ( - 582912\/(q-1) + 3303936)*D(2,q - 1,2)\n + (497664\/(q-1) + 110592)*D(2,q - 1,1)\n - 8100*D(2,q,7) + 28512*D(2,q,6) + 212976*D(2,q,5) - 139968*D(2,q,4)\n   - 1575936*D(2,q,3) - 1603584*D(2,q,2)\n + ( - 136404*q - 761517\/(q-1) - 42759\/(q-2) - 22095\/(q-3) - 7365\/(q-4)\n   + 842490)*D(1,q - 4,1)\n + (341010*q + 1163655\/(q-1) + 64854\/(q-2) + 22095\/(q-3)\n   - 1684980)*D(1,q - 3,2)\n + (144684*q + 297576\/(q-1) - 56556\/(q-2) + 21816\/(q-3) - 366696)*D(1,q - 3,1)\n + ( - 454680*q - 797388\/(q-1) - 36348\/(q-2) + 1684980)*D(1,q - 2,3)\n + ( - 489456*q - 575604\/(q-1) - 9540\/(q-2) + 1253448)*D(1,q - 2,2)\n + (853056*q + 1411440\/(q-1) - 15888\/(q-2) - 3081312)*D(1,q - 2,1)\n + (341010*q + 208434\/(q-1) - 842490)*D(1,q - 1,4)\n + (689544*q + 302484\/(q-1) - 1304568)*D(1,q - 1,3)\n + ( - 1067616*q - 716640\/(q-1) + 2909088)*D(1,q - 1,2)\n + ( - 1528128*q - 794304\/(q-1) + 3600576)*D(1,q - 1,1)\n + 22734*q*D(1,q + 1,6) + 108972*q*D(1,q + 1,5)\n - 41472*q*D(1,q + 1,4) - 931392*q*D(1,q + 1,3)\n - 1520640*q*D(1,q + 1,2) - 663552*q*D(1,q + 1,1)\n + ( - 136404*q + 168498)*D(1,q,5) + ( - 444816*q + 443376)*D(1,q,4)\n   + (482688*q - 816768)*D(1,q,3)\n + (2328192*q - 2827584)*D(1,q,2) + (815616*q - 1472256)*D(1,q,1)\n + ( - 8640*q^2 + 60480*q + 53568\/(q-1) - 1728\/(q-2) - 129600)*D(0,q - 2,1)\n + (17280*q^2 - 86400*q - 25920\/(q-1) + 129600)*D(0,q - 1,2)\n + (138240*q^2 - 691200*q - 207360\/(q-1) + 1036800)*D(0,q - 1,1)\n + ( - 1728*q^2 - 1728*q)*D(0,q + 2,5)\n + ( - 34560*q^2 - 34560*q)*D(0,q + 2,4)\n + ( - 241920*q^2 - 241920*q)*D(0,q + 2,3)\n + ( - 691200*q^2 - 691200*q)*D(0,q + 2,2)\n + ( - 663552*q^2 - 663552*q)*D(0,q + 2,1)\n + (8640*q^2 - 8640*q)*D(0,q + 1,4)\n + (138240*q^2 - 138240*q)*D(0,q + 1,3)\n + (725760*q^2 - 725760*q)*D(0,q + 1,2)\n + (1382400*q^2 - 1382400*q)*D(0,q + 1,1)\n + ( - 17280*q^2 + 51840*q - 43200)*D(0,q,3)\n + ( - 207360*q^2 + 622080*q - 518400)*D(0,q,2)\n + ( - 725760*q^2 + 2177280*q - 1810944)*D(0,q,1)\n + ( - 4416\/(q-1) - 738\/(q-2) - 2946\/(q-4))*J(3,q - 7)\n + ( - 12376\/(q-1) - 7656\/(q-2) + 1656\/(q-3) - 19640\/(q-4))*J(3,q - 6)\n + (377376\/(q-1) + 37632\/(q-2) + 11040\/(q-3) + 188544\/(q-4))*J(3,q - 5)\n + ( - 1218816\/(q-1) + 232704\/(q-2) - 105984\/(q-3))*J(3,q - 4)\n + (1216512\/(q-1) - 552960\/(q-2))*J(3,q - 3)\n + ( - 17664\/(q-1) - 2214\/(q-2) - 2946\/(q-4) + 8100)*J(2,q - 7)\n + ( - 45960\/(q-1) - 16788\/(q-2) + 1656\/(q-3) - 5892\/(q-4)\n   + 38016)*J(2,q - 6)\n + ( - 12400\/(q-1) - 38784\/(q-2) + 3312\/(q-3) + 39280\/(q-4)\n  + 84816)*J(2,q - 5)\n + (60672\/(q-1) - 60864\/(q-2) - 22080\/(q-3) - 188544\/(q-4)\n   - 265536)*J(2,q - 4)\n + (1200384\/(q-1) - 221184\/(q-2) + 105984\/(q-3) + 36864)*J(2,q - 3)\n + ( - 1714176\/(q-1) + 552960\/(q-2) + 110592)*J(2,q - 2)\n + (22734*q + 186816\/(q-1) + 14253\/(q-2) + 7365\/(q-4) - 168498)*J(1,q - 5)\n + ( - 8928*q - 18564\/(q-1) + 48420\/(q-2) - 4140\/(q-3) - 5892\/(q-4)\n   - 25560)*J(1,q - 4)\n + ( - 226656*q - 636624\/(q-1) + 14304\/(q-2) + 3312\/(q-3) + 988992)*J(1,q - 3)\n + (158976*q + 25344\/(q-1) + 41472\/(q-2) - 663552)*J(1,q - 2)\n + (857088*q + 290304\/(q-1) - 635904)*J(1,q - 1)\n + (1728*q^2 - 15552*q - 27648\/(q-1) + 1728\/(q-2) + 43200)*J(0,q - 3)\n + ( - 34560*q^2 + 241920*q + 214272\/(q-1) - 6912\/(q-2) - 518400)*J(0,q - 2)\n + (241920*q^2 - 1209600*q - 359424\/(q-1) + 1810944)*J(0,q - 1)\n + ( - 691200*q^2 + 2073600*q - 1714176)*J(0,q)\n) \/ 41472;\n\nZ_2(q) = ((900*q^2 - 3764*q + 1964\/(q-1) + 1964)*B(0,q - 2,0)\n + ( - 2700*q^2 + 7528*q - 1964\/(q-1) - 1964)*B(0,q - 1,1)\n + ( - 1776*q^2 + 3616*q - 368\/(q-1) - 368)*B(0,q - 1,0)\n - 900*q^2*B(0,q + 1,3) - 3408*q^2*B(0,q + 1,2)\n - 3072*q^2*B(0,q + 1,1) + (2700*q^2 - 3764*q)*B(0,q,2)\n + (5184*q^2 - 6112*q)*B(0,q,1) + (1152*q^2 - 192*q)*B(0,q,0)\n + (156\/(q-1) + 468\/(q-3) + 312)*D(3,q - 6,1)\n + ( - 780\/(q-1) - 624\/(q-2) - 1404\/(q-3) - 1560)*D(3,q - 5,2)\n + (554\/(q-1) - 504\/(q-2) + 2418\/(q-3) + 1108)*D(3,q - 5,1)\n + (1716\/(q-1) + 1872\/(q-2) + 1404\/(q-3) + 3120)*D(3,q - 4,3)\n + ( - 1986\/(q-1) - 2648\/(q-2) - 4758\/(q-3) - 4896)*D(3,q - 4,2)\n + (1604\/(q-1) - 3072\/(q-2) + 8268\/(q-3) + 2824)*D(3,q - 4,1)\n + ( - 2028\/(q-1) - 1872\/(q-2) - 468\/(q-3) - 3120)*D(3,q - 3,4)\n + (2894\/(q-1) + 5768\/(q-2) + 2106\/(q-3) + 6480)*D(3,q - 3,3)\n + (316\/(q-1) - 9072\/(q-2) - 468\/(q-3) - 4376)*D(3,q - 3,2)\n + (6480\/(q-1) + 16576\/(q-2) - 13104\/(q-3) + 10400)*D(3,q - 3,1)\n + (1248\/(q-1) + 624\/(q-2) + 1560)*D(3,q - 2,5)\n + ( - 2176\/(q-1) - 2616\/(q-2) - 3484)*D(3,q - 2,4)\n + ( - 2416\/(q-1) - 240\/(q-2) - 2536)*D(3,q - 2,3)\n + (12416\/(q-1) + 17664\/(q-2) + 21248)*D(3,q - 2,2)\n + ( - 21888\/(q-1) + 5376\/(q-2) - 19200)*D(3,q - 2,1)\n + ( - 312\/(q-1) - 312)*D(3,q - 1,6) + (636\/(q-1) + 636)*D(3,q - 1,5)\n   + (3144\/(q-1) + 3144)*D(3,q - 1,4)\n + ( - 9504\/(q-1) - 9504)*D(3,q - 1,3) + ( - 21504\/(q-1) - 21504)*D(3,q - 1,2)\n + (936*q - 1170\/(q-1) - 312\/(q-2) - 702\/(q-3) - 1560)*D(2,q - 5,1)\n + ( - 2340*q + 2418\/(q-1) + 936\/(q-2) + 702\/(q-3) + 3120)*D(2,q - 4,2)\n + (4496*q - 4362\/(q-1) - 336\/(q-2) - 1638\/(q-3) - 5076)*D(2,q - 4,1)\n + (3120*q - 2574\/(q-1) - 936\/(q-2) - 234\/(q-3) - 3120)*D(2,q - 3,3)\n + ( - 8544*q + 6642\/(q-1) + 1896\/(q-2) + 1170\/(q-3) + 7980)*D(2,q - 3,2)\n + (6176*q - 4824\/(q-1) + 4832\/(q-2) - 4680\/(q-3) - 3968)*D(2,q - 3,1)\n + ( - 2340*q + 1404\/(q-1) + 312\/(q-2) + 1560)*D(2,q - 2,4)\n + (8096*q - 4832\/(q-1) - 1464\/(q-2) - 5564)*D(2,q - 2,3)\n + ( - 544*q + 656\/(q-1) + 5760\/(q-2) + 3536)*D(2,q - 2,2)\n + ( - 22144*q - 3264\/(q-1) - 18048\/(q-2) - 12288)*D(2,q - 2,1)\n + (936*q - 312\/(q-1) - 312)*D(2,q - 1,5) + ( - 3824*q + 1452\/(q-1)\n   + 1452)*D(2,q - 1,4)\n + ( - 5984*q - 480\/(q-1) - 480)*D(2,q - 1,3) + (27904*q - 4800\/(q-1)\n   - 4800)*D(2,q - 1,2)\n + (16896*q + 18432\/(q-1) + 18432)*D(2,q - 1,1)\n   - 156*q*D(2,q,6) + 720*q*D(2,q,5)\n + 3216*q*D(2,q,4) - 6528*q*D(2,q,3) - 26880*q*D(2,q,2) - 18432*q*D(2,q,1)\n + (2010*q^2 - 9136*q + 5025\/(q-1) + 780\/(q-2) + 585\/(q-3)\n   + 5610)*D(1,q - 3,1)\n + ( - 4020*q^2 + 13704*q - 5220\/(q-1) - 780\/(q-2) - 5610)*D(1,q - 2,2)\n + ( - 1428*q^2 - 2316*q + 2532\/(q-1) - 864\/(q-2) + 2100)*D(1,q - 2,1)\n + (4020*q^2 - 9136*q + 1870\/(q-1) + 1870)*D(1,q - 1,3)\n + (4716*q^2 - 3096*q - 12\/(q-1) - 12)*D(1,q - 1,2)\n + ( - 15696*q^2 + 27232*q - 1840\/(q-1) - 1840)*D(1,q - 1,1)\n + 402*q^2*D(1,q + 1,5) + 1644*q^2*D(1,q + 1,4)\n - 1776*q^2*D(1,q + 1,3) - 16128*q^2*D(1,q + 1,2) - 18432*q^2*D(1,q + 1,1)\n + ( - 2010*q^2 + 2284*q)*D(1,q,4)\n + ( - 4860*q^2 + 2836*q)*D(1,q,3)\n + (11328*q^2 - 13856*q)*D(1,q,2)\n + (37248*q^2 - 19776*q)*D(1,q,1)\n + (78\/(q-1) + 234\/(q-3) + 156)*J(3,q - 6)\n + (472\/(q-1) - 96\/(q-2) + 1560\/(q-3) + 944)*J(3,q - 5)\n + ( - 5312\/(q-1) - 640\/(q-2) - 14976\/(q-3) - 10624)*J(3,q - 4)\n + (3072\/(q-1) + 6144\/(q-2) + 6144)*J(3,q - 3)\n + ( - 156*q + 234\/(q-1) + 234\/(q-3) + 312)*J(2,q - 6)\n + ( - 944*q + 1100\/(q-1) - 96\/(q-2) + 468\/(q-3) + 1208)*J(2,q - 5)\n + ( - 2864*q + 2048\/(q-1) - 192\/(q-2) - 3120\/(q-3) + 912)*J(2,q - 4)\n + (1536*q + 5056\/(q-1) + 1280\/(q-2) + 14976\/(q-3) + 10688)*J(2,q - 3)\n + (23808*q - 7680\/(q-1) - 6144\/(q-2) - 10752)*J(2,q - 2)\n + ( - 402*q^2 + 2284*q - 1675\/(q-1) - 585\/(q-3) - 1870)*J(1,q - 4)\n + ( - 72*q^2 + 2576*q - 2364\/(q-1) + 240\/(q-2) + 468\/(q-3) - 2088)*J(1,q - 3)\n + (6144*q^2 - 11648*q + 448\/(q-1) - 192\/(q-2) + 352)*J(1,q - 2)\n + ( - 19200*q^2 + 9984*q + 1152\/(q-1) + 1152)*J(1,q - 1)\n) \/ 1152;\n\nZ_3(q) = (( - 26*q^3 + 88*q^2 - 62*q)*B(0,q - 1,0)\n + ( - 26*q^3 + 26*q^2)*B(0,q + 1,2)\n + ( - 64*q^3 + 64*q^2)*B(0,q + 1,1)\n + (52*q^3 - 114*q^2 + 62*q)*B(0,q,1)\n + (16*q^3 + 24*q^2 - 40*q)*B(0,q,0)\n + ( - 12*q - 24\/(q-2) - 12)*D(3,q - 5,1)\n + (48*q + 72\/(q-2) + 36)*D(3,q - 4,2)\n + ( - 72*q - 124\/(q-2) - 62)*D(3,q - 4,1)\n + ( - 72*q - 72\/(q-2) - 36)*D(3,q - 3,3)\n + (200*q + 244\/(q-2) + 122)*D(3,q - 3,2)\n + ( - 324*q - 424\/(q-2) - 212)*D(3,q - 3,1)\n + (48*q + 24\/(q-2) + 12)*D(3,q - 2,4)\n + ( - 160*q - 108\/(q-2) - 54)*D(3,q - 2,3)\n + (184*q + 24\/(q-2) + 12)*D(3,q - 2,2)\n + (1056*q + 672\/(q-2) + 336)*D(3,q - 2,1)\n - 12*q*D(3,q - 1,5) + 38*q*D(3,q - 1,4) + 92*q*D(3,q - 1,3)\n - 832*q*D(3,q - 1,2) - 1536*q*D(3,q - 1,1)\n + ( - 30*q^2 + 54*q + 36\/(q-2) + 18)*D(2,q - 4,1)\n + (60*q^2 - 96*q - 36\/(q-2) - 18)*D(2,q - 3,2)\n + ( - 176*q^2 + 254*q + 84\/(q-2) + 42)*D(2,q - 3,1)\n + ( - 60*q^2 + 84*q + 12\/(q-2) + 6)*D(2,q - 2,3)\n + (240*q^2 - 336*q - 60\/(q-2) - 30)*D(2,q - 2,2)\n + ( - 312*q^2 + 568*q + 240\/(q-2) + 120)*D(2,q - 2,1)\n + (30*q^2 - 36*q)*D(2,q - 1,4) + ( - 144*q^2 + 182*q)*D(2,q - 1,3)\n + ( - 8*q^2 - 200*q)*D(2,q - 1,2)\n + (1472*q^2 - 1536*q)*D(2,q - 1,1) + ( - 6*q^2 + 6*q)*D(2,q,5)\n + (32*q^2 - 32*q)*D(2,q,4)\n + (104*q^2 - 104*q)*D(2,q,3) + ( - 352*q^2 + 352*q)*D(2,q,2)\n + ( - 768*q^2 + 768*q)*D(2,q,1)\n + ( - 52*q^3 + 208*q^2 - 186*q - 30\/(q-2) - 15)*D(1,q - 2,1)\n + (78*q^3 - 234*q^2 + 171*q)*D(1,q - 1,2)\n + (74*q^3 + 96*q^2 - 194*q)*D(1,q - 1,1)\n + (13*q^3 - 13*q^2)*D(1,q + 1,4)\n + (58*q^3 - 58*q^2)*D(1,q + 1,3)\n + (64*q^3 - 64*q^2)*D(1,q + 1,2)\n + ( - 52*q^3 + 104*q^2 - 52*q)*D(1,q,3)\n + ( - 124*q^3 + 90*q^2 + 34*q)*D(1,q,2)\n + ( - 16*q^3 - 344*q^2 + 360*q)*D(1,q,1)\n + ( - 6*q - 12\/(q-2) - 6)*J(3,q - 5)\n + ( - 48*q - 80\/(q-2) - 40)*J(3,q - 4)\n + (1984*q + 768\/(q-2) + 384)*J(3,q - 3)\n + (6*q^2 - 12*q - 12\/(q-2) - 6)*J(2,q - 5)\n + (48*q^2 - 68*q - 24\/(q-2) - 12)*J(2,q - 4)\n + (216*q^2 - 152*q + 160\/(q-2) + 80)*J(2,q - 3)\n + ( - 800*q^2 - 1248*q - 768\/(q-2) - 384)*J(2,q - 2)\n + (13*q^3 - 65*q^2 + 67*q + 30\/(q-2) + 15)*J(1,q - 3)\n + ( - 8*q^3 - 128*q^2 + 144*q - 24\/(q-2) - 12)*J(1,q - 2)\n + (96*q^2 + 240*q)*J(1,q - 1)\n) \/ 96;\n\\end{verbatim}\n\n\n\n{\\Large \\bf Appendix 8.}\n\nHere and below, the symbol {\\tt JJ(p,q)} in any Appendix\ndesignates ${\\cal J}(p,q)$, whereas {\\tt J(p,q)} designates $J(p,q)$.\nIn computations at $q\\leq 0$, $\\mu_B=${\\tt muB}$=0$ and\n{\\tt JJ(p,q)} should be replaced with {\\tt J(p,q)}.\n\n\n\\begin{verbatim}\nJJ(p,q)=(( 24*BB(p,q)*(1\/(p+3)+1\/(p+2)+1\/(p+1))*( - q^3 + 6*q^2 - 11*q + 6) \n + 24*BB(p+1,q-1)*(1\/(p+3)+1\/(p+2))*( - 5*(p+1)*q^2\n    + 25*(p+1)*q - 30*(p+1) + 2*q^2 - 10*q + 12)\n + 48*BB(p+1,q-1)\/(p+1)*(q^2 - 5*q + 6) \n + 2*BB(p+1,q-2)\/(p+3)*( - 8*(p+2)*(p+1)*q + 24*(p+2)*(p+1)\n    - 21*(p+1)*q^2 + 163*(p+1)*q - 300*(p+1) + 6*q^3 - 42*q^2 + 96*q - 72)\n + 2*BB(p+1,q-2)\/(p+2)*( - 12*(p+3)*(p+1)*q + 36*(p+3)*(p+1)\n    - 21*(p+1)*q^2 + 163*(p+1)*q - 300*(p+1) + 6*q^3 - 42*q^2 + 96*q - 72) \n + 12*BB(p+1,q-2)\/(p+1)*(q^3 - 7*q^2 + 16*q - 12) \n + 24*BB(p+2,q-2)\/(p+3)*(p+1)*( - 11*(p+2)*q + 33*(p+2) + 8*q - 24) \n + 192*BB(p+2,q-2)\/(p+2)*(p+1)*(q - 3) \n + 4*BB(p+2,q-3)\/(p+3)*(p+1)*( - 20*(p+2)*q + 118*(p+2) + 15*q^2 - 81*q + 83) \n + 4*BB(p+2,q-3)\/(p+2)*(p+1)*(12*(p+3) + 15*q^2 - 81*q + 83)\n + BB(p+2,q-4)\/(p+3)*(p+1)*( - 4*(p+2)*q + 46*(p+2) + 21*q^2 - 172*q + 352) \n + BB(p+2,q-4)\/(p+2)*(p+1)*(12*(p+3)*q-48*(p+3)+21*q^2 - 172*q + 352) \n + 336*BB(p+3,q-3)\/(p+3)*(p+2)*(p+1) \n + 4*BB(p+3,q-4)\/(p+3)*(p+2)*(p+1)*(33*q - 152) \n + 4*BB(p+3,q-5)\/(p+3)*(p+2)*(p+1)*(10*q - 53) \n + 6*BB(p+3,q-6)\/(p+3)*(p+2)*(p+1)*(q - 6) \n + 24*JJ(p+1,q-1)*(5*(p+1)*q^2 - 25*(p+1)*q + 30*(p+1)\n    - 2*q^2 + 10*q - 12) \n + 2*JJ(p+1,q-2)*(12*(p+3)*(p+1)*q - 36*(p+3)*(p+1)\n     + 8*(p+2)*(p+1)*q - 24*(p+2)*(p+1) + 21*(p+1)*q^2\n    - 163*(p+1)*q + 300*(p+1) - 6*q^3 + 42*q^2-96*q+72)\n + 24*JJ(p+2,q-2)*(p+1)*(11*(p+2)*q - 33*(p+2) - 8*q + 24) \n + 4*JJ(p+2,q-3)*(p+1)*(30*(p+3)*(p+2)-12*(p+3)+20*(p+2)*q\n    - 118*(p+2) - 15*q^2 + 81*q - 83) \n + JJ(p+2,q-4)*(p+1)*(10*(p+3)*(p+2) - 12*(p+3)*q+48*(p+3)\n    + 4*(p+2)*q - 46*(p+2) - 21*q^2 + 172*q - 352) \n + 24*JJ(p+3,q-3)*(p+2)*(p+1)*(15*(p+3) - 14) \n + 2*JJ(p+3,q-4)*(p+2)*(p+1)*( - 73*(p+3) - 66*q + 304) \n + 4*JJ(p+3,q-5)*(p+2)*(p+1)*( - 10*(p+3) - 10*q + 53)\n + 6*JJ(p+3,q-6)*(p+2)*(p+1)*( - (p+3) - q + 6) \n - 192*JJ(p+4,q-4)*(p+3)*(p+2)*(p+1) \n + 20*JJ(p+4,q-5)*(p+3)*(p+2)*(p+1) \n + 3*JJ(p+4,q-6)*(p+3)*(p+2)*(p+1)\n )\/12+muB^2*(\n  24*BB(p+1,q)*(1\/(p+3)+1\/(p+2)+1\/(p+1))*(q^3 - 6*q^2 + 11*q - 6) \n + 6*BB(p+1,q-1)*(1\/(p+3)+1\/(p+2))*(7*(p+1)*q^2-35*(p+1)*q\n     + 42*(p+1) - 4*q^3 + 26*q^2 - 54*q + 36) \n + 12*BB(p+1,q-1)\/(p+1)*( - 2*q^3 + 13*q^2 - 27*q + 18)\n + 120*BB(p+2,q-1)*(1\/(p+3)+1\/(p+2))*(p+1)*(q^2 - 5*q + 6) \n + 2*BB(p+2,q-2)\/(p+3)*(p+1)*(68*(p+2)*q - 204*(p+2)\n    - 39*q^2 + 149*q - 96) \n + 2*BB(p+2,q-2)\/(p+2)*(p+1)*(12*(p+3)*q - 36*(p+3) - 39*q^2 + 149*q - 96) \n + BB(p+2,q-3)\/(p+3)*(p+1)*(8*(p+2)*q - 58*(p+2) - 63*q^2 + 449*q - 805) \n + BB(p+2,q-3)\/(p+2)*(p+1)*( - 24*(p+3)*q + 84*(p+3)\n    - 63*q^2 + 449*q - 805) \n + 264*BB(p+3,q-2)\/(p+3)*(p+2)*(p+1)*(q - 3) \n + 4*BB(p+3,q-3)\/(p+3)*(p+2)*(p+1)*( - 46*q + 121) \n + 2*BB(p+3,q-4)\/(p+3)*(p+2)*(p+1)*( - 64*q + 273) \n + 6*BB(p+3,q-5)\/(p+3)*(p+2)*(p+1)*( - 4*q + 21) \n + 24*JJ(p+1,q)*( - q^3 + 6*q^2 - 11*q + 6) \n + 6*JJ(p+1,q-1)*( - 7*(p+1)*q^2 + 35*(p+1)*q\n    - 42*(p+1) + 4*q^3 - 26*q^2 + 54*q - 36) \n + 120*JJ(p+2,q-1)*(p+1)*( - q^2 + 5*q - 6) \n + 2*JJ(p+2,q-2)*(p+1)*(-12*(p+3)*q+36*(p+3) - 68*(p+2)*q\n + 204*(p+2) + 39*q^2 - 149*q + 96)\n + JJ(p+2,q-3)*(p+1)*( - 10*(p+3)*(p+2) + 24*(p+3)*q\n    -84*(p+3)-8*(p+2)*q+58*(p+2) + 63*q^2 - 449*q + 805)\n + 264*JJ(p+3,q-2)*(p+2)*(p+1)*( - q + 3)\n + 2*JJ(p+3,q-3)*(p+2)*(p+1)*( - 47*(p+3) + 92*q - 242) \n + 2*JJ(p+3,q-4)*(p+2)*(p+1)*(30*(p+3) + 64*q - 273) \n + 6*JJ(p+3,q-5)*(p+2)*(p+1)*(3*(p+3) + 4*q - 21) \n - 168*JJ(p+4,q-3)*(p+3)*(p+2)*(p+1) \n + 106*JJ(p+4,q-4)*(p+3)*(p+2)*(p+1) \n + 31*JJ(p+4,q-5)*(p+3)*(p+2)*(p+1) \n + 6*JJ(p+4,q-6)*(p+3)*(p+2)*(p+1)\n )\/12+muB^4*(\n  12*BB(p+1,q)*(1\/(p+3)+1\/(p+2)+1\/(p+1))*(q^3 - 6*q^2 + 11*q - 6) \n + 18*BB(p+2,q-1)*(1\/(p+3)+1\/(p+2))*(p+1)*(q^2 - 5*q + 6) \n + BB(p+2,q-2)\/(p+3)*(p+1)*( - 4*(p+2)*q + 12*(p+2) + 63*q^2 - 382*q + 579) \n + BB(p+2,q-2)\/(p+2)*(p+1)*(12*(p+3)*q - 36*(p+3) + 63*q^2 - 382*q + 579) \n + 12*BB(p+3,q-2)\/(p+3)*(p+2)*(p+1)*(q - 3) \n + 2*BB(p+3,q-3)\/(p+3)*(p+2)*(p+1)*(68*q - 239) \n + 18*BB(p+3,q-4)\/(p+3)*(p+2)*(p+1)*(2*q - 9) \n + 12*JJ(p+1,q)*( - q^3 + 6*q^2 - 11*q + 6) \n + 18*JJ(p+2,q-1)*(p+1)*( - q^2 + 5*q - 6) \n + JJ(p+2,q-2)*(p+1)*(-12*(p+3)*q + 36*(p+3) + 4*(p+2)*q\n    - 12*(p+2) - 63*q^2 + 382*q - 579) \n + 12*JJ(p+3,q-2)*(p+2)*(p+1)*( - q + 3) \n + 2*JJ(p+3,q-3)*(p+2)*(p+1)*( - 10*(p+3) - 68*q + 239) \n + 18*JJ(p+3,q-4)*(p+2)*(p+1)*( - (p+3) - 2*q + 9) \n - 6*JJ(p+4,q-3)*(p+3)*(p+2)*(p+1) \n - 61*JJ(p+4,q-4)*(p+3)*(p+2)*(p+1) \n - 18*JJ(p+4,q-5)*(p+3)*(p+2)*(p+1)\n )\/12+muB^6*(\n  7*BB(p+2,q-1)*(1\/(p+3)+1\/(p+2))*(p+1)*( - q^2 + 5*q - 6) \n + 16*BB(p+3,q-2)\/(p+3)*(p+2)*(p+1)*( - q + 3) \n + 2*BB(p+3,q-3)\/(p+3)*(p+2)*(p+1)*( - 4*q + 15) \n + 7*JJ(p+2,q-1)*(p+1)*(q^2 - 5*q + 6) \n + 16*JJ(p+3,q-2)*(p+2)*(p+1)*(q - 3) \n + 2*JJ(p+3,q-3)*(p+2)*(p+1)*((p+3) + 4*q - 15) \n + 9*JJ(p+4,q-3)*(p+3)*(p+2)*(p+1) \n + 6*JJ(p+4,q-4)*(p+3)*(p+2)*(p+1)\n )\/4+muB^8*(\n  BB(p+3,q-2)\/(p+3)*(p+2)*(p+1)*(q - 3) \n + JJ(p+3,q-2)*(p+2)*(p+1)*( - q + 3) \n - JJ(p+4,q-3)*(p+3)*(p+2)*(p+1)\n )\/2)\/(-2)\/(q-1)\/(q-2)\/(q-3);\n\nJJ(-3,q) = (   6*BB(0,q - 6)\n + ( - 18*muB^2 + 40)*BB(0,q - 5)\n + (18*muB^4 - 60*muB^2 + 146)*BB(0,q - 4)\n + ( - 6*muB^6 + 20*muB^4 + 94*muB^2 - 360)*BB(0,q - 3)\n + ( - 21*q^2 + 160*q - 314)*BB(-1,q - 4)\n + ((63*q^2 - 425*q + 731)*muB^2 - 60*q^2 + 324*q - 500)*BB(-1,q - 3)\n + (( - 63*q^2 + 370*q - 543)*muB^4 + (78*q^2 - 322*q + 264)*muB^2\n   - 192*q + 576)*BB(-1,q - 2)\n + ((21*q^2 - 105*q + 126)*muB^6 + ( - 18*q^2 + 90*q - 108)*muB^4\n   + ( - 120*q^2 + 600*q - 720)*muB^2)*BB(-1,q - 1)\n + (9*q^3 - 21*q^2 - 158*q + 420)*BB(-2,q - 2)\n + (( - 18*q^3 + 75*q^2 - 33*q - 90)*muB^2\n   + 156*q^2 - 780*q + 936)*BB(-2,q - 1)\n + ((9*q^3 - 54*q^2 + 99*q - 54)*muB^4\n   + (18*q^3 - 108*q^2 + 198*q - 108)*muB^2)*BB(-2,q)\n + ( - 18*q^3 + 108*q^2 - 198*q + 108)*BB(-3,q)\n + ( - 6*muB^2 - 3)*JJ(1,q - 6)\n + (18*muB^4 - 31*muB^2 - 20)*JJ(1,q - 5)\n + ( - 18*muB^6 + 61*muB^4 - 106*muB^2 + 192)*JJ(1,q - 4)\n + (6*muB^8 - 27*muB^6 + 6*muB^4 + 168*muB^2)*JJ(1,q - 3)\n + (6*q - 36)*JJ(0,q - 6)\n + (( - 24*q + 126)*muB^2 + 40*q - 212)*JJ(0,q - 5)\n + ((36*q - 162)*muB^4 + ( - 128*q + 546)*muB^2\n   + 132*q - 608)*JJ(0,q - 4)\n + (( - 24*q + 90)*muB^6 + (136*q - 478)*muB^4\n   + ( - 184*q + 484)*muB^2 + 336)*JJ(0,q - 3)\n + ((6*q - 18)*muB^8 + ( - 48*q + 144)*muB^6 + (12*q - 36)*muB^4\n   + (264*q - 792)*muB^2)*JJ(0,q - 2)\n + ( - 21*q^2 + 168*q - 306)*JJ(-1,q - 4)\n + ((63*q^2 - 441*q + 747)*muB^2  - 60*q^2 + 244*q + 140)*JJ(-1,q - 3)\n + (( - 63*q^2 + 378*q - 567)*muB^4\n   + (78*q^2 - 162*q - 216)*muB^2 - 456*q + 1368)*JJ(-1,q - 2)\n + ((21*q^2 - 105*q + 126)*muB^6 + ( - 18*q^2 + 90*q - 108)*muB^4\n   + ( - 120*q^2 + 600*q - 720)*muB^2)*JJ(-1,q - 1)\n + (6*q^3 - 246*q + 576)*JJ(-2,q - 2)\n + (( - 12*q^3 + 36*q^2 + 48*q - 144)*muB^2\n   + 144*q^2 - 720*q + 864)*JJ(-2,q - 1)\n + ((6*q^3 - 36*q^2 + 66*q - 36)*muB^4\n   + (12*q^3 - 72*q^2 + 132*q - 72)*muB^2)*JJ(-2,q))\n\/ (12*q^3 - 72*q^2 + 132*q - 72);\n\nJJ(-2,q) = ((21*q^2 - 168*q + 316)*BB(0,q - 4)\n + (( - 63*q^2 + 441*q - 757)*muB^2 + 60*q^2 - 244*q - 20)*BB(0,q - 3)\n + ((63*q^2 - 378*q + 567)*muB^4\n   + ( - 78*q^2 + 162*q + 216)*muB^2 + 456*q - 1368)*BB(0,q - 2)\n + (( - 21*q^2 + 105*q - 126)*muB^6 + (18*q^2 - 90*q + 108)*muB^4\n   + (120*q^2 - 600*q + 720)*muB^2)*BB(0,q - 1)\n + ( - 42*q^2 + 342*q - 648)*BB(-1,q - 2)\n + ((42*q^2 - 210*q + 252)*muB^2 - 120*q^2 + 600*q - 720)*BB(-1,q - 1)\n + (6*muB^2 + 3)*JJ(2,q - 6)\n + ( - 18*muB^4 + 31*muB^2 + 20)*JJ(2,q - 5)\n + (18*muB^6 - 61*muB^4 + 106*muB^2 - 192)*JJ(2,q - 4)\n + ( - 6*muB^8 + 27*muB^6 - 6*muB^4 - 168*muB^2)*JJ(2,q - 3)\n + ( - 6*q + 30)*JJ(1,q - 6)\n + ((24*q - 108)*muB^2 - 40*q + 172)*JJ(1,q - 5)\n + (( - 36*q + 144)*muB^4 + (128*q - 486)*muB^2 - 132*q + 462)*JJ(1,q - 4)\n + ((24*q - 84)*muB^6 + ( - 136*q + 458)*muB^4\n   + (184*q - 578)*muB^2 + 24)*JJ(1,q - 3)\n + (( - 6*q + 18)*muB^8 + (48*q - 144)*muB^6\n   + ( - 12*q + 36)*muB^4 + ( - 264*q + 792)*muB^2)*JJ(1,q - 2)\n + ( - 21*q^2 + 160*q - 304)*JJ(0,q - 4)\n + ((63*q^2 - 425*q + 721)*muB^2 - 60*q^2 + 324*q - 380)*JJ(0,q - 3)\n + (( - 63*q^2 + 370*q - 543)*muB^4 + (78*q^2 - 322*q + 264)*muB^2\n   - 192*q + 576)*JJ(0,q - 2)\n + ((21*q^2 - 105*q + 126)*muB^6 + ( - 18*q^2 + 90*q - 108)*muB^4\n   + ( - 120*q^2 + 600*q - 720)*muB^2)*JJ(0,q - 1)\n + (12*q^3 - 42*q^2 - 110*q + 384)*JJ(-1,q - 2)\n + (( - 24*q^3 + 114*q^2 - 114*q - 36)*muB^2\n   + 168*q^2 - 840*q + 1008)*JJ(-1,q - 1)\n + ((12*q^3 - 72*q^2 + 132*q - 72)*muB^4\n   + (24*q^3 - 144*q^2 + 264*q - 144)*muB^2)*JJ(-1,q))\n\/ (24*q^3 - 144*q^2 + 264*q - 144);\n\nJJ(-1,q) = (  ( - 42*q^2 + 262*q - 408)*BB(0,q - 2)\n + ((42*q^2 - 210*q + 252)*muB^2 - 120*q^2 + 600*q - 720)*BB(0,q - 1)\n + ( - 12*muB^2 - 6)*JJ(3,q - 6)\n + (36*muB^4 - 62*muB^2 - 40)*JJ(3,q - 5) \n + ( - 36*muB^6 + 122*muB^4 - 212*muB^2 + 384)*JJ(3,q - 4)\n + (12*muB^8 - 54*muB^6 + 12*muB^4 + 336*muB^2)*JJ(3,q - 3)\n + (6*q - 24)*JJ(2,q - 6)\n + (( - 24*q + 90)*muB^2 + 40*q - 132)*JJ(2,q - 5)\n + ((36*q - 126)*muB^4 + ( - 128*q + 426)*muB^2 + 132*q - 316)*JJ(2,q - 4)\n + (( - 24*q + 78)*muB^6 + (136*q - 438)*muB^4\n   + ( - 184*q + 672)*muB^2 - 384)*JJ(2,q - 3)\n + ((6*q - 18)*muB^8 + ( - 48*q + 144)*muB^6\n   + (12*q - 36)*muB^4 + (264*q - 792)*muB^2)*JJ(2,q - 2)\n + (21*q^2 - 152*q + 282)*JJ(1,q - 4)\n + (( - 63*q^2 + 409*q - 675)*muB^2 + 60*q^2 - 404*q + 660)*JJ(1,q - 3)\n + ((63*q^2 - 362*q + 519)*muB^4\n   + ( - 78*q^2 + 482*q - 744)*muB^2 - 72*q + 216)*JJ(1,q - 2)\n + (( - 21*q^2 + 105*q - 126)*muB^6 + (18*q^2 - 90*q + 108)*muB^4\n   + (120*q^2 - 600*q + 720)*muB^2)*JJ(1,q - 1)\n + (12*q^3 - 84*q^2 + 192*q - 144)*JJ(0,q - 2)\n + (( - 24*q^3 + 156*q^2 - 324*q + 216)*muB^2\n   + 48*q^2 - 240*q + 288)*JJ(0,q - 1)\n + ((12*q^3 - 72*q^2 + 132*q - 72)*muB^4\n   + (24*q^3 - 144*q^2 + 264*q - 144)*muB^2)*JJ(0,q))\n \/ (24*q^3 - 144*q^2 + 264*q - 144);\n\\end{verbatim}\n\n\n\n{\\Large \\bf Appendix 9.}\n\n\\begin{verbatim}\nJ(p,q) = ( 24*J(p-4,q+4)*(q^3 + 6*q^2 + 11*q + 6) \n+ 2*J(p-3,q+2)*( - 6*q^3 + 21*q^2*(p-3) - 30*q^2\n   + 12*q*(p-1)*(p-3) + 8*q*(p-2)*(p-3) + 5*q*(p-3)\n   - 48*q + 12*(p-1)*(p-3) + 8*(p-2)*(p-3) - 16*(p-3) - 24) \n+ 24*J(p-3,q+3)*(5*q^2*(p-3) - 2*q^2 + 15*q*(p-3) - 6*q + 10*(p-3) - 4) \n+ J(p-2,q)*(p-3)*( - 21*q^2 - 12*q*(p-1) + 4*q*(p-2)\n    + 4*q + 10*(p-1)*(p-2) - 30*(p-2)) \n+ 4*J(p-2,q+1)*(p-3)*( - 15*q^2 + 20*q*(p-2) - 39*q\n    + 30*(p-1)*(p-2) - 12*(p-1) - 38*(p-2) + 1) \n+ 24*J(p-2,q+2)*(p-3)*(11*q*(p-2) - 8*q + 11*(p-2) - 8) \n+ 6*J(p-1,q-2)*(p-2)*(p-3)*( - q - (p-1) + 2) \n+ 4*J(p-1,q-1)*(p-2)*(p-3)*( - 10*q - 10*(p-1) + 13) \n+ 2*J(p-1,q)*(p-2)*(p-3)*( - 66*q - 73*(p-1) + 40) \n+ 24*J(p-1,q+1)*(p-2)*(p-3)*(15*(p-1) - 14) \n+ (3*J(p,q-2) + 20*J(p,q-1))*(p-1)*(p-2)*(p-3) \n+ 6*D(p-3,q+3,1)*(4*q^3 - 7*q^2*(p-3) + 22*q^2\n    - 21*q*(p-3) + 38*q - 14*(p-3) + 20) \n+ 24*D(p-3,q+4,1)*( - q^3 - 6*q^2 - 11*q - 6) \n+ D(p-2,q+1,1)*(p-3)*(63*q^2 + 24*q*(p-1) - 8*q*(p-2)\n    + 55*q - 10*(p-1)*(p-2) + 12*(p-1) + 26*(p-2) + 17) \n+ 2*D(p-2,q+2,1)*(p-3)*(39*q^2-12*q*(p-1) - 68*q*(p-2)\n    + 163*q - 12*(p-1) - 68*(p-2) + 124) \n+ 120*D(p-2,q+3,1)*(p-3)*( - q^2 - 3*q - 2) \n+ 6*D(p-1,q-1,1)*(p-2)*(p-3)*(4*q + 3*(p-1) - 5) \n+ 2*D(p-1,q,1)*(p-2)*(p-3)*(64*q + 30*(p-1) - 17) \n+ 2*D(p-1,q+1,1)*(p-2)*(p-3)*(92*q - 47*(p-1) + 126) \n- 264*D(p-1,q+2,1)*(p-2)*(p-3)*(q + 1) \n+ (6*D(p,q-2,1) + 31*D(p,q-1,1) + 106*D(p,q,1)\n    - 168*D(p,q+1,1))*(p-1)*(p-2)*(p-3) \n- 12*D(p-3,q+4,2)*(q + 1)*(q + 2)*(q + 3)\n+ D(p-2,q+2,2)*(p-3)*(-63*q^2 - 12*q*(p-1) + 4*q*(p-2)\n    - 122*q - 12*(p-1) + 4*(p-2) - 59) \n+ 18*D(p-2,q+3,2)*(p-3)*( - q^2 - 3*q - 2) \n+ 18*D(p-1,q,2)*(p-2)*(p-3)*( - 2*q - (p-1) + 1) \n+ 2*D(p-1,q+1,2)*(p-2)*(p-3)*( - 68*q - 10*(p-1) - 33) \n- 12*D(p-1,q+2,2)*(p-2)*(p-3)*(q + 1) \n- (18*D(p,q-1,2) + 61*D(p,q,2) + 6*D(p,q+1,2))*(p-1)*(p-2)*(p-3) \n+ 3*(p-3)*(  7*D(p - 2,q + 3,3)*(q + 1)*(q + 2)\n- 2*D(p - 1,q + 2,4)*(q+1)*(p-2) \n+ 16*D(p - 1,q + 2,3)*(q+1)*(p-2) \n+ 2*D(p - 1,q + 1,3)*(4*q+p)*(p-2) \n+(- 2*D(p,q + 1,4) + 9*D(p,q + 1,3)\n    + 6*D(p,q,3))*(p-1)*(p-2)\n))\/192\/(p-1)\/(p-2)\/(p-3);\n\\end{verbatim}\n\n{\\Large \\bf Appendix 10.}\n\\begin{eqnarray}\\hspace*{-12 mm}\n&&  J(4,-1) = - 523385\/2976768 \\; Y_{11} - 104677\/91392 \\; Y_{10} - 17130665\/26790912 \\; Y_9 \\\\ \\nonumber && \n + 523385\/459648 \\; Y_8 + 11014687\/5515776 \\; Y_7 - 104677\/31256064 \\; Y_5 \\\\ \\nonumber && \n - 53544991\/62512128 \\; Y_4-3 \\; Y_0 + 11316787\/7354368 \\;  (2\\pi)^{-2}-3\/4 \\; F_0 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(5,-1) =    881852823875\/3876752130048 \\; Y_{11} + 176370564775\/119023091712 \\; Y_{10} \\\\ \\nonumber && \n + 28894631548079\/34890769170432 \\; Y_9 - 881852823875\/598616137728 \\; Y_8 \\\\ \\nonumber && \n - 17115981580297\/7183393652736 \\; Y_7 + 176370564775\/40705897365504 \\; Y_5 \\\\ \\nonumber && \n + 86132421399433\/81411794731008 \\; Y_4 + 25\/8 \\; Y_0  \\\\ \\nonumber && \n + (25\/32\\; F_0 - 20532421134805\/9577858203648 ) (2\\pi)^{-2};\\\\ \\nonumber && \n  J(6,-1) =  - 28542210469686553\/93496923371077632 \\; Y_{11} \\\\ \\nonumber && \n - 28542210469686553\/14352597885911040 \\; Y_{10} - 2087037111021330529\/4207361551698493440 \\; Y_9 \\\\ \\nonumber && \n + 28542210469686553\/14437024932298752 \\; Y_8 + 2217244724616422759\/866221495937925120 \\; Y_7 \\\\ \\nonumber && \n - 28542210469686553\/4908588476981575680 \\; Y_5 - 12694496481519981287\/9817176953963151360 \\; Y_4  \\\\ \\nonumber && \n - 3163\/960 \\; Y_0 + (3108094517023801819\/1154961994583900160 - 3163\/3840 \\; F_0 ) (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(7,-1) = 318815426788929230845117\/821909762068921372901376 \\; Y_{11} \\\\ \\nonumber && \n + 318815426788929230845117\/126170358212334421278720 \\; Y_{10} \\\\ \\nonumber &&\n + 11542347686773160604861797\/36985939293101461780561920 \\; Y_9 \\\\ \\nonumber && \n - 318815426788929230845117\/126912536790054035521536 \\; Y_8 \\\\ \\nonumber && \n - 22134243295873177135745923\/7614752207403242131292160 \\; Y_7 \\\\ \\nonumber && \n + 318815426788929230845117\/43150262508618372077322240 \\; Y_5 \\\\ \\nonumber && \n + 135786390066815073864737539\/86300525017236744154644480 \\; Y_4 + 14099\/3840 \\; Y_0 \\\\ \\nonumber && \n - 34028993605943331376331719\/10153002943204322841722880 \\;  (2\\pi)^{-2} + 14099\/15360 \\; F_0 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(8,-1) = - 9235026172226013567109728738337\/18732046758141338739230512250880 \\; Y_{11} \\\\ \\nonumber && \n - 9235026172226013567109728738337\/2875533493574328315232754073600 \\; Y_{10} \\\\ \\nonumber && \n - 55876713828271512392254716439529\/842942104116360243265373051289600 \\; Y_9 \\\\ \\nonumber && \n + 9235026172226013567109728738337\/2892448396477706717087064391680 \\; Y_8 \\\\ \\nonumber && \n + 582625738707581471281711807374367\/173546903788662403025223863500800 \\; Y_7 \\\\ \\nonumber && \n - 9235026172226013567109728738337\/983432454802420283809601893171200 \\; Y_5 \\\\ \\nonumber && \n - 3802005063193739791001963690889631\/1966864909604840567619203786342400 \\; Y_4 \\\\ \\nonumber && \n - 150949\/35840 \\; Y_0 + 965476530505339393064578836261427\/ \\\\ \\nonumber && \n   231395871718216537366965151334400 \\;  (2\\pi)^{-2} - 150949\/143360 \\; F_0 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(9,-1) =  \\\\ \\nonumber && \n   10189310117043029375946591507272718551\/16263612564540505685529672268508037120 \\; Y_{11} \\\\ \\nonumber && \n + 10189310117043029375946591507272718551\/2496607191925077627164642672797286400 \\; Y_{10} \\\\ \\nonumber && \n - 150087551903588632763072238483785668913\/731862565404322755848835252082861670400 \\; Y_9 \\\\ \\nonumber && \n - 10189310117043029375946591507272718551\/2511293116583460436736199394401976320 \\; Y_8 \\\\ \\nonumber && \n - 598626882055818268392045217522764699881\/150677586995007626204171963664118579200 \\; Y_7 \\\\ \\nonumber && \n + 10189310117043029375946591507272718551\/853839659638376548490307794096671948800 \\; Y_5 \\\\ \\nonumber && \n + 4095937462034066852819132682453485174633\/1707679319276753096980615588193343897600 \\; Y_4 \\\\ \\nonumber && \n + 8535263\/1720320 \\; Y_0 + 8535263\/6881280 \\; F_0 \\;  (2\\pi)^{-2} \\\\ \\nonumber && \n - 1051450108870580974979233285323814292021\/200903449326676834938895951552158105600 \\; (2\\pi)^{-2} ; \\\\ \\nonumber \n&& J(4,-2) =  775\/124032 \\; Y_{11} + 155\/3808 \\; Y_{10} + 2315881\/1116288 \\; Y_9 - 775\/19152 \\; Y_8 \\\\ \\nonumber && \n - 535775\/229824 \\; Y_7 + 155\/1302336 \\; Y_5 + 1268159\/2604672 \\; Y_4 + 4 \\; Y_0 \\\\ \\nonumber && \n - 256019\/306432 \\;  (2\\pi)^{-2} + \\; F_0 \\;  (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(5,-2) = - 35198207035\/161531338752 \\; Y_{11} - 7039641407\/4959295488 \\; Y_{10} \\\\ \\nonumber && \n - 1590044736319\/1453782048768 \\; Y_9 + 35198207035\/24942339072 \\; Y_8 \\\\ \\nonumber && \n + 824164813817\/299308068864 \\; Y_7 - 7039641407\/1696079056896 \\; Y_5 \\\\ \\nonumber && \n - 3786323755001\/3392158113792 \\; Y_4 - 4 \\; Y_0 + (886011822725\/399077425152 - \\; F_0 )\\; (2\\pi)^{-2}; \\\\ \\nonumber && \n  J(6,-2) = 11638328158211917\/35061346264154112 \\; Y_{11} \\\\ \\nonumber && \n + 11638328158211917\/5382224207216640 \\; Y_{10} + 1835049614849615189\/1577760581886935040 \\; Y_9 \\\\ \\nonumber && \n - 11638328158211917\/5413884349612032 \\; Y_8 - 1118098215139974931\/324833060976721920 \\; Y_7 \\\\ \\nonumber && \n + 11638328158211917\/1840720678868090880 \\; Y_5 + 5661105033256044691\/3681441357736181760 \\; Y_4 \\\\ \\nonumber && \n + 181\/40 \\; Y_0 + (181\/160 \\; F_0 - 1412418713096881687\/433110747968962560 )\\; (2\\pi)^{-2} ; \\\\ \\nonumber && \n  J(7,-2) = - 16095494538985786829003\/34246240086205057204224 \\; Y_{11} \\\\ \\nonumber && \n - 16095494538985786829003\/5257098258847267553280 \\; Y_{10} \\\\ \\nonumber && \n - 1220878670422477200084451\/1541080803879227574190080 \\; Y_9 \\\\ \\nonumber && \n + 16095494538985786829003\/5288022366252251480064 \\; Y_8 \\\\ \\nonumber && \n + 1258484886934090324074869\/317281341975135088803840 \\; Y_7 \\\\ \\nonumber && \n - 16095494538985786829003\/1797927604525765503221760 \\; Y_5 \\\\ \\nonumber && \n - 1435246543741057501042993\/719171041810306201288704 \\; Y_4 - 2447\/480 \\; Y_0  \\\\ \\nonumber &&  \n + (1817967522495901075073489\/423041789300180118405120 - 2447\/1920 \\; F_0 )\\;(2\\pi)^{-2}; \\\\ \\nonumber && \n  J(8,-2) = 97496623093573646692443907729\/156100389651177822826920935424 \\; Y_{11} \\\\ \\nonumber && \n + 97496623093573646692443907729\/23962779113119402626939617280 \\; Y_{10} \\\\ \\nonumber && \n + 703823889446906512753183459525\/1404903506860600405442288418816 \\; Y_9 \\\\ \\nonumber && \n - 97496623093573646692443907729\/24103736637314222642392203264 \\; Y_8 \\\\ \\nonumber && \n - 6768519700158561511510642407343\/1446224198238853358543532195840 \\; Y_7 \\\\ \\nonumber && \n + 97496623093573646692443907729\/8195270456686835698413349109760 \\; Y_5 \\\\ \\nonumber && \n + 41525287801127597648514834574639\/16390540913373671396826698219520 \\; Y_4 \\\\ \\nonumber && \n + 79489\/13440 \\; Y_0 + 79489\/53760 \\; F_0 \\;  (2\\pi)^{-2} \\\\ \\nonumber && \n - 2132153450344177531160317231847\/385659786197027562278275252224 \\;  (2\\pi)^{-2} ; \\\\ \\nonumber && \n  J(9,-2) = - 184720162027043790012174961950856949\/225883507840840356743467670395944960 \\; Y_{11} \\\\ \\nonumber && \n - 184720162027043790012174961950856949\/34675099887848300377286703788851200 \\; Y_{10} \\\\ \\nonumber && \n - 1182390536568106894749732117915566173\/10164757852837816053456045167817523200 \\; Y_9 \\\\ \\nonumber && \n + 184720162027043790012174961950856949\/34879071063659172732447213811138560 \\; Y_8 \\\\ \\nonumber && \n + 11666341875097592955381754264025444939\/2092744263819550363946832828668313600 \\; Y_7 \\\\ \\nonumber && \n - 184720162027043790012174961950856949\/11858884161644118729032052695787110400 \\; Y_5 \\\\ \\nonumber && \n - 76075734513749258577040135726310510027\/23717768323288237458064105391574220800 \\; Y_4 \\\\ \\nonumber && \n - 501267\/71680 \\; Y_0 - 501267\/286720 \\; F_0 \\;  (2\\pi)^{-2} \\\\ \\nonumber && \n + 19693107617724772864308858550443517679\/2790325685092733818595777104891084800 \\; (2\\pi)^{-2}; \\nonumber \n\\end{eqnarray}\n\\newpage\n\n{\\Large \\bf Appendix 11.}\n\\begin{verbatim}\nJ(p,q) = -(  6*D(p - 1,q + 4,4)*( - p^2*q - 3*p^2 + 5*p*q + 15*p - 6*q - 18) \n+ 6*D(p,q + 3,4)*( - p^3 + 6*p^2 - 11*p + 6)\n + 21*D(p-2,q+5,3)*(p*q^2 + 7*p*q + 12*p - 3*q^2 - 21*q - 36) \n + 48*D(p - 1,q + 4,3)*(p^2*q + 3*p^2 - 5*p*q - 15*p + 6*q + 18) \n + 6*D(p-1,q+3,3)*(p-2)*(p-3)*(p+4*q+8) \n + 27*D(p,q + 3,3)*(p^3 - 6*p^2 + 11*p - 6)\n + 18*D(p,q+ 2,3)*(p^3 - 6*p^2 + 11*p - 6) \n + 12*D(p-3,q+6,2)*( - q^3 - 12*q^2 - 47*q - 60) \n + 18*D(p-2,q+5,2)*( - p*q^2 - 7*p*q - 12*p + 3*q^2 + 21*q + 36) \n + D(p - 2,q + 4,2)*( - 8*p^2*q - 24*p^2\n    - 63*p*q^2 - 346*p*q - 471*p + 189*q^2 + 1110*q + 1629) \n + 12*D(p-1,q+4,2)*( - p^2*q - 3*p^2 + 5*p*q + 15*p - 6*q - 18) \n + 2*D(p-1,q+3,2)*( - 10*p^3 - 68*p^2*q - 109*p^2\n    + 340*p*q + 735*p - 408*q - 954) \n + 6*D(p,q + 3,2)*( - p^3 + 6*p^2 - 11*p + 6) \n + 18*D(p - 1,q + 2,2)*( - p^3 - 2*p^2*q + 3*p^2 + 10*p*q + 4*p - 12*q - 12) \n + 61*D(p,q + 2,2)*( - p^3 + 6*p^2 - 11*p + 6) \n + 18*D(p,q + 1,2)*( - p^3 + 6*p^2 - 11*p + 6) \n + 24*D(p-3,q+6,1)*( - q^3 - 12*q^2 - 47*q - 60) \n + 6*D(p-3,q+5,1)*( - 7*p*q^2 - 49*p*q - 84*p\n + 4*q^3 + 67*q^2 + 321*q + 468) \n + 120*D(p - 2,q + 5,1)*( - p*q^2 - 7*p*q - 12*p + 3*q^2 + 21*q + 36) \n + 2*D(p - 2,q + 4,1)*( - 80*p^2*q - 240*p^2\n    + 39*p*q^2 + 707*p*q + 1770*p - 117*q^2 - 1401*q - 3150) \n + 264*D(p - 1,q + 4,1)*( - p^2*q - 3*p^2 + 5*p*q + 15*p - 6*q - 18) \n + D(p - 2,q + 3,1)*( - 10*p^3 + 16*p^2*q + 130*p^2\n    + 63*p*q^2 + 251*p*q - 21*p - 189*q^2 - 897*q - 837) \n + 2*D(p - 1,q + 3,1)*( - 47*p^3 + 92*p^2*q\n    + 592*p^2 - 460*p*q - 2067*p + 552*q + 2142) \n + 168*D(p,q + 3,1)*( - p^3 + 6*p^2 - 11*p + 6) \n + 2*D(p - 1,q + 2,1)*(30*p^3 + 64*p^2*q - 69*p^2\n    - 320*p*q - 225*p + 384*q + 486) \n + 106*D(p,q + 2,1)*(p^3 - 6*p^2 + 11*p - 6) \n + 6*D(p - 1,q + 1,1)*(3*p^3 + 4*p^2*q - 15*p^2 - 20*p*q + 18*p + 24*q) \n + 31*D(p,q + 1,1)*(p^3 - 6*p^2 + 11*p - 6) \n + 6*D(p,q,1)*(p^3 - 6*p^2 + 11*p - 6) \n + 24*J(p - 4,q + 6)*(q^3 + 12*q^2 + 47*q + 60) \n + 24*J(p - 3,q + 5)*(5*p*q^2 + 35*p*q + 60*p - 17*q^2 - 119*q - 204) \n + 2*J(p - 3,q + 4)*(20*p^2*q + 60*p^2 + 21*p*q^2\n    + p*q - 186*p - 6*q^3 - 129*q^2 - 423*q - 270) \n + 24*J(p - 2,q + 4)*(11*p^2*q + 33*p^2 - 63*p*q - 189*p + 90*q + 270) \n + 4*J(p - 2,q + 3)*(30*p^3 + 20*p^2*q - 190*p^2\n    - 15*p*q^2 - 199*p*q + 231*p + 45*q^2 + 417*q + 207) \n + 24*J(p - 1,q + 3)*(15*p^3 - 104*p^2 + 235*p - 174) \n + J(p - 2,q + 2)*(10*p^3 - 8*p^2*q - 106*p^2\n    - 21*p*q^2 - 52*p*q + 240*p+63*q^2+228*q-36)\n + 2*J(p - 1,q + 2)*( - 73*p^3 - 66*p^2*q + 346*p^2\n    + 330*p*q - 343*p - 396*q - 114) \n + 192*J(p,q + 2)*( - p^3 + 6*p^2 - 11*p + 6) \n + 4*J(p-1,q+1)*( - 10*p^3 - 10*p^2*q + 53*p^2 + 50*p*q - 75*p - 60*q + 18) \n + 20*J(p,q+1)*(p^3 - 6*p^2 + 11*p - 6) \n + 6*J(p-1,q)*( - p^3 - p^2*q + 6*p^2 + 5*p*q - 11*p - 6*q + 6) \n )\/3\/(p-1)\/(p-2)\/(p-3);\n\\end{verbatim}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\nThe solar atmosphere contains a wide variety of chromospheric ejections that cover a large range\nof scales: from the smallest ones with maximum size of a few megameters, such as penumbral \nmicrojets \\citep[e.g.,][]{Katsukawa:2007wk, Drews:2017}, or spicules \\citep[][among others]\n{Hansteen+DePontieu2006, de-Pontieu:2007kl, Pereira:2012dz}; up to ejections that can reach, \nin extreme cases, several tens of megameters, like surges \\citep[e.g.,][]{canfield1996, Kurokawa2007, \nGuglielmino:2010lr,YangH:2014} and macrospicules \\citep{Bohlin1975, Georgakilas, Murawski2011, \nKayshap2013}. Surges, in particular, are often associated with magnetic flux emergence from the solar \ninterior. They are typically observed as darkenings in images taken in the {H$\\alpha$}\\ blue\/red wings with line-of-sight \n(LOS) velocities of a few to several tens of km s$^{-1}$, and are usually related to other explosive \nphenomena like EUV and X-ray jets, UV bursts and Ellerman bombs \\citep[see]\n[hereafter NS2017, and references therein]{Nobrega-Siverio:2017a}. Although \nobservationally known for several decades now, the understanding of surges has progressed \nslowly and various aspects like, e.g., their impact on\n  the transition region (TR) and corona concerning the mass and energy budget, are still poorly known.\n\nFrom the theoretical point of view, the first explanation of the surge phenomenon came through 2.5D \nnumerical models \\citep{Shibata1992a,Yokoyama:1995uq,Yokoyama:1996kx}, where a cold ejection was \nidentified next to a hot jet as a consequence of a magnetic reconnection process between the magnetic \nfield in plasma emerged from the interior and the preexisting coronal field. \\cite{Nishizuka:2008zl} used \na similar numerical setup to associate the surge with jet-like features seen in {\\ion{Ca}{2}}\\ H+K observations by \nmeans of morphological image comparisons. Further 2.5D models that include\nthe formation of a cool chromospheric ejection are those of \\cite{jiang2012}\n(canopy-type coronal magnetic field), and \\cite{YangL:2013, YangL:2018}, who\nstudy the cool jets resulting from the interaction between moving magnetic features at\nthe base of their experiment and the preexisting ambient field in the\natmosphere. Turning to three dimensional models, in the magnetic flux emergence experiment of\n\\cite{Moreno-Insertis:2013aa}, a dense wall-like surge appeared surrounding the emerged \nregion with temperatures from $10^4$~K to a few times $10^5$~K and speeds around $50$ km s$^{-1}$. \n\\cite{MacTaggart2015} found similar velocities for the surges in their 3D model of flux emergence in \nsmall-scale active regions. The availability of a radiation-MHD code like Bifrost \n\\citep{Gudiksen:2011qy} has opened up the\npossibility of much more detailed modeling of the cool ejections than\nbefore. Bifrost has a realistic treatment of the material properties of the plasma,\ncalculates the radiative transfer in the photosphere and chromosphere \nand includes the radiative and heat conduction entropy sources in the corona. Using that code, \\cite{Nobrega-Siverio:2016}, hereafter NS2016,\nargued that entropy sources play an important  \nrole during the surge formation and showed that a relevant fraction of the surge could not be obtained in \nprevious and more idealized experiments. \n\n\nThe realistic treatment of surges may require an even larger degree\n  of complication. The solar atmosphere is a highly dynamical environment;\nthe evolution sometimes occurs on short timescales that bring different\natomic species out of equilibrium ionization, thus complicating both the\nmodeling and the observational diagnostics\n\\citep[e.g.,][]{Griem:1964,Raymond:1978,Joselyn:1979, Hansteen:1993}.  For\nhydrogen, for instance, using 2D numerical experiments,\n\\cite{Leenaarts:2007sf, Leenaarts:2011qy} found that the temperature\nvariations in the chromosphere can be much larger than for statistical\nequilibrium (SE), which has an impact on, e.g., its coolest regions (the\nso-called cool pockets). For helium, \\citet{golding2014,golding:2016}\ndescribed how nonequilibrium (NEQ) ionization leads to higher temperatures\nin wavefronts and lower temperatures in the gas between shocks. For heavy\nelements, \\cite{Bradshaw:2003,Bradshaw:2006, Bradshaw:2011, Reep:2017}\nshowed, through 1D hydrodynamic simulations, that there are large departures\nfrom SE balance in cooling coronal loops, nanoflares and other impulsive\nheating events that affects the EUV emissivity. Through 3D\n  experiments, \\cite{Olluri:2013fu} found that deduced electron densities for {\\ion{O}{4}}\\ can be up to\nan order of magnitude higher when NEQ effects are taken into\naccount.  Also in 3D, \\cite{olluri:2015} discussed the importance of the NEQ ionization of \ncoronal and TR lines to reproduce \nabsolute intensities, line widths, ratios, among others, \nobserved by, e.g., \\cite{Chae:1998, Doschek:2006, Doschek:2008qy}. \n\\cite{De-Pontieu2015} were able to explain the correlation between\nnon-thermal line broadening and intensity of TR lines only\nwhen including NEQ ionization in their 2.5D numerical\nexperiments. \\cite{Martinez-Sykora:2016obs} studied the statistical\nproperties of the ionization of silicon and oxygen in different solar\ncontexts: quiet Sun, coronal hole, plage, quiescent active region, and\nflaring active region, finding similarities with the observed intensity\nratios only if NEQ effects are taken into account. Given their highly\ntime-dependent nature and the relevance of the heating and cooling mechanisms\nin their evolution, surges are likely to be affected by NEQ ionization. \nMotivated by this fact, NS2017 included the NEQ\nionization of silicon to compare synthetic {\\ion{Si}{4}}\\ spectra of two 2.5D\nnumerical experiments with surge observations obtained by \nthe \\textit{Interface Region Imaging Spectrograph}\n\\citep[\\textit{IRIS},][]{De-Pontieu:2014vn} and the Swedish 1-m Solar Telescope\n\\citep[SST,][]{Scharmer:2003ve}. The results showed that the experiments were\nable to reproduce major features of the observed surge; nonetheless, the\ntheoretical aspects to understand the enhanced {\\ion{Si}{4}}\\ emissivity within the\nnumerical surge and its properties were not addressed in that publication. \n\n\n\n\n\\begin{figure*}\n\\epsscale{1.18}\n\\plotone{figure1.pdf}\n\\caption{\nLeft: Horizontal averages for the initial stratification of $\\rho$, $P_{g}$, and $T$ normalized to their \nphotospheric values at $z = 0$~Mm, namely, $\\rho_{ph} = 3.1\\times10^{-7}$~g cm$^{-3}$,\n$P_{g_{ph}} = 1.1 \\times 10^{5}$~erg cm$^{-3}$ and $T_{ph}= 5.7\\times10^3$~ K. Solid black line represents the stratification for the vertical coronal field experiment; the red dotted, for the slanted one. The horizontal and vertical dotted lines mark the reference normalization values at $z=0$ Mm. Right: 2D maps for the initial temperature with magnetic field lines in black for the vertical experiment (top) and slanted experiment (bottom). The maps only show temperatures below $2\\times10^4$ K (although the range varies from 1660 K up to $\\sim$ 1 MK) and heights between $-2.6 \\leq z \\leq 8.0$~Mm (the top of the domain reaches $z=30$ Mm). The solar surface is roughly at $z=0$ Mm (white dashed horizontal line). \\label{figure1}}\n\\end{figure*}  \n\nThe aim of the present paper is to provide theoretical explanations\nconcerning the relevance of the NEQ ionization for surges and the\ncorresponding impact on the emissivity of TR lines. We use\n2.5D numerical experiments carried out with the Bifrost code\n\\citep{Gudiksen:2011qy} including the module developed by \\cite{olluri:2013aa}\nthat solves the time-dependent rate equations to calculate the ionization\nstates of different elements, thus allowing for departures from SE. Here we apply this module to determine the\nionization levels of silicon and oxygen. We conclude that consideration of \nNEQ is necessary to get the proper population levels of the ions and, consequently,\nthe right emissivity to interpret observations. A statistical\nanalysis of temperature is provided to constrain the plasma properties\ninvolved in the emissivity of relevant lines of {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ within the\nsurges.  Through detailed Lagrange tracing, we are able to determine the\norigin of the emitting plasma and the role of the optically thin radiation\nand thermal conduction to explain the departure of SE of the relevant\nions. Furthermore, we compute synthetic profiles to understand previous\nobservational results and predict future ones,\nhighlighting the surge regions that are more likely to be detected\nand addressing the importance of the angle of LOS.\n\nThe layout of the paper is as follows. Section \\ref{sec:2} describes the physical and numerical models. \nSection \\ref{sec:3} explains the general features of the time evolution of the experiments. In Section \n\\ref{sec:4}, we show the main results of the paper splitting the section in a) the relevance of the NEQ \nionization of {\\ion{Si}{4}}\\,, and also {\\ion{O}{4}}, in surges (Section \\ref{sec:4.1}); b) the consequences of the NEQ \nionization for the surge plasma emitting in those TR lines, analyzing its properties and \ncompare them with a generic quiet TR (Section \\ref{sec:4.2}); and c) the origin of the NEQ plasma, \naddressing the role of the entropy sources (Section \\ref{sec:4.3}). In Section\n\\ref{sec:5}, we have calculated absolute intensities and synthetic spectral \nprofiles for diagnostic purposes and comparison with observations, emphasizing also the importance of \nthe surge geometry and LOS. Finally, Section \\ref{sec:6} contains a summary and conclusions.\n\n\n\\Needspace{5\\baselineskip}\n\\section{The physical and numerical model}\\label{sec:2}\nWe have run two 2.5D numerical flux emergence experiments in which surges are a natural \nconsequence of magnetic reconnection processes. Those two experiments \nwere also used by NS2017 and \\cite{Rouppe2017} to compare the synthetic profiles with \nthe complex profiles observed with \\textit{IRIS}\\ and SST.\n\nThis section is divided into two parts: (1) the numerical code, and (2) the description of the \nmodel underlying our experiments.\n\n\\Needspace{5\\baselineskip}\n\\subsection{The numerical code}\\label{sec:2.1}\nThe two experiments have been carried out with the 3D radiation-MHD (R-MHD)\nBifrost code \\citep{Gudiksen:2011qy, Carlsson:2012uq, Hayek:2010ac}, which treats \nthe radiative transfer from the photosphere to the corona and thermal conduction \nin a self-consistent manner (see also NS2016 for further details of this\ncode applied to surge experiments). Furthermore, we have enabled in the code a module \ndeveloped by \\citep{olluri:2013aa} to follow the NEQ ionization \nstates of elements with atomic \nnumber greater than 2. This module solves the rate equations for those elements using \nthe temperature, mass density, electronic number density $n_e$ and velocity values of the simulation without modifying the results of the R-MHD calculation, so there is no feedback, e.g., on the energy equation terms such as the optically thin losses (see the discussion in Section \\ref{sec:6.1}). In particular, we have employed it to calculate the NEQ ionization fraction for silicon and oxygen, using abundances from \\cite{Asplund:2009}, 7.52 and 8.69, respectively, in the customary astronomical scale\nwhere 12 corresponds to hydrogen. \n\n\\Needspace{5\\baselineskip}\n\\subsection{Description of the models}\\label{sec:2.2}\n\n\\Needspace{5\\baselineskip}\n\\subsubsection{Physical domain and initial condition}\n\nIn the two experiments, we began with a statistically stationary 2D snapshot that spans\n from the uppermost layers of the solar interior to the corona, and whose physical \n domain is $0.0$~Mm $\\leq x \\leq$ $32.0$~Mm and $-2.6$~Mm $\\leq z \\leq$ \n $30.0$~Mm, where $z=0$~Mm corresponds to the solar surface. The grid is \n uniform in the $x$-direction with $\\Delta x=31$~km, but it is nonuniform in the \n vertical direction in order to better resolve the photosphere and chromosphere:\n the vertical grid spacing is $20$ km  from the photosphere to the transition\n region, and increases gradually in the corona up to $147$ km at the top of the domain.\n\nThe left panel in Figure~\\ref{figure1} contains the horizontal averages for the initial \ndensity, $\\rho$, gas pressure, $P_g$, and temperature, $T$, for both experiments\nnormalized to photospheric values, namely, $\\rho_{ph} = 3.1\\times10^{-7}$~g cm$^{-3}$,\n$P_{g_{ph}} = 1.1 \\times 10^{5}$~erg cm$^{-3}$ and $T_{ph}= 5.7\\times10^3$~ K. \nThe corona has a temperature around $1$ MK and a magnetic field with a strength \nof $10$ G, with the difference that one of the experiments \n(hereafter \\textit{the vertical experiment}) has a vertical \nmagnetic field in the corona \nwhile in the other (\\textit{the slanted experiment}), the magnetic field in the corona is inclined \n$30$\\degree\\ with respect to the vertical direction (see magnetic field lines superimposed in black \nin the 2D temperature maps for the initial snapshot in Figure~\\ref{figure1}). \n\n\n\\Needspace{5\\baselineskip}\n\\subsubsection{Chemical elements calculated in NEQ and their spectral lines}\n\nWe have used the NEQ module of\n  \\cite{olluri:2013aa}  mentioned in the \\nameref{sec:introduction}  \nto compute the nonequilibrium ionization of silicon \nin both numerical experiments. Furthermore, in the vertical experiment we also \ncalculate the NEQ ionization of oxygen, with the goal of predicting \nfuture observational results.  Once the NEQ populations are obtained, we are able \nto compute the emissivity using\n\\begin{eqnarray}\n\t\\epsilon_{\\lambda} & = &   \\frac{h\\, c}{4\\, \\pi\\, \\lambda}\\, n_u\\, A_{ul},\n \\label{eq:emissivity}\n\\end{eqnarray}\nwhere $h$ is the Planck's constant, $c$ the light speed, $\\lambda$ is the\nwavelength of the spectral line, $n_u$ the population density of the upper\nlevel of the transition (i.e., the number density of emitters), and $A_{ul}$ the Einstein coefficient for\nspontaneous de-excitation given by\n\\begin{eqnarray}\n\tA_{ul} & = &   \\frac{8\\, \\pi^2\\, e^2\\, }{m_e\\, c} \\frac{1}{\\lambda^2}\\, \\frac{g_l}{g_u}\\, f_{lu},\n\\label{eq:einstein}\n\\end{eqnarray}\nwhere $e$ is the electron charge, $m_e$ the electron mass, $g_l$ and $g_u$ the statistical \nweights of the lower and upper states respectively, and $f_{lu}$ the oscillator strength. \nThe units used in this paper for the emissivity $\\epsilon$ are erg \ncm$^{-3}$ sr$^{-1}$ s$^{-1}$. For the sake of compactness, we will refer to it in the following \nas $\\epsilon_{_{CGS}}$.\n\nSince we are interested in understanding the response of the TR to \nchromospheric phenomena like surges, we have chosen the following \\textit{IRIS}\\ lines: \n{\\ion{Si}{4}}\\ 1402.77 $\\textup{\\AA}$, which is the weakest of the two silicon resonance \nlines; and {\\ion{O}{4}}\\ 1401.16 $\\textup{\\AA}$, the strongest of the forbidden oxygen lines \nthat \\textit{IRIS}\\ is able to observe. The corresponding formation temperature peaks in \nstatistical equilibrium, $T_{_{SE}}$, and other relevant parameters to calculate\nthe Einstein coefficient (Equation \\ref{eq:einstein}) and the corresponding emissivity \n(Equation \\ref{eq:emissivity}) of these lines are shown in Table \\ref{table1}.\nUnder optically thin conditions, {\\ion{Si}{4}}\\ 1393.76 \n$\\textup{\\AA}$ is twice stronger than 1402.77 $\\textup{\\AA}$, so the results we obtain in\nthis paper can also be applied to {\\ion{Si}{4}}\\ 1393.76 $\\textup{\\AA}$. Furthermore, the study of\n{\\ion{Si}{4}}\\ 1402.77 $\\textup{\\AA}$ can provide theoretical support to our previous paper NS2017. \nIn turn, the choice of the 1401.16 $\\textup{\\AA}$ line for\noxygen is because the {\\ion{O}{4}}\\ lines \nare faint and require longer exposure times to be observed \\citep{De-Pontieu:2014vn}. \nThus, in order to make any prediction that could be corroborated in future \\textit{IRIS}\\ \nanalysis, we focus on the strongest of the oxygen lines, which has a better chance \nto be detected. For simplicity, hereafter we refer to the {\\ion{Si}{4}}\\ 1402.77 $\\textup{\\AA}$ \nand {\\ion{O}{4}}\\ 1401.16 $\\textup{\\AA}$ emissivities as the {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivities, \nrespectively.\n\n\n\\setcounter{table}{0}\n\\begin{deluxetable}{c|ccccc}\n\\caption{Relevant parameters for the studied emission lines}\\label{table1}\n\\tablehead{\n\\colhead{Line} & \\colhead{$T_{_{SE}}$ (K)} & \\colhead{$g_u$} & \\colhead{$g_l$} & \\colhead{$f_{lu}$} & \\colhead{$n_u\/\\epsilon_{\\lambda}$}}\n\\startdata\n{\\ion{Si}{4}}\\ 1402.77 \\AA & $10^{4.9}$ & $2$ & $2$ & $2.7\\times10^{-1}$ & $974$\\\\\n\\hline\n{\\ion{O}{4}}\\ 1401.16  \\AA & $10^{5.2}$ & $6$ & $4$ & $5.1\\times10^{-7}$ & $7.62\\times10^{8}$ \\\\\n\\enddata\n\\end{deluxetable}\n\n\n\\Needspace{5\\baselineskip} \n\\subsubsection{Boundary conditions}\nWe are imposing periodicity at the side boundaries; \nfor the vertical direction, characteristic conditions are implemented at the\ntop, whereas an open boundary is maintained at the bottom keeping a fixed value\nof the entropy of the incoming plasma. Additionally, in order to\nproduce flux emergence, we inject a twisted magnetic tube through the bottom\nboundary following the method described by \\cite{Martinez-Sykora:2008aa}. \nThe parameters of the tube (specifically, the initial location of\nthe axis, $x_0$ and $z_0$; the field strength there, $B_0$; the tube radius\n$R_0$; and the amount of field line twist $q$) are identical in both\nexperiments and given in Table \\ref{table2}. The total axial magnetic flux is\n$\\Phi_0 = 6.3 \\times 10^{18}$~Mx, which is in the range of  \nan ephemeral active region \\citep{Zwaan:1987yf}. Details about this kind of\nsetup are provided in the paper by NS2016.\n\n\\begin{table}[h!]\n\\renewcommand{\\thetable}{\\arabic{table}}\n\\centering\n\\caption{Parameters of the initial twisted magnetic tube for both experiments} \\label{table2}\n\\begin{tabular}{ccccc}\n\\tablewidth{0pt}\n\\hline\n\\hline\n$x_0$ (Mm) & $z_0$ (Mm) & $R_0$ (Mm) & $q$ (Mm$^{-1}$) & $B_0$ (kG) \\\\\n\\hline\n\\decimals\n15.0 & -2.8 & 0.10 & 2.4 & 20     \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\\begin{figure*}\n\\epsscale{1.18}\n\\plotone{figure2.pdf}\n\\caption{Image (taken from NS2017) showing 2D \ntemperature maps for the context of the surge experiments and the regions of interest. \nA) The vertical coronal magnetic field experiment at $t=65.0$ minute. \nB) The slanted coronal magnetic case \nat $t=64.3$ minute. Additionally, magnetic field lines (green), \ntemperature contours for the $T_{_{SE}}=7.9 \\times 10^{4}$ of {\\ion{Si}{4}}\\ K (blue), and \nfor $T=1.2 \\times 10^{6}$ K (red) are added. \\label{figure2}}\n\\end{figure*}  \n\n\n\\Needspace{5\\baselineskip}\n\\section{General features of the time evolution of the experiments}\\label{sec:3}\nThe numerical experiments start with the injection of the twisted magnetic tube \nthrough the bottom boundary ($t=0$ minute). Within the convection zone, the tube rises\nwith velocities of $\\lesssim 2$ km s$^{-1}$ and suffers deformations due to the\nconvection flows, mainly in the regions where the downflows are located. The twisted tube \ncontinues rising until it reaches the surface. There, the magnetized plasma accumulates\nuntil it develops a buoyancy instability ($t \\approx 40$ minute) in a similar way as\nexplained by NS2016.\n\n\nThe subsequent phases of evolution are characterized by the emergence and \nexpansion of the magnetized plasma into the solar atmosphere, producing a dome-like \nstructure of cool and dense matter ($t \\sim 50$ minute). During the \nexpansion process, the dome interior becomes rarefied due to gravitational flows. \nSimultaneously, the magnetic field of the emerged plasma collides with the preexisting \ncoronal ambient field and, as a consequence, non-stationary magnetic reconnection occurs,\nforming and ejecting several plasmoids. Our vertical experiment \nhas recently been used by \\cite{Rouppe2017} to show that the {\\ion{Si}{4}}\\ spectral synthesis of those\nplasmoids is able to reproduce the highly broadened line profiles,\noften with non-Gaussian and triangular shapes, seen in \\textit{IRIS}\\ observations.\n\nAs an indirect consequence of the magnetic reconnection, a surge is obtained\nin both experiments. This is illustrated in Figure \\ref{figure2} through temperature \nmaps with overlying magnetic field lines for each experiment: panel A, the vertical \nexperiment at $t=65.0$ minute, and panel B, the slanted one at $t=64.3$ minute. Those\nare representative instants when the surge is clearly distinguishable as an elongated \nstructure detached from the dome. For later\nreference, different regions have been marked in the figure that\n  will be seen below to correspond to prominent features of the surge\nin terms of NEQ ionization and brightness in the spectra: the internal footpoint, which is located at the base of a wedge created by the detachment process that separates the surge from the dome (NS2016); the external footpoint, which is\njust the external boundary of the surge, and the flanks and top of the crests.\nAlthough not directly discussed in this paper, another region is probably worth mentioning, namely the \nhot jet: in the vertical experiment, it is shown clearly through the red temperature contour of \n$T=1.2 \\times 10^{6}$ K; in the slanted one, the temperatures of the high-speed collimated ejection \nare not distinguishable from the rest of the corona. The difference between both\nexperiments may be due to the fact that the slanted case has a denser emerged\ndome and, perhaps, the entropy sources in it are less efficient in \nheating the plasma that passes through the magnetic reconnection site.\n\n\n\n\n\\Needspace{5\\baselineskip}\n\\section{The role of the nonequilibrium (NEQ) ionization}\\label{sec:4}\nThe importance of the NEQ ionization is studied in this section from a triple\nperspective: (a) the comparison of the NEQ number densities with those calculated under the SE approximation\n(Section~\\ref{sec:4.1}); (b) the consequences for the emissivity of the\nplasma (Section~\\ref{sec:4.2}); and (c) the key mechanisms that cause the\ndeparture from statistical equilibrium in the surge plasma\n(Section~\\ref{sec:4.3}).\n\n\n\\Needspace{5\\baselineskip}\n\\subsection{The SE and NEQ number densities}\\label{sec:4.1}\nThe results of the current paper are obtained by solving the equation rates for\nthe relevant ionization states of Si and O, i.e., taking into account\nnonequilibrium effects using the \\citet{olluri:2013aa} module mentioned in\nearlier sections: the number densities of emitters $n_u$ thus calculated will be\nindicated with the symbol $n_{_{NEQ}}$. In order to test the accuracy of the SE\napproximation, we have also calculated the $n_u$ that would be obtained \nimposing statistical equilibrium in the \\citet{olluri:2013aa} module: those will be \nindicated with the symbol $n_{_{SE}}$. The accuracy or otherwise of the SE \napproximation is measured here through the following ratio:\n\n\\begin{eqnarray}\n\tr & = &  \\frac{n_{_{SE}} - n_{_{NEQ}}} { n_{_{SE}} + n_{_{NEQ}}  },\n\\label{eq:ratio}\n\\end{eqnarray}\nThe parameter $r$ varies between -1 and 1; its meaning is as follows:\n\n\n\\begin{enumerate}[a)]\n\n\\item If $r \\approxeq 0$, the number density of emitters obtained imposing SE would be approximately\nequal to the one allowing NEQ rates ($n_{_{SE}} \\approxeq n_{_{NEQ}}$), so in those regions the SE approximation to calculate the state of ionization would be valid.\n\n\\item If clearly $r < 0$, this means that $n_{_{SE}} < n_{_{NEQ}}$, so the approximation of SE ionization would underestimate the real \npopulation. As $r$ becomes more negative, the NEQ\neffects would be more prominent and the SE approximation \nwould become less accurate. In the extreme case ($r = -1$), the assumption of SE  would mistakenly result in an absence of ions in the ionization state of interest!\n\n\\item On the other hand, if $r > 0$,  it follows that $n_{_{SE}} > n_{_{NEQ}}$,  so the computation \nof the ionization in SE would be wrong again, but in this case because it would overestimate the real population. When $r = 1$, SE would give as a result a totally fictitious population, since\nthe full NEQ calculation indicates that there are no ions!\n\n\\end{enumerate}\n\n\n\\begin{figure}\n\\epsscale{1.05}\n\\plotone{figure3.pdf}\n\\caption{2D maps of the ratio $r$ from Equation (\\ref{eq:ratio}) for the\n  {\\ion{Si}{4}}\\ population for (A) the slanted experiment at $t=64.3$ minute and (B)\n  the vertical one at $t=65.0$ minute. Panel C shows $r$ for the\n  {\\ion{O}{4}}\\ population in the vertical experiment at $t=65.0$ minute. \n  A gray color mask is overplotted where the emissivity obtained from\n  the NEQ computation is $\\epsilon_{_{CGS}} < 10^{-10}$. Below each 2D map,\n  the median M of $\\vert r \\vert$ in the high-emissivity region (i.e., outside\n  of the mask) is shown. Solid and dashed\n  vertical lines delimit the ETR and QTR regions, respectively. The\n  horizontal line in the M panels marks the average value of M within the\n  QTR. \\label{figure3}}\n\\end{figure} \n\n \nThe ratio $r$ is plotted in Figure~\\ref{figure3} for\n  the two experiments described in this paper. The upper panels in each block\n  contain 2D maps of $r$, namely, for (A) {\\ion{Si}{4}}\\ in the slanted experiment at\n  $t=64.3$ minute; (B) {\\ion{Si}{4}}\\ in the vertical experiment at $t=65$ minute;\n  and (C) {\\ion{O}{4}}\\ also in the vertical experiment at $t=65$ minute. To limit\n  the diagram to the relevant regions, a grey mask is overplotted and only those pixels with emissivity obtained from\n  the NEQ computation above a threshold ($\\epsilon_{_{CGS}} \\geq 10^{-10}$) are being shown. \n The bottom panel in each block contains a line plot for the median M of the absolute value of $r$\n   in the regions not covered by the mask in each column. Using\n  the absolute value of the ratio elucidates the areas where NEQ ionization is \n important, either because SE underestimates ($r < 0$) or overestimates \n ($r > 0$) the real number density of emitters $n_u$. In the figure, two regions can be clearly distinguished:\n  \n  \n\n\n1. The \\textit{Quiet Transition Region} (hereafter QTR). We define it as \nthe transition region that has not been perturbed by the flux emergence and subsequent surge \nand\/or jet phenomena. The horizontal extent of the QTR is marked in the figure with dashed vertical lines \nand corresponds to the region located between $0.0 \\leq x \\leq 2.0$ and $22.0 \\leq x \\leq 32.0$ Mm, \nfor the slanted experiment; and $0.0 \\leq x \\leq 3.0$ and $26.5 \\leq x \\leq 32.0$ Mm, for the vertical one. \nIn this domain, $r$ mostly shows negative values (blue color in the image) in a thin \nlayer in the transition region ($z \\sim 2$ Mm). The corresponding M value is on average \nbetween $0.2$ and $0.3$ (horizontal dashed line in red in the panels), which indicates that both {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ \nsuffer significant departures from statistical equilibrium.\n\n\n2. The second region corresponds to the main result of this section: the emerged \ndomain, namely, the dome and surge, are severely affected by the NEQ ionization both for silicon \nand oxygen (see the dark blue color which corresponds to $r\\approx-1$, and corresponding M value close to 1). \nThe value $r\\approx-1$ is found either in cold regions with $T \\sim 2 \\times 10^4$ K and\nalso in hot domains ($T \\sim 5 \\times 10^5$ K). Also, on the left of \nthe surge, we also find some regions where $r > 0$ (red), especially in the slanted experiment,\nwhich indicates that the SE approximation is overestimating the real population. Since our main goal is \nto study the surge, we focus on its surroundings, and, in particular, on the domain marked in the figure with solid lines, i.e.,  $8.5 \\leq x \\leq 21.0$ Mm, for the slanted experiment, and $11.0 \\leq x \\leq 25.5$ Mm, for the vertical one. In the following, we refer to this range as the \\textit{Enhanced Transition Region} (ETR). \n In the associated 1D panels, we see that M shows larger values than in the QTR; in fact, the median reaches values close to one in many places of the ETR . There are some specific locations within the ETR where M shows substantially lower values, e.g., $x=16.2$ Mm or $x=18.1$ Mm in the B and C panels . Looking at the 2D panels, we realize that in those locations, part of the TR has $r$ close to zero  (white patch above the blue line).  Consequently, the median in that vertical columns decreases. Note, however, that even in those locations the M values in the ETR are larger than, or at least comparable to, the largest ones found in the QTR. This finding highlights the relevance of including the NEQ calculation for eruptive phenomena like surges, since without it, the calculated {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ populations would be totally erroneous. This would translate into wrong emissivity values and therefore mistaken synthesis diagnostics. \n\n\n\n\n\n\n\n\n\\begin{figure*}\n\\epsscale{1.0}\n\\plotone{figure4.pdf}\n\\caption{\nMaps of the 2D emissivity $\\epsilon$ for A) {\\ion{Si}{4}}\\ in the slanted experiment, \nB) {\\ion{Si}{4}}\\ in the vertical one, and C) {\\ion{O}{4}}\\ also for the vertical experiment. Diamonds have been superimposed on the region of high emissivity to mark the position of $z_{max}$ (see Equation \\ref{eq:H}). \nA color scale at the right column of the figure contains the translation from emissivity to number densities of emitters $n_u$. In each of the blocks, secondary panels for $\\epsilon$, $T$, and $n_e$ have been inserted that use $H$ as vertical scale. Additionally, a panel of the ratio of the {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivities, $R_{\\epsilon}$, is added for B and C blocks. All the maps only show places where $\\epsilon_{_{CGS}} \\geq 10^{-8}$. The instants in the panels and the vertical lines are\nthe same as in Figure \\ref{figure3}. The accompanying animation shows the time evolution of the\nthree experiments from the early stages of the surge until the its decay phase.\\\\\n(An animation of the figure is available)}\\label{figure4}\n\\end{figure*} \n\n\n\\Needspace{5\\baselineskip}\n\\subsection{Characterizing the plasma in NEQ}\\label{sec:4.2}\nOnce we have studied the NEQ effects on the two different domains of our \nexperiments (ETR, QTR), we now turn to the associated question of the \nemissivity, in particular, for the {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ lines. To that end, we start by showing 2D\nmaps of the emissivity in Figure \\ref{figure4} (top panel in each block) for the same \ninstants as in Figure \\ref{figure3}. In this case, we have constrained the\nmaps to values of $\\epsilon_{_{CGS}} > 10^{-8}$, just to \nfocus on the layer with the largest emission, which is the natural candidate to be observed. \nSince the emitting layer is really thin, we are adding small 2D maps at the bottom of each block containing a blow-up of the emissivity, $\\epsilon$, and, additionally, of the temperature, $T$; electronic number density, $n_e$; and the ratio between the {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivities, $R_{\\epsilon}$. More precisely, for each vertical column we define a height coordinate $H$ centered at the position [called $z_{max}(x)$ in the following] of the \nmaximum emissivity in that column:\n\\begin{eqnarray}\n\tH & = &   z - z_{max},\n\\label{eq:H}\n\\end{eqnarray}\nand use it, instead of $z$, in the maps. For\n  clarity, in the top panel we have indicated the location of $z_{max}$ at\n  selected columns using symbols. Since emissivities can be converted into number densities of emitters $n_u$ via simple multiplication with a constant factor\n  (Equation~\\ref{eq:emissivity} and Table~\\ref{table1}), a\n  color bar with $n_u$ both for {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ has been added in the figure. \n\n\n  By comparing the two emissivity panels in each block of\n  Figure \\ref{figure4} (see also associated movie), we find that the region of high emissivity\n  at the footpoints and crests of the surge covers a larger vertical range\n  than in other regions. This is mainly  caused by the varying mutual angle of the vertical\n  with the local tangent to the TR, so, in some sense, it is a line-of-sight\n  (LOS) effect: full details of different LOS effects are discussed in Section\n  \\ref{sec:5.2}. Inspecting the lower panels of $\\epsilon$ of each block, some locations (e.g.,\n  the internal footpoint, $x \\approx 15$ Mm, in {\\ion{O}{4}}\\ at $t=65.0$ minute)\n  are seen to have enhanced emissivity by a factor 2 or 3 in comparison to the\n  maximum values usually seen at positions of the QTR and ETR; nonetheless, \n  this behavior is sporadic as seen in the accompanying movie. \n  \n  We also notice that both {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ show similar values of emissivity, in spite of the \n  huge contrast in the corresponding number density of emitters (see second \n  color scale at the right-top corner of the image). This is due to the difference in the \n  oscillator strengths $f_{lu}$, which for {\\ion{O}{4}}\\ is six orders of magnitude weaker than for \n  {\\ion{Si}{4}}\\ (see Table \\ref{table1}). In panels B4 and C4, we have plotted \n  the emissivity ratio of {\\ion{Si}{4}}\\ to {\\ion{O}{4}}, $R_{\\epsilon}$, finding that the typical values \n  in the locations with the highest emissivity within the ETR are around 2 \n  (although it can reach up to factors around 5), while in the QTR\n  the average $R_{\\epsilon}$ is close to 1. On the other hand, in the locations with low emissivity \n  and high temperature, especially in the \n  QTR, we appreciate that $R_{\\epsilon}$ is lower than unity, which is not\n  surprising since {\\ion{O}{4}}\\ can be found at higher temperatures. Note\n  that this is a ratio of emissivities and does not correspond to the intensity ratio commonly\n  used for density diagnostics (e.g., \\citealp{Hayes:1987,Feldman:2008,Polito2016}).  \n\n  \n  \n  \n  We cannot find in the emissivity maps the same sort of drastic contrast \n  between QTR and ETR that we found for the $r$ parameter in the previous \n  section; nonetheless, we do appreciate differences between both\n  regions in terms of temperature and electronic number density: the range of  $T$ and \n  $n_e$ in the ETR is larger than in the QTR. This is especially evident in the hot and \n  low-density part, where the $T$ and $n_e$ of the ETR reach values around 1 MK and \n  $10^9$ cm$^{-3}$, respectively (Note that the $n_e$ provided is obtained from local \n  thermodynamic equilibrium (LTE) since the ionization of the main contributors for electrons, such as \n  hydrogen and helium, are computed in LTE according to the equation-of-state table of Bifrost). \n  In order to further explore those differences, we\n    resort to a statistical study of the values of emissivity and temperature\n    in the different regions (QTR, ETR) and for the two ions, which is presented in the\n    following. The statistics is based on all plasma elements with\n    $\\epsilon_{_{CGS}} \\geq 10^{-8}$ in the time span between surge formation\n    ($t = 55.0$ minute) and decay ($t = 70.7$ minute). The resulting sample\n    contains $4 \\times 10^6$ elements. Figure~\\ref{figure5} shows the\n    corresponding Joint Probability Density Functions (JPDFs) for \n    emissivity $\\epsilon$ and temperature $T$ for the vertical\n    experiment. For {\\ion{Si}{4}}\\ we could also show results for the statistical distributions \n    for the slanted experiment, but the resulting JPDFs are very similar to those \n    presented here. This similarity suggests that,\n    although the vertical and slanted experiments differ in \n    terms of magnetic configuration, size of the emerged dome, and shape of the \n    surge (compare the two panels of Figure \\ref{figure2}), the results described below\n    could be applicable to different surge scenarios. In the following, \n    we explain the results first for {\\ion{Si}{4}}\\ (Section \\ref{sec:4.2.1}), and then \n    for {\\ion{O}{4}}\\ (Section \\ref{sec:4.2.2}).\n\n\n\n\n\\Needspace{5\\baselineskip}\n\\subsubsection{Plasma emitting in {\\ion{Si}{4}}}\\label{sec:4.2.1}\n\\begin{figure}\n\\epsscale{1.18}\n\\plotone{figure5.pdf}\n\\caption{JPDFs of emissivity and temperature in the QTR and ETR in\n    the vertical experiment for {\\ion{Si}{4}}\\ (top row) and {\\ion{O}{4}}\\ (bottom row) for\n    the time range $t=55.0$ minute to $t=70.7$ minute. The size of the sample is $4 \\times\n  10^6$ elements. The white lines are isocontours of probability\n    equal to $10^{-4}$ in the ETR distribution. The areas\n  marked by ovals and arrows are discussed in the text.\\label{figure5}}\n\\end{figure}  \n\n\nWe start analyzing the QTR and ETR distributions for the {\\ion{Si}{4}}\\ emissivity\n(see top row of Figure \\ref{figure5}). The main result is that in the region\nwith the largest emissivity values the ETR is more densely populated than the\nQTR (see the region marked by a pink oval around $\\epsilon_{_{CGS}} \\sim\n10^{-5.6}$), i.e., the boundaries of the surges are more likely to show\nsignal in {\\ion{Si}{4}}\\ observations than the QTR. This helps explain why, in the\n\\textit{IRIS}\\ observations of our previous paper NS2017, we could detect the surge\nas an intrinsically brighter structure than the rest of the TR. Furthermore,\nboth the QTR and ETR have the greatest values of emissivity in the\ntemperature range between $10^{5.0}$ and $10^{5.1}$ K. This differs from what\none would expect in SE, where the maximum emissivity is located at the peak\nformation temperature ($T_{_{SE}} = 10^{4.9}$ K, see Table \\ref{table1}),\nagain an indicator of the importance of taking into account NEQ effects. Additionally,\nthe distribution both for QTR and ETR is more spread in temperature than what one would expect from a\ntransition region distribution computed in SE (see, e.g., Figure 15 of the paper by \\citealp{olluri:2015}). The\nmass density found both for QTR and ETR around the maximum {\\ion{Si}{4}}\\ emissivity\nis $\\rho \\sim 6.3 \\times 10^{-15}$ g cm$^{-3}$.\n\n\nAs part of the analysis, we have also found other features worth mentioning:\n\n\n\\begin{figure*}\n\\epsscale{1.18}\n\\plotone{figure6.pdf}\n\\caption{Density map showing the basic distribution of the (approximately) 6000 Lagrange \ntracers used in the text, distributed into two parts drawn with yellow and\nred dots corresponding to the two populations discussed in\nSection~\\ref{sec:4.3}. The accompanying animation shows the evolution of the\nLagrange tracers in the two experiments from  \nearly stages of the surge formation ($t \\approx 55.0$ minute) until the its decay \nphase ($t \\approx 70.0$ minute).\\\\\n(An animation of this figure is available.)}\n\\label{figure6}\n\\end{figure*}  \n\n\\begin{itemize}\n\n\\item The ETR has a broader temperature distribution than the QTR. \nIn order to illustrate this fact, all the panels of Figure \\ref{figure5} contain isolines \nin white for the probability $10^{-4}$ in the ETR. \nThe comparison of those contours with the QTR distribution shows that\nthe ETR has a wider distribution\nin temperature, specially above $10^{5.5}$ K. Although not shown in the figure,  the mass density values for \n most of the emitting plasma (more precisely: the mass density values with probability\n above $10^{-4}$) are constrained to similar ranges \n for both the QTR and ETR: approximately at [$2.0 \\times 10^{-15}$, $7.9 \\times 10^{-14}$] g cm$^{-3}$, \n for the vertical experiment; and at [$1.0 \\times 10^{-15}$, $6 .0 \\times 10^{-14}$] g cm$^{-3}$, for the slanted one.\n\n\n\n\\item A secondary probabilty maximum is located $\\epsilon_{_{CGS}} \\sim 10^{-7.2}$ and \nT $\\sim 10^{4.3}$ K (see the arrows\nin the panels). This corresponds to the temperature of the second ionization of helium \naccording to the LTE equation-of-state of Bifrost:  the energy deposited\nin the plasma is used to ionize the element instead of heating the plasma.\nIncluding the NEQ ionization of helium should scatter the density probability in temperatures, as shown by \n\\cite{golding:2016} in the TR of their numerical experiments\n(the equivalent to our QTR); nevertheless, the NEQ computation of helium and \na detailed discussion of their effects are out of scope of this paper. \n\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\n\n\\Needspace{5\\baselineskip}\n\\subsubsection{Plasma emitting in {\\ion{O}{4}}}\\label{sec:4.2.2}\nFocusing now on the statistical properties of the {\\ion{O}{4}}\\ emission (see bottom row \nof Figure \\ref{figure5}), we see that, like for {\\ion{Si}{4}}, the probability distributions for \nboth QTR and ETR  differ from what we could expect for a SE distribution, \nsince they are centered at temperatures between $10^{4.9}$ and $10^{5.0}$ K\ninstead of $T_{_{SE}} = 10^{5.2}$ K (see Table \\ref{table1}). Furthermore,\nalso here, the ETR and QTR distributions are broader in temperature than what one would \nexpect from SE. Further noteworthy features of the plasma emitting in {\\ion{O}{4}}\\ are:\n\n\\begin{itemize}\n\n\\item The QTR exhibits larger probability than the ETR ($> 10^{-3}$) in\nthe maximum values of the emissivities ($\\epsilon_{_{CGS}} = 10^{-5.8}$); \nnevertheless, this fact changes around $\\epsilon_{_{CGS}} = 10^{-6.0}$,\nwhere the ETR shows larger emissivity (compare the region within the colored oval). \nDue to this complex behavior, we need to integrate the emissivity to know whether the \nETR can be detected as a brighter structure compared to the QTR. In Section  \n\\ref{sec:5} we discuss this fact analyzing the obtained synthetic profiles.\n\n\n\\item  The ETR shows emissivity in {\\ion{O}{4}}\\ in a larger range of temperatures than \nthe QTR, which is akin to the result for {\\ion{Si}{4}}\\ described in Section\n\\ref{sec:4.2.1}. This difference in the ranges is apparent mainly in hot coronal\n temperatures comparing the probability contours of the ETR (in white) \n with the QTR distribution.\n\n\\item We find the same secondary probability maximum as in the {\\ion{Si}{4}}\\\npanels at the temperature of the second ionization of helium (see the pinks arrows).\n\n\\item Comparing the {\\ion{O}{4}}\\ panels with the {\\ion{Si}{4}}\\ ones, we see that the {\\ion{O}{4}}\\ \ndistribution is more populated in hot temperatures, and correspondingly, \nlower densities. This is something we could expect since the ionization of this particular\noxygen ion occurs at higher temperatures than {\\ion{Si}{4}}.\n\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\Needspace{5\\baselineskip}\n\\subsection{Lagrange tracing: how the entropy sources affect the NEQ ionization}\\label{sec:4.3}\nWe focus now on the role of the entropy sources in the emissivity and \nNEQ ionization. To that end, we follow in time $ \\approx 6000$ plasma elements of the \nETR through Lagrange tracing. In the following, we explain the set up for the Lagrange elements\n(Section \\ref{sec:4.3.1}) and the results obtained from their tracing (Section \\ref{sec:4.3.2}).\n\n\n\\subsubsection{The choice of the Lagrange elements}\\label{sec:4.3.1}\n\nThe Lagrange elements are selected at a given instant, corresponding to an\n  intermediate evolutionary stage when the surge is clearly \n  distinguishable as a separate structure from the dome. The\n    selected instants are $t=64.3$ minute for the slanted experiment and\n    $t=65.0$ minute for the vertical experiment, which are the same times\n    used for Figures \\ref{figure2}, \\ref{figure3}, and \\ref{figure4}. In order to focus on \n    the domain in and near the surge, we limit the selection to\n  the rectangular areas: $12.0 \\leq x \\leq 25.2$, $2.2 \\leq z \\leq\n  15.0$ (vertical experiment); and $7.0 \\leq x \\leq 19.0$, $5.0 \\leq z \\leq\n  15.0$, and $19.0 \\leq x \\leq 20.5$, $1.5 \\leq z \\leq 15.0$ (slanted\n  experiment). On those rectangles we lay a grid with uniform spacing $\\Delta\n  x = \\Delta z =40$~ km: the Lagrange elements are chosen among the\n  pixels in that grid. A further criterion is then introduced: we are\n  interested in studying the origin and evolution of the plasma elements with\n  strong emission in {\\ion{Si}{4}}\\ and {\\ion{O}{4}}. Thus, a lower bound in the {\\ion{Si}{4}}\\ and\n  {\\ion{O}{4}}\\ emissivity is established, namely $\\epsilon_{_{CGS}} > 10^{-10}$,\n  discarding all the pixels with emissivities below that value at the\n  instants mentioned in the previous bullet point. The resulting choice of\n  Lagrange tracers is shown in Figure \\ref{figure6} as red and yellow dots\n  (the colors serve to distinguish the populations described below). Once the distribution is \n  set, we then follow the tracers backward in\n  time for 10 minutes, to study their origin, and forward in time for 5.7\n  minutes, to see the whole surge evolution until the decay phase, with a\n  high temporal cadence of $0.2$~seconds (see the accompanying animation to\n  Figure~\\ref{figure6}).\n\n\n\n\\subsubsection{Plasma populations and role of the entropy sources}\\label{sec:4.3.2}\nStudying the time evolution of the Lagrange tracers, in particular their\nthermal properties, one can distinguish two populations that are the source of \nthe {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emission: one originating in the emerged dome (yellow plasma population in\nFigure~\\ref{figure6}) and the other one originating in the corona (red\npopulation). By carefully inspecting the tracers of each population, we find that\ntheir behavior is well defined: the major difference between the elements within\nthe same population is not the nature or order of the physical events described below, \nbut rather the starting time of the evolution for each tracer. Figure~\\ref{figure7} contains the time evolution of\ndifferent quantities as measured following a representative Lagrange element of\n  each population, namely temperature, $T$, (green); {\\ion{Si}{4}}\\ emissivity,\n$\\epsilon_{_{Si\\ IV}}$, (dark blue); {\\ion{O}{4}}\\ emissivity, $\\epsilon_{_{O\\ IV}}$,\n(light blue); characteristic time of the optically thin losses,\n$\\tau_{thin}$, (black); and characteristic time for the thermal conduction,\n$\\tau_{Spitz}$ (red). \n\n\n\n\n\\begin{figure}\n\\epsscale{1.15}\n\\plotone{figure7.pdf}\n\\caption{Time evolution of key physical quantities for representative\n  Lagrange tracers in the vertical experiment of Figure~\\ref{figure5}.  Top:\n  Lagrange element coming from the emerged dome (yellow population in\n  Figure~\\ref{figure6}). Bottom: Lagrange originating in the corona (red\n  population in Figure~\\ref{figure6}).  The curves show (left\n    ordinate axis) the logarithm of temperature $T$ (green); of the\n    characteristic time of the optically thin losses $\\tau_{thin}$ (black);\n    and of the characteristic time for the thermal conduction $\\tau_{Spitz}$\n    (red); and (right ordinate axis) the logarithm of the {\\ion{Si}{4}}\\ emissivity\n    (dark blue); and of the {\\ion{O}{4}}\\ emissivity (light blue). All quantities are\n    in CGS units. \\label{figure7}}\n\\end{figure}  \n\n\\textit{The first population} (top panel of Figure~\\ref{figure7}, corresponding to the\n  elements marked in yellow in Figure~\\ref{figure6}), starts as\n    cool and dense plasma coming from the emerged dome with extremely low\n    emissivity (see the curves for the temperature in green, and for the\n    emissivities in dark and light blue). At some point\n  that plasma approaches the reconnection site and passes through the current\n  sheet, thereby suffering strong Joule and viscous heating and\n    quickly reaching TR temperatures. The sharp\n    spike in the {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivity (blue curves) around $t\\sim\n    62.5$ minute corresponds to this phase: the temperature increase leads to\n    the appearance of those ionic species, but, as the plasma continues being\n    heated, it reaches high temperatures (maximum around $10^6$ K) and the\n    number densities $n_u$ of {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ decrease again. At those high\n  temperatures the entropy sinks become efficient, with short\n  characteristic times: see the red ($\\tau_{Spitz}$) and black\n  ($\\tau_{thin}$) curves. The plasma thus enters a phase of gradual\n    cooling, going again through TR temperatures, renewed\n    formation of the {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ ions, and increase in the corresponding\n    emissivity (broad maximum in the blue curves in the right half of the\n    panel). The plasma elements, finally, cool down to chromospheric\n  temperatures, with the emissivity decreasing again to very low\n  values. \n\n\n\n\n\n The defining feature of \\textit{the second population}  (bottom panel of\n   Figure~\\ref{figure7}, red dots in Figure~\\ref{figure6}) is that it\n   originates in the corona as apparent in the temperature curve\n   (green). This population starts at heights far above the reconnection\n   site, with standard coronal temperature and density. During the magnetic\n   reconnection process, its associated field line changes connectivity,\n   becoming attached to the cool emerged region. Consequently, a steep\n   temperature gradient arises along the field line, so the thermal\n   conduction starts to cool down the plasma; given the temperature range,\n   also the optically thin losses contribute to the cooling, although to a\n   lesser extent (see the $\\tau_{Spitz}$ and $\\tau_{thin}$ curves around\n   $t\\sim 61$ minute, in red and black, respectively). The temperature drops\n   to values around T $\\sim 10^{5.5}$ K, which, according to the\n   JPDFs of Figure~\\ref{figure5}, makes it sufficiently likely that the\n   {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivities from the Lagrange element are high. This\n   explains the large increase, by a few orders of magnitude, in the blue\n   emissivity curves around $t\\sim 61$ minute \n   (although a small factor  $\\sim 4$  is due to the simultaneous \n   increase in the mass density, which is reflected in a linear fashion in the emissivity). \n   This cooling to TR\n   temperatures, however, is short lived: as the plasma element itself passes\n   near the current sheet, it can\n   be heated because of the Joule and viscous terms and the temperature\n   climbs again to values where the emissivities are low: hence the sharp\n   spike in the blue curves between $t\\sim 61$ and $t\\sim 61.5$ minute. There\n   ensues a phase of gradual cooling from $t \\sim 64$ onward, similar to what\n   happened to the previous population, with characteristic cooling times of\n   a few to several hundred seconds (see red and black curves), passage\n   through TR temperatures, broad maximum in the emissivity\n   curves and with the plasma finally reaching chromospheric temperatures. \n\n\n\n\\begin{figure}\n\\epsscale{1.18}\n\\plotone{figure8.pdf}\n\\caption{\nJPDFs for the {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivity of the Lagrange tracers\n  over $15.7$ minutes versus the \ncharacteristic time of the optically thin losses ($\\tau_{thin}$, left column) and the thermal conduction \n($\\tau_{Spitz}$, right column). \\label{figure8}}\n\\end{figure}  \n\n\n\n\\begin{figure*}\n\\epsscale{1.21}\n\\plotone{figure9.pdf}\n\\caption{Synthetic profiles for the vertical experiment at different times for various LOS ($\\theta$). \nA) 2D map of the $\\epsilon_{_{CGS}}$ for {\\ion{O}{4}}\\ to show the context; B) synthetic spectral \nintensity for {\\ion{Si}{4}}, and ; C) synthetic spectral \nintensity for {\\ion{O}{4}}. To identify the LOS of each panel, we have added the symbols $0$, $-$, and\n$+$ respectively to $\\theta = 0, -15$ and $15$ \\degree. In the $\\theta \\neq 0$ rows, $z_P$ and $x_P$ are, respectively, the vertical and horizontal \ncoordinates of the rotated figures. The animation of this figure shows \nthe time evolution of the surge from its origin ($t = 61$ minute) up to its decay phase \n($t = 70.7$ minute) in the vertical experiment for the three LOS.\\\\\n(An animation of this figure is available.)} \\label{figure9}\n\\end{figure*}   \n\nIn our previous paper (NS2016) we found that surges were constituted by four\ndifferent populations according to their thermal evolution. In the current\npaper, we see that only two of them, labelled Populations B and D in the\nNS2016 paper, are behind the enhanced emissivity of TR lines\nlike those from silicon and oxygen discussed here. The other two populations\ndescribed by NS2016 (A and C) keep cool chromospheric temperatures during\ntheir evolution and do not play any role for the TR elements.\n\n\n\n\nUsing the Lagrange tracing method developed here, we can produce conclusive\nevidence of enhanced {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivity and occurrence of fast evolution \ndue to short-time scales in the entropy sources associated with heat conduction \nor optically thin radiative cooling. Figure~\\ref{figure8} contains double PDFs for $\\epsilon{_{CGS}}$\nversus $\\tau_{thin}$ (left panels) and $\\tau_{Spitz}$ (right panels) using\nas statistical sample the values of those quantities for all Lagrange\ntracers along their evolution. The choice of the ionic species ({\\ion{Si}{4}}, {\\ion{O}{4}}) and\nexperiment (slanted, vertical) in the panels is as in Figures \\ref{figure3} and \\ref{figure4}.\nThe figure clearly shows that when the entropy sources act on short time\nscales, the ({\\ion{Si}{4}}, {\\ion{O}{4}}) emissivities are large. In fact, the maximum\nvalues of $\\epsilon{_{CGS}}$ correspond to characteristic cooling times\nbetween $20-100$~s for $\\tau_{thin}$ and between $4-40$~s for\n$\\tau_{Spitz}$. Those\n   changes are fast enough for the ionization levels of those elements to be\n   far from statistical equilibrium.\n\n\n\n\\Needspace{5\\baselineskip}\n\\section{Observational consequences}\\label{sec:5}\nIn the paper by NS2017,  different observed {\\ion{Si}{4}}\\ signatures\nwithin the surge were analyzed. Moreover, counterparts to the observational features were\nidentified in the synthetic spectral profiles obtained from the numerical\nmodel; however, a theoretical analysis to understand the origin of the\nspectral features and the reason for the\nbrightness in the various regions of the surge was not addressed. In the\nfollowing subsection a theoretical \nstudy is carried out trying to quantify the impact of the NEQ ionization of silicon\nand oxygen on the spectral and total intensities and the observational consequences thereof\n(Section \\ref{sec:5.1}). Then, given the involved geometry of the\n  surge, the particular LOS for the (real or synthetic) observation turns out to be\ncrucial for the resulting total intensity and spectra. This is studied in Section \\ref{sec:5.2}.\n\n\n\n\n\\Needspace{5\\baselineskip}\n\\subsection{Synthetic profiles}\\label{sec:5.1}\nFigure \\ref{figure9} contains the synthetic profiles obtained by integrating the \nemissivity along the line of sight for different wavelengths in the {\\ion{Si}{4}}\\ $1402.77$ \n\\AA\\ and {\\ion{O}{4}}\\ $1401.16$ \\AA\\ spectral region and for the vertical experiment. The \nthree rows of the figure correspond to different inclination angles $\\theta$ for the \nLOS: from top to bottom, 0\\degree, -15\\degree\\ and 15\\degree, respectively. The \npanels in each row contain A) the context of the experiment through a 2D map of \nthe {\\ion{O}{4}}\\ emissivity; B) the synthetic spectral intensity for {\\ion{Si}{4}}\\ with the spectral \ndimension in ordinates and in the form of Doppler shifts from the central \nwavelength; and C) the corresponding synthetic spectral intensity for {\\ion{O}{4}}. \nThose spectra are obtained taking into account the Doppler shift due to the plasma \nvelocity and applying a spatial and spectral PSF (Gaussian) degradation \nas explained in detail by \\cite{Martinez-Sykora:2016obs}, their section 3.1, and by \nNS2017, their section 2.2. In this way, we will be able to directly\ncompare the results with \\textit{IRIS}\nobservations. In order to ease the identification of the LOS in each panel, \nwe have added to the labels on it, the symbols $0$, $-$, and $+$ respectively to $\\theta = 0, -15$ and $15$ \\degree. In the middle and bottom rows of the image, $x_P$ and \n$z_P$ are, respectively, the horizontal and vertical coordinates of the rotated figures. \n\nTo extend the analysis, it is also of interest to\nconsider the total emitted intensity for each vertical column, i.e.,\n\\begin{eqnarray}\n\tI_{\\epsilon} (x) & = &   \\int_{z_0}^{z_f}{ \\epsilon\\; dz}\\;,\n\\label{eq:ie}\n\\end{eqnarray}\nwhich, following Equation \\ref{eq:emissivity}, is equal to the column density \nof the emitting species along the LOS except for a constant factor). Equation (\\ref{eq:ie})\nhas been calculated separately for {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ and with the\nemissivities obtained assuming either NEQ or SE ionization, to\nbetter gauge the importance of disregarding the NEQ \neffects. The results are shown in the middle and bottom\npanel of Figure \\ref{figure10} for {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ , respectively.  The top\npanel of the image contains the 2D map of the emissivity for {\\ion{Si}{4}}\\ for\ncontext identification.  Combining Figures \\ref{figure9} and \\ref{figure10},\nwe are able to discern and describe characteristic features of the spectral\nprofiles.\n\n\n\\begin{itemize}\n\n\\item \nThe most prominent feature in the synthetic profiles of Figure \\ref{figure9}\nis the brightening associated with the location of the internal footpoint of\nthe surge. In the corresponding movie, we can see how that footpoint is\nformed at around $t=64$ minute, as the surge detaches from the emerged\ndome. During those instants, the associated synthetic profiles are\ncharacterized by large intermittent intensities and\nbidirectional behavior with velocities of tens of km s$^{-1}$, as apparent,\ne.g., in the B0 and C0 panels at $x \\in [15,16]$ Mm. In\n{\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ (B and C panels), we find that the internal footpoint is\nusually the brightest region, although there are some instants in which the\nbrightest points can be located in the crests or the external footpoint. This\nis a potentially important result from the observational point of view\nbecause it can help us to unravel the spatial geometry of the surge in future\nobservational studies: if strong brightenings are detected in {\\ion{Si}{4}}\\ and also\nin {\\ion{O}{4}}\\ within the surge, it would be reasonable to think that they\ncorrespond to the internal footpoint of the surge. In this region, \nthe intensity ratio between {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ ranges between 2 and 7, approximately. Note\nthat, in general, the intensity ratio values vary depending on the observed region \nand features \\citep{Martinez-Sykora:2016obs}.\n\n\nIn Section \\ref{sec:5.2} we will see that LOS effects play a major role in causing \nthe large brightness of the internal footpoint (and other bright features) compared to \nthe rest of the surge. Here we consider the parallel question of the role of NEQ: what would be obtained for the intensity of the internal footpoint if SE were assumed? Comparing the values for {\\ion{Si}{4}}\\ in Figure\n\\ref{figure10} (middle panel, $13.8 \\le x \\le 16.1$), there is roughly a\nfactor $2$, in the average, between the NEQ and SE intensities; for {\\ion{O}{4}}\\ (bottom\npanel), there is no major difference in the intensity between both\ncalculations. One could conclude that while NEQ is important for\n  the {\\ion{Si}{4}}\\ diagnostics, SE could be applied for the\n  {\\ion{O}{4}}\\ case; nonetheless, even in the latter case, although the\n  NEQ and SE intensities are similar, one would make a mistake in the\n  determination of derived quantities like the number densities of emitters (Section\n\\ref{sec:4.1}) and temperatures (Section \\ref{sec:4.2}).\n\n \nFurther distinctive brightenings in the spectral profiles appear at the site \nof the crests and of the external footpoint of the surge. The brightness of those\n  regions is clear in the {\\ion{Si}{4}}\\ profiles (B0, B$-$ and B$+$\npanels of Figure \\ref{figure9}) through their large intensity, and is sometimes\ncomparable or greater than that of the internal footpoint (see, e.g., the\nlocations at $x \\sim 13$, $18$, $23$ and $25$ Mm in the top row, or, $x \\sim\n14, 19$ and $25$ Mm in the bottom row). In the {\\ion{O}{4}}\\ profiles (C0, C$-$ and C$+$ panels),\nalthough faint in comparison with {\\ion{Si}{4}}\\ (around a factor 5 less intense), \nmost of those features are still slightly brighter than the rest of the surge. This difference in intensities\nbetween {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ can also be used to understand the observations: if\nstrong signals are observed in {\\ion{Si}{4}}\\ associated with some moderate signal in\n{\\ion{O}{4}}, it could indicate that we are detecting the crests or the external\nfootpoint. Concerning the NEQ\/SE comparison of the intensity (Figure\n\\ref{figure10}), the crests and footpoints show the same behavior as the\ninternal footpoint.\n\n\n\n\\item The intensity of the rest of the surge is small in comparison\n  with that of the footpoints and crests just described, so we\n  wonder whether one could see it as a bright structure in actual\n    observations and distinguish it from the rest of the TR.\n  In the middle panel of Figure \\ref{figure10}, comparing the $I_{\\epsilon}$\n  values for {\\ion{Si}{4}}\\ within the ETR against those in the QTR, we can see that\n  all the regions in the surge have a higher intensity than the\n  QTR; outside of the brightest features, the excess emission of\n    {\\ion{Si}{4}}\\ in the surge may be just a factor 2 or 3 above the QTR, but that\n  can provide enough contrast to tell the two regions apart\n  observationally, as found in the NS2017 paper. Note, importantly,\n   that there is a large difference between the NEQ and SE results\n  for {\\ion{Si}{4}}\\ in the surge, up to a factor 10, so\n  the SE assumption would seriously underestimate the\n  intensity. In fact, in most of the places {\\ion{Si}{4}}\\ would be similar or fainter\n   than {\\ion{O}{4}}\\ if SE were valid as shown also, e.g. by \\cite{Dudik:2014} and \\cite{Dzifcakova:2017}.\n  On the other hand, for {\\ion{O}{4}}\\ (bottom panel), the prominent features are the footpoints and crests\n  of the surge, while the other parts have $I_{\\epsilon}$\n  comparable to the QTR. As a consequence and from an observational point of view, while in {\\ion{Si}{4}}\\ we\n  could find enhanced emission in the whole surge (which is compatible with\n  the NS2017 observation), for {\\ion{O}{4}}\\ only the brightest regions would stand\n  out, namely, the internal footpoints and, to a lesser extent, the crests and external footpoints. \n  \n  The underlying reason for the enhanced brightness of all those features, \n  mainly the footpoints and crests of the surges, is not just the presence of  \n  additional numbers of ions due to NEQ effects: the complex geometry of\n  the surge TR has important consequences when integrating the emissivity\n  along the line of sight to obtain intensities . This is discussed in the\n  next section. \n  \n  \n\\end{itemize}\n\n\n\n\\begin{figure}\n\\epsscale{1.21}\n\\plotone{figure10.pdf}\n\\caption{Top panel: 2D map of the {\\ion{Si}{4}}\\ emissivity. The\nvertically integrated intensity $I_{\\epsilon}$ is shown both for {\\ion{Si}{4}}\\ (middle panel) and\n{\\ion{O}{4}}\\ (bottom panel) assuming NEQ ionization (light blue curve) and SE (purple).\nSolid and dashed lines are overplotted in the image to \ndelimit the ETR and QTR regions as in previous figures. Dotted vertical lines are\nsuperimposed in the middle and bottom panel correspondingly to the cuts shown\nin Figure \\ref{figure11}.} \\label{figure10}\n\\end{figure}   \n\n\\Needspace{5\\baselineskip}\n\\subsection{The role of the LOS}\\label{sec:5.2}\nThe observation of\n  TR lines generated in the surge strongly depends on the particular\n  LOS. We show this here through two different effects:\n\n\\begin{enumerate}[a)]\n\n\\item The alignment of the LOS with respect to the orientation of the surge's\n  TR.  We can appreciate this effect, e.g., in A0 panel of Figure \\ref{figure9}. \n  There, considering,\n  e.g., the crests situated at $x =13.5$, $x=18$ or $x=23$ Mm, we see that a\n  vertical LOS grazing the left side of the crest will include contributions\n  from a much larger length of the TR than if the crossing were perpendicular\n  or nearly so. The same can be said of the external footpoint at $x=24.8$ Mm\n  and also of the internal footpoint around $x = 16$ Mm.  This effect clearly\n  enhances the brightenings seen in those values of $x$ in panels B0 and C0.\n  Further evidence can be found by checking the $I_{\\epsilon}$ curves in the\n  middle and bottom panels of Figure \\ref{figure10}; in fact, since\n    the effect is purely geometrical, the contribution to brightness can be seen both in the\n    NEQ and SE curves in the two panels of the figure. \n    Varying the angle of the LOS, we can reach enhancement factors \n    between 2 and 4; nonetheless, discerning which part of that factor is exclusively due to the \n    LOS is complicated, since variations in the angle of integration imply integrations along\n    slightly different emitting layers. Additionally, the\n  inclination of the LOS with respect to the surge's TR may be important for\n  the apparent horizontal size of the brightenings. This can be seen through\n  comparison of the three rows of Figure \\ref{figure9}: considering the size\n  of the brightening associated with the internal footpoint, we see that it\n  covers a larger horizontal range in the $\\theta=0$\\degree\\ and $\\theta=15$\\degree\\ cases (top and bottom rows), than in the $\\theta=-15$\\degree\\ case\n  (middle row), since the good alignment of the latter is lost in the former.\n\n\\item The multiple crossings of the TR by individual lines of sight. \nGiven that the TR of the surge is folded, there are horizontal ranges in\nwhich the LOS crosses it more than once (typically three times, in some\nlimited ranges even five times). Given the optically thin\napproximation, the emitted intensity in those lines of sight may be\na few, or several, times larger than the value in a single crossing. \n\n\\end{enumerate}\n\n\nTo further illustrate those two effects, we use Figure\n  \\ref{figure11}.  The top panel contains a 2D map of the {\\ion{Si}{4}}\\ emissivity\nin which vertical cuts in different regions of interest are overplotted\nthrough colored and labelled lines.  The corresponding {\\ion{Si}{4}}\\ emissivity\ndistribution along those cuts is shown in panel B. Additionally, a similar\nplot but for the {\\ion{O}{4}}\\ distribution is shown in panel C. Those vertical\ncuts are also shown in panels B and C of Figure \\ref{figure10} for comparison\npurposes. The light and dark\ngreen cuts (numbers 3 and 5) are typical examples of the effect of\nthe tangency between the LOS and the surge's TR at the crests: note the\nenhanced width of the maximum on the right in those two cuts due to tangency\neffects. Those cuts and\nthe one in black (number 2) further show the effect of\nmultiple crossings of the TR by the LOS. In particular, the LOS\ndrawn in black crosses the folded TR near the internal footpoint four times\n($5.5 < z < 8$ Mm), and an additional time at the the top of the surge ($z\n\\sim 11$ Mm). For the sake of completeness, we have added a dashed\nline for the crests (3 and 5) in the middle and lower panels showing the\nemissivity if SE had been assumed: the thickness of the high-emissivity TR\nwould be much smaller.  Finally, in contrast to all\nthe foregoing cases, the rest of the surge (pink line, label 4) and the QTR\n(red and blue lines, labels 1 and 6, respectively) show a simpler geometry,\nthere is no TR-LOS alignment and there is just a single crossing.\n\n\n\n\n\\begin{figure}\n\\epsscale{1.21}\n\\plotone{figure11.pdf}\n\\caption{Illustration of the multiple crossings of the transition\n    region by a single LOS. A) 2D map of the {\\ion{Si}{4}}\\ emissivity with colored\n    and labelled lines overplotted in regions of interest. The lines\n    corresponds to different vertical cuts used in panels B and C. B)\n    {\\ion{Si}{4}}\\ emissivity versus the height $Z$ for the different vertical cuts\n    shown in the 2D map. C) Like panel B but for the {\\ion{O}{4}}\\ \n    emissivity. Additionally, for the cuts labelled 3 and 5 we have added as a dashed line the\n    corresponding emissivity if SE had been assumed.\n} \\label{figure11}\n\\end{figure}   \n\n\\Needspace{5\\baselineskip}\n\\section{Discussion}\\label{sec:6}\nIn this paper, we have carried out two 2.5D radiative-MHD numerical experiments \nof magnetic flux emergence through granular convection and into the solar\natmosphere. The experiments were performed with Bifrost, including an \nextra module of the code that computes the nonequilibrium ionization (NEQ)\nof silicon and oxygen. The time evolution of the two experiments leads to the \nformation of a cool and dense surge. We have studied the relevance\nof the NEQ ionization for the presence of {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ in the\nperiphery of the surge \nand how it affects the corresponding emissivities. The properties\n of the surge plasma emitting in {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ were then\ncharacterized and compared \nwith those of the general TR plasma outside of the surge. We\nhave also analyzed the role of the heat  \nconduction and optically-thin radiative cooling in the NEQ ionization. \nFurthermore, through forward modelling, \nwe have understood different features\nof the synthetic spectral profiles of {\\ion{Si}{4}}\\ and {\\ion{O}{4}}, explaining the importance of the\nshape of the transition region surrounding the surge\ncombined with the different possible angles of the LOS and providing\npredictions for future observational studies.  \n\n\nIn the following, we first address the implications of the\nimportance of the NEQ ionization in \nnumerical experiments of eruptive phenomena in which heating and \ncooling are key mechanisms. (Section \\ref{sec:6.1}). We then discuss \ntheir relevance for present and future observations (Section\n\\ref{sec:6.2}). \n\n\n\n\\Needspace{5\\baselineskip}\n\\subsection{On the importance of the nonequilibrium (NEQ) ionization}\\label{sec:6.1}\n\nThe main result of this paper is that the envelope of the emerged\ndomain, more specifically, \nof the dome and surge, are strongly affected by NEQ ionization (Section \\ref{sec:4.1}). \nFocusing on the boundaries of the surge, comparing the number densities of emitters\ncomputed via detailed solution of the NEQ rate equations with those obtained \nassuming statistical equilibrium (SE) we have concluded that the SE assumption \nwould produce an erroneous result in the population of {\\ion{Si}{4}}\\ and {\\ion{O}{4}}, mainly \nbecause it leads to a gross underestimate of the number density of emitters. The transition \nregion outside of the flux emergence site is also affected by NEQ, but to a smaller \nextent. \n\nThe above result has consequences in the corresponding emissivity (Section\n\\ref{sec:4.2}) and therefore, in the interpretation of the observations.  By\nmeans of statistical analysis, we have shown that the boundaries of the surge\nhave greater values of the {\\ion{Si}{4}}\\ emissivity than the region outside of the\nflux emergence site.  Correspondingly, we have given the name\n\\textit{enhanced transition region} (ETR) to the former and \\textit{quiet\n  transition region} (QTR) to the latter (Section \\ref{sec:4.2.1}).  This\ndifference is part of the explanation of why the surge is a brighter\nstructure than the rest of the transition region in the \\textit{IRIS}\\ observations\nby NS2017. Furthermore, the joint probability distributions for emissivity\nand temperature are not centered at the peak formation temperature of\n{\\ion{Si}{4}}\\ or {\\ion{O}{4}}\\ in SE (see $T_{_{SE}}$ in Table \\ref{table1} and Section\n\\ref{sec:4.2.2}). This reinforces earlier results\n  (e.g., \\citealp{olluri:2015}) about the inaccuracies inherent in the process of deducing\n  temperature values from observations in transition region lines using SE\n  considerations.\n\nIn Section \\ref{sec:4.3} we have found that there are two different\npopulations concerning the thermal evolution that leads to the {\\ion{Si}{4}}\\ and\n{\\ion{O}{4}}\\ emissivity. They have very different origins (one in the emerged plasma\ndome, the other in coronal heights) but both are characterized by the\nfact that they go through a period of rapid thermal change caused by the\noptically thin losses and thermal conduction: the maximum values of the\n{\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ emissivity in them are related to short characteristic\ncooling times: $20-100$ s for $\\tau_{thin}$ and around $4-40$ s for\n$\\tau_{Spitz}$. Those characteristic times are compatible with the\ntheoretical results by \\cite{Smith:2010}, who found that for typical\ndensities of the active corona and the transition region, the solar plasma\ncan be affected by NEQ effects if changes occur on timescales shorter than\nabout $10-100$ s.  Those results highlight the role of optically thin losses\nand thermal conduction because a) they provide the physical mechanism to\ndiminish the entropy and, consequently, obtain plasma with the adequate\ntemperatures to form ions of {\\ion{Si}{4}}\\ or {\\ion{O}{4}}\\ ($\\sim 10^5$ K ); and b) they\nare fast enough to produce important departures from SE. On the other hand,\nthe ion populations calculated through the present NEQ module in Bifrost are\nnot used in the energy equation of the general R-MHD calculation (see Section\n\\ref{sec:2.1}), so this could underestimate the effects of the entropy sinks\nin the experiments. In fact, \\cite{Hansteen:1993} found deviations of more\nthan a factor two in the optically thin losses when considering\nnonequilibrium effects in his loop model, so $\\tau_{thin}$ could be even more\nefficient.\n\n\n\nOur results indicate that surges, although traditionally described as\nchromospheric phenomena, show important emission in transition region lines\ndue to the NEQ ionization linked to the quick action of the cooling\n  processes, so the response of the transition region is intimately tied to\nthe surge dynamics and energetics. In fact, the same statement may apply for\nother eruptive phenomena, in which impulsive plasma heating and cooling\noccurs \\citep[see][for a review of NEQ processes in the solar\n  atmosphere]{Dudik:2017rv}. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\Needspace{5\\baselineskip}\n\\subsection{Understanding observations and predictions for the future}\\label{sec:6.2}\n\n  From the number density of emitters and emissivity\n  results of Section~\\ref{sec:4} we gather that calculating heavy element\n  populations directly through the rate equations instead of via the\n  assumption of statistical equilibrium can be important to understand the\n  observations of surges (and of other fast phenomena which reach TR\n  temperatures in their periphery like spicules \\citealt{DePontieu:2017l1},\n  or UV bursts \\citealt{Hansteen:2017ib}). In that section, analyzing the {\\ion{Si}{4}}\\\n  and {\\ion{O}{4}}\\ emissivities of the plasma elements, we find that the ratio between them is approximately 2 in the regions with the highest emission within the ETR. Even though the intensity ratio is more commonly used (\\citealp{Hayes:1987,Feldman:2008,Polito2016}, among others), the emissivity ratio can also be a valuable tool to understand the behavior of the ions in different regions of the Sun. In particular, we see that in the ETR this ratio is larger than in the transition region that has not been perturbed by the\nflux emergence and subsequent surge and\/or jet phenomena (QTR).\n  \n  To provide theoretical support to the NS2017\nobservations and predictions for future ones, in Section \\ref{sec:5} we have\n  therefore computed the synthetic profiles of {\\ion{Si}{4}}\\ 1402.77\n\\AA\\ and {\\ion{O}{4}}\\ 1401.16 \\AA, taking into account the Doppler shift because of\nthe plasma velocity and degrading the spatial and spectral resolutions to the\n\\textit{IRIS}\\ ones.  A line-of-sight\n  integration of the emissivities has also been carried out, to provide a\n  measure for the total intensity emitted by the different regions of the\n  surge. The strongest brightenings in {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ have been located at\n  the site of the internal footpoint, followed by the crests and the external\n  footpoint (Figure \\ref{figure9}, Section \\ref{sec:5.1}). The intensity\n  ratio between {\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ in those regions is, approximately, 5 (although it\n  can range from 2 and 7).  Those values are between those for a coronal hole \n  and a quiescent active region obtained by \\cite{Martinez-Sykora:2016obs}, which\n  is consistent since we are mimicking an initial stratification similar to a coronal\n  hole in which a total axial magnetic flux in the range of an \n  ephemeral active region \\citep{Zwaan:1987yf} has been injected. The comparison\n  of the total intensity for the NEQ and SE cases (Figure \\ref{figure10},\n  Section \\ref{sec:5.1}) leads to a further indication of the importance of\nusing NEQ equations to determine the number density of {\\ion{Si}{4}}: the NEQ\ncalculation yields intensities coming out of the surge which are a factor\nbetween $2$ and $10$ larger than when SE is assumed. For {\\ion{O}{4}}, instead, the\nNEQ and SE calculations yield similar integrated intensities.  For {\\ion{O}{4}}, therefore, \nthe NEQ calculations are important mainly to determine derived quantities like \nnumber densities of emitters $n_u$ (Section \\ref{sec:4.1}) and temperatures (Section \\ref{sec:4.2}).\nIn addition, we have found that for {\\ion{Si}{4}}\\ all the regions in the surge have\na greater intensity than the QTR: this can explain why the surge\ncan be observationally distinguished from the QTR, as found in\nthe NS2017 paper. \n\nThe high brightness of various features in the surge has been seen\n  to arise in no small part from different LOS effects tied to the peculiarly irregular shape\n  of its TR, and, in particular, to its varying inclination and the folds\n  that develop in it (Section~\\ref{sec:5.2}). On the one hand, whenever LOS\n  and tangent plane to the TR are not mutually orthogonal, the issuing\n  intensity collects emissivity from a larger number of plasma elements in\n  the TR (alignment effect); on the other hand, given that the surge's TR is\n  variously folded, forming crests and wedges, the LOS crosses the emitting\n  layer multiple times (multiple-crossing effect). The alignment and\n  multiple-crossing effects are quite evident in the footpoints and\n  crests. This explains their remarkable brightness and makes clear their\n  potential as beacons to indicate the presence of those special features in\n  surge-like phenomena when observed in TR lines like {\\ion{Si}{4}}\\ (Figure\n  \\ref{figure11}). Additionally, the multiple crossings can also have an\nimpact on the observed Doppler shifts since we could be integrating various\nemitting layers with different dynamics. So, when confronted with a TR \nobservation of a region where a surge is\n  taking place, the detection of strong brightenings in\n{\\ion{Si}{4}}\\ and {\\ion{O}{4}}\\ could help unravel the geometry of\nthe surge. Furthermore, since the internal footpoint of\nthe surge is close to the reconnection site, we might also find observational\nevidences of reconnection in the neighborhood.  This provides theoretical\nsupport to the location of the brightenings in the \\textit{IRIS}\\ observations by\nNS2017. In addition, if strong signals are observed in {\\ion{Si}{4}}\\ related to some\nmoderate signal in {\\ion{O}{4}}, they could correspond to the crests and the external\nfootpoint of the surge; nonetheless, the rest of the surge structure could be\nonly differentiated from the transition region in {\\ion{Si}{4}}.\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\\ \\vspace{-2mm} \n\\acknowledgments We gratefully acknowledge financial support\nby the Spanish Ministry of Economy and Competitiveness (MINECO) through\nprojects AYA2011-24808 and AYA2014-55078-P, as well as by NASA through \ngrants NNX16AG90G, NNH15ZDA001N, NNX17AD33G, and by NSF grant \nAST1714955 and contract NNG09FA40C (\\textit{IRIS}).\nWe also acknowledge the computer resources and assistance provided at the \nMareNostrum (BSC\/CNS\/RES, Spain) and TeideHPC (ITER, Spain) \nsupercomputers, where the calculations have been carried out, and at the\nPleiades cluster (computing projects s1061, s1472 and s1630 from the High \nEnd Computing division of NASA), where relevant code developments\nhave been made. Finally, the authors are grateful to Dr. Peter R. Young for his \nsuggestions during the \\textit{Hinode}-11\/\\textit{IRIS}-8 science meeting,\nand also to Dr.~Jaroslav Dud\\'ik for his constructive comments to improve the paper.\n\n \n\n\n\\bibliographystyle{apj} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\numberwithin{equation}{section}\n\nOur goal in this paper is to prove a concentration inequality for product spaces which is somewhat different in spirit when compared\nwith the well-known concentration inequalities discovered by Talagrand \\cite{Tal1,Tal2}. Roughly speaking, it asserts that under\nsome mild  regularity conditions, every random variable defined on the product of sufficiently many probability spaces exhibits\npseudorandom behavior.\n\nTo state this inequality we need to introduce some pieces of notation. Let $n$ be a positive integer and let\n$(\\Omega_1,\\calf_1,\\mathbb{P}_1),\\dots,(\\Omega_n,\\calf_n,\\mathbb{P}_n)$ be a finite sequence of probability spaces.\nBy $(\\bbo,\\bcalf,\\bbp)$ we denote their product. More generally, for every nonempty subset $I$ of $\\{1,\\dots,n\\}$ by\n$(\\bbo_I,\\bcalf_{\\!I},\\mathbf{P}_{\\!I})$ we denote the product of the spaces $\\langle(\\Omega_i,\\calf_i,\\mathbb{P}_i):i\\in I\\rangle$.\nIn particular, we have\n\\begin{equation} \\label{e1.1}\n\\boldsymbol{\\Omega}=\\prod_{i=1}^n\\Omega_i \\ \\text{ and } \\ \\boldsymbol{\\Omega}_I=\\prod_{i\\in I} \\Omega_i.\n\\end{equation}\n(By convention, $\\boldsymbol{\\Omega}_{\\emptyset}$ stands for the empty set.)\n\nNow let $f\\colon\\boldsymbol{\\Omega}\\to\\rr$ be an integrable random variable and $I\\subseteq \\{1,\\dots,n\\}$ such that $I$ and\n$I^{\\mathsf{c}}\\coloneqq \\{1,\\dots,n\\}\\setminus I$ are nonempty. For every $\\mathbf{x}\\in\\boldsymbol{\\Omega}_I$ let\n$f_{\\mathbf{x}}\\colon\\boldsymbol{\\Omega}_{I^{\\mathsf{c}}}\\to\\rr$ be the section of $f$ at $\\bx$, that is,\n$f_{\\mathbf{x}}(\\mathbf{y})=f\\big( (\\mathbf{x},\\mathbf{y})\\big)$ for every $\\by\\in\\bbo_{I^{\\mathsf{c}}}$.\nFubini's theorem asserts that the random variable $\\mathbf{x}\\mapsto \\ave(f_{\\mathbf{x}})$ is integrable and satisfies\n\\begin{equation} \\label{e1.2}\n\\int \\ave(f_{\\mathbf{x}}) \\, d\\mathbf{P}_{\\!I} = \\ave(f).\n\\end{equation}\nBeyond this basic information, not much can be said at this level of generality. This random variable is rather amorphous.\n\nHowever, our main result shows that if $f\\in L_p(\\bbo,\\bcalf,\\bbp)$ for some $p>1$ and $n$ is sufficiently large,\nthen one can find a set $I$ of coordinates of cardinality proportional to $n$, such that the random variable\n$\\bbo_I\\ni \\mathbf{x}\\mapsto \\ave(f_{\\mathbf{x}})$ is highly concentrated around its mean. Specifically, we have the following theorem.\n\\begin{thm} \\label{t1}\nLet $0<\\ee\\mik 1$ and $1<p\\mik 2$, and set\n\\begin{equation} \\label{e1.3}\nc(\\ee,p)= \\frac{1}{4}\\,\\ee^{\\frac{2(p+1)}{p}} (p-1).\n\\end{equation}\nAlso let $n$ be a positive integer with $n\\meg 2\/c(\\ee,p)$ and let $(\\bbo,\\bcalf,\\bbp)$ be the product of a finite\nsequence $(\\Omega_1,\\calf_1,\\mathbb{P}_1),\\dots,(\\Omega_n,\\calf_n,\\mathbb{P}_n)$ of probability spaces. Then for every\n$f\\in L_p(\\bbo,\\bcalf,\\bbp)$ with $\\|f\\|_{L_p}\\mik 1$ there exists an interval $J$ of\\, $\\{1,\\dots,n\\}$\nwith $J^{\\mathsf{c}}\\neq\\emptyset$ and\n\\begin{equation} \\label{e1.4}\n|J|\\meg c(\\ee,p) \\, n\n\\end{equation}\nsuch that for every nonempty $I\\subseteq J$ we have\n\\begin{equation} \\label{e1.5}\n\\mathbf{P}_{\\!I} \\big( \\{ \\mathbf{x}\\in\\boldsymbol{\\Omega}_I: |\\ave(f_{\\bx}) - \\ave(f)|\\mik \\ee \\}\\big) \\meg 1-\\ee.\n\\end{equation}\n\\end{thm}\nOf course, the case of random variables in $L_p(\\bbo,\\bcalf,\\bbp)$ for $p>2$ is reduced to the case $p=2$. In other words, Theorem \\ref{t1}\nis valid for any $p>1$. Also notice that Theorem 1 can be reformulated as follows.\n\\begin{Th1'}\nLet $\\ee, p, n$ be as in Theorem \\emph{\\ref{t1}} and let $X_1,\\dots,X_n$ be a finite sequence of independent random variables defined\non a probability space $(\\Omega,\\mathcal{F},\\mathbb{P})$. Let $Y$ be another random variable which can be expressed as\n$Y=F(X_1,\\dots,X_n)$ for some measurable function $F$, and assume that $\\ave(|Y|^p)\\mik 1$. Then there exists an interval $J$ of\\,\n$\\{1,\\dots,n\\}$ with $J^{\\mathsf{c}}\\neq\\emptyset$ and satisfying \\eqref{e1.4}, such that for every nonempty $I\\subseteq J$ we have\n\\begin{equation} \\label{e1.6}\n\\mathbb{P}\\big( |\\ave(Y\\, |\\, \\mathcal{F}_I)-\\ave(Y)|\\mik \\ee \\big)\\meg 1-\\ee\n\\end{equation}\nwhere\\, $\\ave(Y\\, |\\, \\mathcal{F}_I)$ stands for the conditional expectation of\\, $Y$ with respect to the\n$\\sigma\\text{-algebra}$ $\\mathcal{F}_I\\coloneqq \\sigma\\big(\\{X_i: i\\in I\\}\\big)$.\n\\end{Th1'}\nWe proceed to discuss another consequence of Theorem \\ref{t1} which is of ``geometric\" nature. Let $\\bbo$ be as in Theorem \\ref{t1}\nand let $A$ be a measurable event of $\\bbo$. Also let $I\\subseteq\\{1,\\dots,n\\}$ such that $I$ and $I^{\\mathsf{c}}$ are nonempty, and\nobserve that if $f$ is the indicator function of $A$, then for every $\\bx\\in\\bbo_I$ the quantity $\\ave(f_{\\bx})$ is the probability of the section\n$A_{\\bx}=\\{\\by\\in\\bbo_{I^{\\mathsf{c}}}: (\\bx,\\by)\\in A\\}$ of $A$ at $\\bx$. Taking into account this remark, we obtain the following corollary.\n\\begin{cor} \\label{c2}\nLet $\\ee, p, n$ and $(\\bbo,\\bcalf,\\bbp)$ be as in Theorem \\emph{\\ref{t1}}. Then for every $A\\in\\bcalf$ there exists an interval $J$\nof\\, $\\{1,\\dots,n\\}$ with $J^{\\mathsf{c}}\\neq\\emptyset$ and satisfying \\eqref{e1.4}, such that for every nonempty $I\\subseteq J$ we have\n\\begin{equation} \\label{e1.7}\n\\mathbf{P}_{\\!I} \\Big( \\big\\{ \\mathbf{x}\\in\\boldsymbol{\\Omega}_I: |\\bbp_{\\!I^{\\mathsf{c}}}(A_{\\bx}) - \\bbp(A)|\\mik \\ee \\bbp(A)^{1\/p}\n\\big\\}\\Big) \\meg 1-\\ee.\n\\end{equation}\n\\end{cor}\nVersions of Corollary \\ref{c2} for subsets of the product of certain finite probability spaces were proved in \\cite{DKT1,DKT2}\nand were applied to combinatorial problems (we will briefly comment on these applications in Subsection 4.1, and for a more complete\nexposition we refer the reader to \\cite{DK}). Theorem \\ref{t1} was motivated by these results and was found in an effort to abstract\ntheir probabilistic features. We expect that Theorem \\ref{t1} will in turn facilitate further applications, possibly even beyond the\ncombinatorial context of \\cite{DKT1,DKT2}.\n\nWe also note that Corollary \\ref{c2} does not hold true for $p=1$ (thus, the range of $p$ in Theorem \\ref{t1} is optimal).\nTo see this, let $n$ be an arbitrary positive integer and for every $i\\in \\{1,\\dots,n\\}$ let $(\\Omega_i,\\calf_i,\\mathbb{P}_i)$\nbe a probability space with the property that there exists a measurable event $A_i$ of $\\Omega_i$ with $\\mathbb{P}_i(A_i)=1\/2$.\nAs above, we denote by $(\\bbo,\\bcalf,\\bbp)$ the product of the spaces\n$(\\Omega_1,\\calf_1,\\mathbb{P}_1),\\dots,(\\Omega_n,\\calf_n,\\mathbb{P}_n)$ and we set $A=A_1\\times\\cdots\\times A_n\\in \\bcalf$.\nNotice that if $I$ is a subset of $\\{1,\\dots,n\\}$ such that $I$ and $I^{\\mathsf{c}}$ are nonempty, then for every $\\bx\\in\\bbo_I$\nwe have $\\bbp_{\\!I^{\\mathsf{c}}}(A_{\\bx})=0$ if $\\bx\\notin\\prod_{i\\in I} A_i$ while $\\bbp_{\\!I^{\\mathsf{c}}}(A_{\\bx})=2^{-n+|I|}$\nif $\\bx\\in\\prod_{i\\in I} A_i$. It follows, in particular, that for every $\\bx\\in\\bbo_I$ we have\n$|\\bbp_{\\!I^{\\mathsf{c}}}(A_{\\bx})-\\bbp(A)|\\meg \\bbp(A)$ and so if $p=1$, then the probability of no section of $A$ can approximate\nthe probability of $A$ with the desired accuracy.\n\nSome final remarks on the proof of Theorem \\ref{t1} which is based on a certain estimate for martingale difference sequences.\nMartingales are, of course, very useful tools for obtaining concentration inequalities (see, e.g., \\cite{L,MS} and the references\ntherein). However, the most interesting part of the argument is how one locates the desired interval $J$. This is achieved\nwith a variant of Szemer\\'{e}di's regularity lemma~\\cite{Sz}, especially as described by Tao in \\cite{Tao}.\n\n\\subsubsection*{Acknowledgments}\nWe would like to thank the anonymous referees for their comments and remarks, and for suggesting Theorem 1$'$.\n\n\n\\section{An estimate for martingale difference sequences}\n\nRecall that a finite sequence $(d_i)_{i=1}^n$ of random variables is said to be a \\emph{martingale difference sequence} if it is of the form\n\\begin{equation} \\label{e2.1}\nd_i=f_i-f_{i-1}\n\\end{equation}\nwhere $(f_i)_{i=1}^n$ is a martingale and $f_0=0$. Clearly, for any $p\\meg 1$, every martingale difference sequence in $L_p$ is a monotone basic\nsequence. Also notice that martingale difference sequences are orthogonal in $L_2$. Hence, for every martingale difference sequence\n$(d_i)_{i=1}^n$ in $L_2$ we have\n\\begin{equation} \\label{e2.2}\n\\Big( \\sum_{i=1}^n \\|d_i\\|^2_{L_2} \\Big)^{1\/2} = \\big\\| \\sum_{i=1}^n d_i\\big\\|_{L_2}.\n\\end{equation}\nWe will need the following extension of this basic fact.\n\\begin{prop} \\label{p3}\nLet $(\\Omega,\\calf,\\mathbb{P})$ be a probability space and $1<p \\mik 2$. Then for every martingale difference\nsequence $(d_i)_{i=1}^n$ in $L_p(\\Omega,\\calf,\\mathbb{P})$ we have\n\\begin{equation} \\label{e2.3}\n\\Big( \\sum_{i=1}^n \\|d_i\\|^2_{L_p} \\Big)^{1\/2} \\mik \\Big(\\frac{1}{p-1}\\Big)^{1\/2} \\cdot \\big\\| \\sum_{i=1}^n d_i\\big\\|_{L_p}.\n\\end{equation}\n\\end{prop}\nThe estimate in \\eqref{e2.3} is optimal, and was recently proved by Ricard and Xu \\cite{RX} who deduced it (via an elegant pseudo-differentiation argument)\nfrom the following sharp uniform convexity inequality for $L_p$ spaces.\n\\begin{prop}[\\cite{BCL}] \\label{p4}\nLet $(\\Omega,\\Sigma,\\mu)$ be an arbitrary measure space and $1<p \\mik 2$. Then for every $x,y\\in L_p(\\Omega,\\Sigma,\\mu)$ we have\n\\begin{equation} \\label{e2.4}\n\\|x\\|_{L_p}^2 + (p-1) \\|y\\|_{L_p}^2 \\mik \\frac{\\|x+y\\|_{L_p}^2+\\|x-y\\|_{L_p}^2}{2}.\n\\end{equation}\n\\end{prop}\nFor details, as well as noncommutative extensions, we refer to \\cite{RX}.\n\n\n\\section{Proof of Theorem \\ref{t1}}\n\nLet $n$ be a positive integer and let $(\\Omega_1,\\calf_1,\\mathbb{P}_1),\\dots,(\\Omega_n,\\calf_n,\\mathbb{P}_n)$ be a finite\nsequence of probability spaces. Recall that by $(\\bbo,\\bcalf,\\bbp)$ we denote their product.\nFor notational simplicity, for every $m\\in \\{1,\\dots,n\\}$ we shall denote by $(\\bbo_m,\\bcalf_m,\\bbp_m)$ the\nproduct of the spaces $(\\Omega_1,\\calf_1,\\mathbb{P}_1),\\dots,(\\Omega_m,\\calf_m,\\mathbb{P}_m)$.\nNotice that the $\\sigma$-algebra $\\bcalf_m$ is not comparable to $\\bcalf$, but it may be ``lifted\" to the full\nproduct $\\bbo$ using the natural projection $\\pi_m\\colon \\bbo\\to \\bbo_m$. Specifically, for every $m\\in \\{1,\\dots,n\\}$ we set\n\\begin{equation} \\label{e3.1}\n\\cals_m=\\{ \\pi_m^{-1}(A): A\\in\\bcalf_m\\}.\n\\end{equation}\nObserve that $\\cals_m=\\{ A\\times \\Omega_{m+1}\\times\\cdots\\times \\Omega_n: A\\in\\bcalf_m\\}$ if $m<n$ while $\\cals_n=\\bcalf$.\nIt follows, in particular, that $(\\cals_m)_{m=1}^n$ is an increasing sequence of sub-$\\sigma$-algebras of $\\bcalf$, and so\nfor every $1<p\\mik 2$ and every $f\\in L_p(\\bbo,\\bcalf,\\bbp)$ with $\\|f\\|_{L_p}\\mik 1$ the sequence\n$\\ave(f \\, | \\, \\cals_1),\\dots, \\ave(f \\, | \\, \\cals_n)$ is a finite martingale which is contained in the unit ball of\n$L_p(\\bbo,\\bcalf,\\bbp)$. We have the following property which is satisfied by all finite martingales of this form.\n\\begin{lem} \\label{l5}\nLet $0<\\theta\\mik 1$ and $1<p\\mik 2$, and let $n$ be a positive integer with\n\\begin{equation} \\label{e3.2}\nn\\meg 8\\,\\theta^{-2}(p-1)^{-1}.\n\\end{equation}\nAlso let $(\\Omega, \\calf,\\mathbb{P})$ be a probability space and $(\\cala_m)_{m=1}^n$ an increasing finite sequence\nof $\\text{sub-}\\sigma\\text{-algebras}$ of $\\calf$. Finally, let $g\\in L_p(\\Omega,\\calf,\\mathbb{P})$ with $\\|g\\|_{L_p}\\mik 1$.\nThen there exist $i,j\\in\\{1,\\dots,n-1\\}$ with\n\\begin{equation} \\label{e3.3}\nj-i\\meg \\big(4^{-1}\\theta^2 (p-1)\\big)\\, n\n\\end{equation}\nsuch that\n\\begin{equation} \\label{e3.4}\n\\|\\ave(g \\, | \\, \\cala_j)-\\ave(g \\, | \\, \\cala_i)\\|_{L_p}\\mik \\theta.\n\\end{equation}\nIn particular, for every $m,l\\in \\{i,\\dots,j\\}$ we have $\\|\\ave(g \\, | \\, \\cala_m)-\\ave(g \\, | \\, \\cala_l)\\|_{L_p}\\mik 2\\theta$.\n\\end{lem}\nThe argument in the proof of Lemma \\ref{l5} is, essentially, the $L_p$-version of the ``energy increment strategy\" devised\nin the proof of Theorem 2.11 in \\cite{Tao}. Further applications of this $L_p$-method are given in \\cite{DKK}.\n\\begin{proof}[Proof of Lemma \\emph{\\ref{l5}}]\nWe argue by contradiction. So, assume that for every pair $i,j\\in \\{1,\\dots,n-1\\}$ satisfying \\eqref{e3.3} we have that\n$\\|\\ave(g \\, | \\, \\cala_j)-\\ave(g \\, | \\, \\cala_i)\\|_{L_p} >\\theta$. Set $\\ell=\\lfloor \\theta^{-2}(p-1)^{-1}\\rfloor+1$ and notice\nthat $\\lfloor (n-2)\/\\ell\\rfloor\\meg 1$. Moreover, for every $k\\in \\{1,\\dots,\\ell+1\\}$ let $i_k=(k-1)\\lfloor (n-2)\/\\ell\\rfloor+1$.\nWith these choices, for every $k\\in\\{1,\\dots,\\ell\\}$ we have $1\\mik i_k < i_{k+1}\\mik n-1$ and\n\\begin{equation} \\label{e3.5}\ni_{k+1}-i_k = \\Big\\lfloor \\frac{n-2}{\\ell}\\Big\\rfloor \\meg \\frac{n}{2\\ell} \\meg \\Big( \\frac{\\theta^2 (p-1)^2}{4} \\Big) n\n\\end{equation}\nwhich implies, by our assumption that the lemma is false, that\n\\begin{equation} \\label{e3.6}\n\\|\\ave(g \\, | \\, \\cala_{i_{k+1}})-\\ave(g \\, | \\, \\cala_{i_k})\\|_{L_p} >\\theta.\n\\end{equation}\nWe set $d_1=\\ave(g \\, | \\, \\cala_{i_1})$ and $d_{k+1}=\\ave(g \\, | \\, \\cala_{i_{k+1}})-\\ave(g \\, | \\, \\cala_{i_k})$\nfor every $k\\in\\{1,\\dots,\\ell\\}$, and we observe that the sequence $(d_k)_{k=1}^{\\ell+1}$ is a martingale difference sequence\nin $L_p(\\Omega,\\calf,\\mathbb{P})$. Therefore, by Proposition \\ref{p3}, we obtain that\n\\begin{eqnarray} \\label{e3.7}\n\\ \\ \\ \\ \\ 1 \\!\\!\\!\\! & < & \\sqrt{p-1}\\theta\\sqrt{\\ell} \\ \\stackrel{\\eqref{e3.6}}{<} \\sqrt{p-1} \\Big(\\sum_{k=1}^{\\ell}\n\\| \\ave(g \\, | \\, \\cala_{i_{k+1}})-\\ave(g \\, | \\, \\cala_{i_k})\\|_{L_p}^2\\Big)^{1\/2} \\\\\n& \\mik & \\sqrt{p-1} \\Big(\\sum_{k=1}^{\\ell+1} \\|d_k\\|_{L_p}^2\\Big)^{1\/2} \\stackrel{\\eqref{e2.3}}{\\mik}\n\\big\\| \\sum_{k=1}^{\\ell+1} d_k \\big\\|_{L_p} = \\|\\ave(g \\, | \\, \\cala_{i_{\\ell+1}})\\|_{L_p}  \\mik \\|g\\|_{L_p} \\nonumber\n\\end{eqnarray}\nwhich contradicts, of course, our hypothesis that $\\|g\\|_{L_p}\\mik 1$.\n\nFinally, let $1\\mik i < j\\mik n$ and notice that for every $i\\mik l\\mik m \\mik j$ we have\n\\[ \\ave(g \\, | \\, \\cala_m)-\\ave(g \\, | \\, \\cala_l)\\! =\\! \\ave(\\ave(g \\, | \\, \\cala_j)-\\ave(g \\, | \\, \\cala_i)\\, | \\, \\cala_m) -\n\\ave(\\ave(g \\, | \\, \\cala_j)-\\ave(g \\, | \\, \\cala_i)\\, | \\, \\cala_l) \\]\nwhich yields that $\\|\\ave(g \\, | \\, \\cala_m)-\\ave(g \\, | \\, \\cala_l)\\|_{L_p}\\mik 2 \\|\\ave(g \\, | \\, \\cala_j)-\\ave(g \\, | \\, \\cala_i)\\|_{L_p}$.\nThe proof of Lemma \\ref{l5} is completed.\n\\end{proof}\nWe will also need the following lemma. In its proof, and in the rest of this paper, we will follow the common practice\nwhen proving inequalities and we will ignore measurability issues since they can be resolved with standard arguments.\n\\begin{lem} \\label{l6}\nLet $n$ be a positive integer and $(\\Omega_1,\\calf_1,\\mathbb{P}_1),\\dots,(\\Omega_n,\\calf_n,\\mathbb{P}_n)$ a finite sequence\nof probability spaces, and denote by $(\\bbo,\\bcalf,\\bbp)$ their product. Also let $I\\subseteq \\{1,\\dots,n\\}$ and assume that\n$I$ and $I^{\\mathsf{c}}$ are nonempty. Then for every $p\\meg 1$ and every $g,h\\in L_p(\\bbo,\\bcalf,\\bbp)$ we have\n\\begin{equation} \\label{e3.8}\n\\int \\|g_{\\bx}-h_{\\bx}\\|_{L_1}^p \\, d\\bbp_{\\!I} \\mik \\|g-h\\|_{L_p}^p.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nNotice first that, by Fubini's theorem,\n\\begin{equation} \\label{e3.9}\n\\|g-h\\|_{L_p}^p = \\int \\Big( \\int |g_{\\bx}-h_{\\bx}|^p \\, d\\bbp_{\\!I^{\\mathsf{c}}} \\Big) \\, d\\bbp_{\\!I}.\n\\end{equation}\nOn the other hand, by Jensen's inequality, for every $\\bx\\in\\bbo_I$ we have\n\\begin{equation} \\label{e3.10}\n\\|g_{\\bx}-h_{\\bx}\\|_{L_1}^p = \\Big( \\int |g_{\\bx}-h_{\\bx}| \\, d\\bbp_{\\! I^{\\mathsf{c}}} \\Big)^p \\mik\n\\int |g_{\\bx}-h_{\\bx}|^p \\, d\\bbp_{\\! I^{\\mathsf{c}}}\n\\end{equation}\nand so, taking the average over all $\\bx\\in \\bbo_I$ and using \\eqref{e3.9}, we obtain the desired estimate.\n\\end{proof}\nWe are ready to complete the proof of Theorem \\ref{t1}.\n\\begin{proof}[Proof of Theorem \\emph{\\ref{t1}}]\nWe fix $f\\in L_p(\\bbo,\\bcalf,\\bbp)$ with $\\|f\\|_{L_p}\\mik 1$ and we set\n\\begin{equation} \\label{e3.11}\n\\theta=\\ee^{\\frac{p+1}{p}}.\n\\end{equation}\nSince $n\\meg 2\/c(\\ee,p)$, by \\eqref{e1.3} and \\eqref{e3.11}, we see that $n\\meg 8\\, \\theta^{-2}(p-1)^{-1}$.\nHence, by Lemma \\ref{l5} applied to the random variable $f$ and the filtration $(\\cals_m)_{m=1}^n$,\nthere exist $i,j\\in\\{1,\\dots,n-1\\}$ satisfying \\eqref{e3.3} and such that\n\\begin{equation} \\label{e3.12}\n\\|\\ave(f \\, | \\, \\cals_j)-\\ave(f \\, | \\, \\cals_i)\\|_{L_p}\\mik \\theta.\n\\end{equation}\nWe set $J=\\{i+1,\\dots,j\\}$ and we claim that the interval $J$ is as desired. To this end notice, first, that $J^{\\mathsf{c}}\\neq\\emptyset$.\nMoreover, by \\eqref{e3.3} and the choice of $c(\\ee,p)$ and $\\theta$ in \\eqref{e1.3} and \\eqref{e3.11} respectively, we have\n\\begin{equation} \\label{e3.13}\n|J|=j-i\\meg \\big( 4^{-1} \\theta^2 (p-1) \\big)\\, n = c(\\ee,p) \\, n.\n\\end{equation}\nNext, let $I$ be a nonempty subset of $J$ and set\n\\begin{equation} \\label{e3.14}\ng=\\ave(f \\, | \\, \\cals_j) \\ \\text{ and } \\ h=\\ave(f \\, | \\, \\cals_i).\n\\end{equation}\nWe have the following claim.\n\\begin{claim} \\label{c7}\nFor every $\\bx\\in\\bbo_I$ we have $\\ave(g_{\\bx})=\\ave(f_{\\bx})$ and $\\ave(h_{\\bx})=\\ave(f)$.\n\\end{claim}\n\\begin{proof}[Proof of Claim \\emph{\\ref{c7}}]\nFix $\\bx\\in \\bbo_I$ and set $\\mathcal{I}=\\{1,\\dots,i\\}$ and $\\mathcal{J}=\\{1,\\dots,j\\}$.\n\nFirst we argue to show that $\\ave(g_{\\bx})=\\ave(f_{\\bx})$. Indeed, observe that $I\\subseteq J\\subseteq \\mathcal{J}$ and so,\nby \\eqref{e3.14} and Fubini's theorem, we see that for every $\\by\\in \\bbo_{\\mathcal{J}\\setminus I}$ the function\n$g_{(\\bx,\\by)}\\colon \\bbo_{\\mathcal{J}^{\\mathsf{c}}}\\to \\rr$ is constant and equal to $\\ave(f_{(\\bx,\\by)})$. Therefore,\n\\begin{eqnarray} \\label{e3.15}\n\\ave(g_{\\bx})=\\int g_{\\bx} \\, d\\bbp_{\\! I^{\\mathsf{c}}} & = &\n\\int \\Big(\\int g_{(\\bx,\\by)} \\, d\\bbp_{\\! \\mathcal{J}^{\\mathsf{c}}} \\Big) \\, d\\bbp_{\\! \\mathcal{J}\\setminus I} \\\\\n& = & \\int \\ave(f_{(\\bx,\\by)}) \\, d\\bbp_{\\! \\mathcal{J}\\setminus I} \\nonumber \\\\\n& = & \\int\\Big(\\int f_{(\\bx,\\by)} \\, d\\bbp_{\\! \\mathcal{J}^{\\mathsf{c}}}\\Big) \\, d\\bbp_{\\! \\mathcal{J}\\setminus I} \\nonumber \\\\\n& = & \\int f_{\\bx} \\, d\\bbp_{\\! I^{\\mathsf{c}}}= \\ave(f_{\\bx}). \\nonumber\n\\end{eqnarray}\n\nWe proceed to show that $\\ave(h_{\\bx})=\\ave(f)$. As above we notice that, by \\eqref{e3.14} and Fubini's theorem, for every\n$\\bz\\in \\bbo_{\\mathcal{I}}$ the function $h_{\\bz}\\colon\\bbo_{\\mathcal{I}^{\\mathsf{c}}}\\to \\rr$ is constant and equal to $\\ave(f_{\\bz})$.\nSince $\\mathcal{I}\\cap I=\\emptyset$, the function $h_{(\\bx,\\bz)}\\colon\\bbo_{(\\mathcal{I}\\cup I)^{\\mathsf{c}}}\\to \\rr$ is also constant\nand equal to $\\ave(f_\\bz)$. Hence,\n\\begin{eqnarray} \\label{e3.16}\n\\ave(h_{\\bx}) = \\int h_{\\bx} \\, d\\bbp_{\\! I^{\\mathsf{c}}} & = &\n\\int\\Big(\\int h_{(\\bx,\\bz)} \\,  d\\bbp_{\\! (\\mathcal{I}\\cup I)^{\\mathsf{c}}}\\Big) \\, d\\bbp_{\\! \\mathcal{I}} \\\\\n& = & \\int \\ave(f_{\\bz}) \\, d\\bbp_{\\! \\mathcal{I}} = \\ave(f) \\nonumber\n\\end{eqnarray}\nand the proof of Claim \\ref{c7} is completed.\n\\end{proof}\nBy Claim \\ref{c7}, for every $\\bx\\in\\bbo_I$ we have\n\\begin{equation} \\label{e3.17}\n|\\ave(f_{\\bx})-\\ave(f)| = \\big| \\int (g_{\\bx}-h_{\\bx}) \\, d\\bbp_{\\! I^{\\mathsf{c}}}\\big| \\mik \\|g_{\\bx}-h_{\\bx}\\|_{L_1}\n\\end{equation}\nand so\n\\begin{equation} \\label{e3.18}\n\\int |\\ave(f_{\\bx})-\\ave(f)|^p \\, d\\bbp_{\\! I} \\mik \\int \\|g_{\\bx}-h_{\\bx}\\|^p_{L_1} \\, d\\bbp_{\\! I}.\n\\end{equation}\nIt follows by Lemma \\ref{l6}, \\eqref{e3.12}, \\eqref{e3.14} and the previous estimate that\n\\begin{equation} \\label{e3.19}\n\\int |\\ave(f_{\\bx})-\\ave(f)|^p \\, d\\bbp_{\\! I} \\mik \\theta^p.\n\\end{equation}\nTherefore, by Markov's inequality, we conclude that\n\\begin{equation} \\label{e3.20}\n\\bbp_{\\! I} \\big( \\{ \\bx\\in\\bbo_I: |\\ave(f_{\\bx})-\\ave(f)| \\meg \\theta^{\\frac{p}{p+1}} \\} \\big)\n\\mik \\frac{\\theta^p}{\\theta^{p^2\/(p+1)}} = \\theta^{\\frac{p}{p+1}}\n\\end{equation}\nwhich is equivalent to saying, by the choice of $\\theta$ in \\eqref{e3.11}, that\n\\begin{equation} \\label{e3.21}\n\\bbp_{\\! I} \\big( \\{ \\bx\\in\\bbo_I: |\\ave(f_{\\bx})-\\ave(f)| \\mik \\ee \\} \\big) \\meg 1-\\ee.\n\\end{equation}\nThe proof of Theorem \\ref{t1} is completed.\n\\end{proof}\n\n\n\\section{Comments}\n\n\\subsection*{4.1}\n\nFor every positive integer $n$ and every finite set $A$ with $|A|\\meg 2$ let\n\\begin{equation} \\label{e4.1}\nA^n=\\{(a_1,\\dots,a_n): a_1,\\dots, a_n\\in A\\}\n\\end{equation}\nand let $\\mathbb{P}$ be the uniform probability measure on the hypercube $A^n$. Moreover, for every nonempty\nsubset $I$ of $\\{1,\\dots,n\\}$ by $\\mathbb{P}_{\\! A^I}$ we denote the uniform probability measure on\n$A^I\\coloneqq \\{(a_i)_{i\\in I}: a_i\\in A \\text{ for every } i\\in I\\}$. We have the following lemma.\n\\begin{lem} \\label{l8}\nLet $k,m$ be positive integers with $k\\meg 2$ and $0<\\eta\\mik 1$. Also let $A$ be a set with $|A|=k$ and let $n$ be a positive integer with\n\\begin{equation} \\label{e4.2}\nn\\meg \\frac{16m k^{3m}}{\\eta^3}\n\\end{equation}\nThen for every subset $D$ of $A^n$ there exists an interval $I\\subseteq \\{1,\\dots,n\\}$ with $|I|=m$ such that for every $t\\in A^I$ we have\n\\begin{equation} \\label{e4.3}\n|\\mathbb{P}_{\\! A^{I^{\\mathsf{c}}}}(D_t)-\\mathbb{P}(D)|\\mik \\eta\n\\end{equation}\nwhere  $D_t=\\{s\\in A^{I^{\\mathsf{c}}}: (t,s)\\in D\\}$ is the section of $D$ at $t$.\n\\end{lem}\nA simpler version of Lemma \\ref{l8} was proved in \\cite{DKT1} and was used as a tool in a proof of the density Hales--Jewett\ntheorem \\cite{FK}; closely related applications were also obtained in \\cite{DKT2} (see also \\cite{DK}). Of course, the main point\nin Lemma \\ref{l8} is that by demanding a large---but not necessarily dense---set $I$ of coordinates, one can upgrade Theorem \\ref{t1}\nand guarantee that the probability of \\textit{every} section of $D$ along elements of $A^I$ is essentially equal to the probability of $D$.\nWe proceed to the proof.\n\\begin{proof}[Proof of Lemma \\emph{\\ref{l8}}]\nWe view $A$ and $A^n$ as discrete probability spaces equipped with their uniform probability measures. Then notice that the\nprobability space $A^n$ is the product of $n$ many copies of $A$. Next we set $\\ee = \\eta k^{-m}2^{-1\/3}$ and we observe that,\nby \\eqref{e1.3} and \\eqref{e4.2}, we have $n\\meg m\\big(2\/c(\\ee,2)\\big)$. Hence, by Corollary \\ref{c2} applied to the set $D$,\nthe constant $\\ee$ and $p=2$, there exists an interval $J\\subseteq \\{1,\\dots,n\\}$ with $J^{\\mathsf{c}}\\neq \\emptyset$ and\n$|J|\\meg 2m$, and satisfying \\eqref{e1.7} for every nonempty $I\\subseteq J$. We select an interval $I\\subseteq J$ with $|I|=m$\nand we claim that $I$ is as desired. Indeed, by the choice of $\\ee$, we have\n\\begin{equation} \\label{e4.4}\n\\mathbb{P}_{\\! A^I}\\big( \\{ t\\in A^I: |\\mathbb{P}_{\\! A^{I^{\\mathsf{c}}}}(D_t)-\\mathbb{P}(D)|\\mik \\ee \\} \\big) \\meg 1-\\ee\n\\meg 1- k^{-m}2^{-1\/3} > 1-\\frac{1}{|A^I|}\n\\end{equation}\nwhich implies that $|\\mathbb{P}_{\\! A^{I^{\\mathsf{c}}}}(D_t)-\\mathbb{P}(D)|\\mik \\ee$ for every $t\\in A^I$. Since $\\ee\\mik\\eta$\nwe conclude that the estimate in \\eqref{e4.3} is satisfied and the proof is completed.\n\\end{proof}\n\n\\subsection*{4.2}\n\nThere is a natural extension of Theorem \\ref{t1} which deals simultaneously with a family of random variables.\nAlthough in applications one usually encounters only finite families of random variables (see, e.g., \\cite{DKT2}),\nthe cleanest formulation of this extension is for stochastic processes indexed by the sample space of a probability\nspace $(T,\\Sigma,\\mu)$. Specifically, we have the following theorem.\n\\begin{thm} \\label{t9}\nLet $0<\\ee\\mik 1$ and $1<p\\mik 2$, and set\n\\begin{equation} \\label{e4.5}\nc'(\\ee,p) = \\frac{1}{4}\\, \\ee^{\\frac{2(2p+1)}{p}} (p-1).\n\\end{equation}\nAlso let $n$ be a positive integer with $n\\meg 2\/c'(\\ee,p)$ and let $(\\bbo,\\bcalf,\\bbp)$ be the product of a finite sequence\n$(\\Omega_1,\\calf_1,\\mathbb{P}_1),\\dots,(\\Omega_n,\\calf_n,\\mathbb{P}_n)$ of probability spaces. Finally, let $(T,\\Sigma,\\mu)$\nbe a probability space and $F\\colon T\\times \\bbo\\to \\rr$ a random variable with $\\|F\\|_{L_p}\\mik 1$. Then there exist\n$G\\in\\Sigma$ with $\\mu(G)\\meg 1-\\ee$ and an interval $J$ of $\\{1,\\dots,n\\}$  with $J^{\\mathsf{c}}\\neq\\emptyset$ and\n\\begin{equation} \\label{e4.6}\n|J|\\meg c'(\\ee,p)\\, n\n\\end{equation}\nsuch that for every $t\\in G$ and every nonempty $I\\subseteq J$ we have\n\\begin{equation} \\label{e4.7}\n\\mathbf{P}_{\\!I} \\big( \\{ \\mathbf{x}\\in\\boldsymbol{\\Omega}_I: |\\ave(F_{(t,\\bx)}) - \\ave(F_t)|\\mik \\ee \\}\\big) \\meg 1-\\ee.\n\\end{equation}\n\\end{thm}\nThe proof of Theorem \\ref{t9} is similar to the proof of Theorem \\ref{t1} and so we will briefly sketch the argument.\nFirst, for every $m\\in\\{1,\\dots,n\\}$ we define\n\\begin{equation} \\label{e4.8}\n\\widetilde{\\cals}_m= \\Sigma\\otimes \\cals_m\n\\end{equation}\nwhere $\\cals_m$ is as in \\eqref{e3.1}. Each $\\widetilde{\\cals}_m$ is a sub-$\\sigma$-algebra of $\\Sigma\\otimes\\bcalf$ and, moreover,\nthe finite sequence $(\\widetilde{\\cals}_m)_{m=1}^n$ is increasing. Hence, by Lemma \\ref{l5} applied to $F$,\nthe filtration $(\\widetilde{\\cals}_m)_{m=1}^n$ and $\\theta=\\ee^{(2p+1)\/p}$, there exist $i,j\\in \\{1,\\dots,n-1\\}$ with\n\\begin{equation} \\label{e4.9}\nj-i \\meg \\big( 4^{-1}\\, \\theta^2 (p-1)\\big)\\, n \\stackrel{\\eqref{e4.5}}{=} c'(\\ee,p)\\, n\n\\end{equation}\nand such that\n\\begin{equation} \\label{e4.10}\n\\| \\ave(F \\, | \\, \\widetilde{\\cals}_j) - \\ave(F \\, | \\, \\widetilde{\\cals}_i)\\|_{L_p} \\mik \\ee^{\\frac{2p+1}{p}}.\n\\end{equation}\nSet $g=\\ave(F \\, | \\, \\widetilde{\\cals}_j)$ and $h=\\ave(F \\, | \\, \\widetilde{\\cals}_i)$, and notice that,\nby Lemma \\ref{l6} and \\eqref{e4.10},\n\\begin{equation} \\label{e4.11}\n\\int \\|g_t-h_t\\|_{L_1}^p \\, d\\mu \\mik \\ee^{2p+1}.\n\\end{equation}\nTherefore, by Markov's inequality, there exists $G\\in\\Sigma$ with $\\mu(G)\\meg 1-\\ee$ such that\n\\begin{equation} \\label{e4.12}\n\\|g_t-h_t\\|_{L_1}\\mik \\ee^2\n\\end{equation}\nfor every $t\\in G$. The set $G$ and the interval $J\\coloneqq \\{i+1,\\dots,j\\}$ are as desired. Indeed, let $I$ be a nonempty subset of $J$.\nObserve that $g_t=\\ave(F_t \\, | \\, \\cals_j)$ and $h_t=\\ave(F_t \\, | \\, \\cals_i)$ for every $t\\in T$ which implies, by Claim \\ref{c7}, that\n$\\ave(g_{t,\\bx})=\\ave(F_{t,\\bx})$ and $\\ave(h_{t,\\bx})=\\ave(F_t)$ for every $t\\in T$ and every $\\bx\\in\\bbo_I$. Taking into account these\nobservations, we conclude that the estimate in \\eqref{e4.7} follows from \\eqref{e4.12} and a second application of Markov's inequality.\n\n\\subsection*{4.3}\n\nRecall that a Banach space $X$ is said to be \\textit{uniformly convex} if for every $\\ee>0$ there exists $\\delta>0$ such that for\nevery $x,y\\in X$ with $\\|x\\|_X=\\|y\\|_X=1$ and $\\|x-y\\|_X\\meg \\ee$ we have that $\\|(x+y)\/2\\|_X\\mik 1-\\delta$. A classical result due to\nJames \\cite{J} and, independently, V. Gurarii and N. Gurarii \\cite{GG}, implies that for every uniformly convex Banach space $X$ and\nevery $p>1$ there exist $q\\meg 2$ and a constant $C>0$ such that for every $X$-valued martingale difference sequence $(d_i)_{i=1}^n$ we have\n\\begin{equation} \\label{e4.13}\n\\Big( \\sum_{i=1}^n \\|d_i\\|_{L_p(X)}^q \\Big)^{1\/q} \\mik C \\big\\| \\sum_{i=1}^n d_i \\big\\|_{L_p(X)}.\n\\end{equation}\n(See, also, \\cite{Pi} for a proof and a detailed presentation of related material.) Using this estimate and arguing precisely as in Section 3,\nwe obtain the following vector-valued version of Theorem \\ref{t1}.\n\\begin{thm} \\label{t10}\nFor every uniformly convex Banach space $X$, every $0<\\ee\\mik 1$ and every $p>1$ there exists a constant $c(X,\\ee,p)>0$ with the following\nproperty. Let $n$ be a positive integer with $n\\meg c(X,\\ee,p)^{-1}$ and let $(\\bbo,\\bcalf,\\bbp)$ be the product of a finite sequence\n$(\\Omega_1,\\calf_1,\\mathbb{P}_1),\\dots,(\\Omega_n,\\calf_n,\\mathbb{P}_n)$ of probability spaces. If $f\\colon\\bbo\\to X$ is a random variable\nwith $\\|f\\|_{L_p(X)}\\mik 1$, then there exists an interval $J$ of $\\{1,\\dots,n\\}$ with $J^{\\mathsf{c}}\\neq\\emptyset$ and\n\\begin{equation} \\label{e4.14}\n|J| \\meg c(X,\\ee,p)\\, n\n\\end{equation}\nsuch that for every nonempty $I\\subseteq J$ we have\n\\begin{equation} \\label{e4.15}\n\\mathbf{P}_{\\!I} \\big( \\{ \\mathbf{x}\\in\\boldsymbol{\\Omega}_I: \\|\\ave(f_{\\bx}) - \\ave(f)\\|_X\\mik \\ee \\}\\big) \\meg 1-\\ee.\n\\end{equation}\n\\end{thm}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAnswer Set Programming (ASP) is a widely used problem solving\napproach. It offers declarative languages that can be used \nto formalize actions, planning, and agent policies, in an expressive setting\n(e.g.\\ direct and indirect action effects)\n\\cite{lif99c,bara-2003,DBLP:journals\/aim\/ErdemGL16}),\nand has led to dedicated action languages \\cite{lifschitz08}.\nThis and the availability of efficient solvers\nmakes ASP a convenient tool for representing and reasoning about actions.\n\nConsider a scenario in which a robot may be in an unknown grid-cell environment with\nobstacles and aim to find a missing person (Fig.~\\ref{fig:scenario}). \nIt acts according to a policy, which tells it\n\\begin{wrapfigure}{r}{3cm}\n\n\\centering\n\n\\vspace*{-.5\\baselineskip}\n\n\\includegraphics[height=1.7cm]{person-search-initial-4x4.png}\n\n\\vspace*{-.5\\baselineskip}\n\n\\caption{\\quad Missing person search in an unkown environment}\n\\label{fig:scenario}\n\\vspace*{-\\baselineskip}\n\n\\end{wrapfigure}\nwhere to move next,\ndepending on the current observations (free \/ blocked cells) and\npossible memory of past observations, until the person is found. To\nthis end, an action domain with a policy description, formalized in an\nASP program, is evaluated in each step. \nNaturally, we wonder whether the policy works, i.e.,\nthe person is always found, regardless of actual obstacle\nlocations. This can generate a large state space (for an $n{\\times}n$\ngrid, of size larger than $2^{n{\\times}n}$) and simple approaches such\nas searching for a run in which the policy fails\nquickly become infeasible.\n\nTo overcome this, we aim at using abstraction, which is a well-known\napproach to reduce problem complexity. In a deliberate loss of\ninformation, the problem is approximated to achieve a smaller or\nsimpler state space, at the price of spurious\ncounterexamples to the behavior \\cite{clarke03}. In planning, abstraction\nis mostly focused on relaxing the model, by omitting preconditions of\nactions and details of the domain model\n\\cite{giunchiglia1992theory,knoblock1994automatically,sacerdoti1974planning}.\nCartesian abstraction \\cite{seipp2013counterexample} refines\nin the spirit of \\cite{clarke03} failure states of abstract\ntrajectories, starting from a trivial abstraction; the classical\nplanning setting, however, disregards incomplete\ninitial states\n(a known source of complexity).\nThese works do not consider policies with background knowledge that can do decision-making with information beyond action effects.\n\nIn the area of ASP-based action languages, abstraction has not been\nconsidered so far, and neither in the broader ASP context.\nIn order to exploit abstraction for reasoning about action\ndescriptions and policies in ASP, we need an abstraction method for\nASP programs that offers the following features. First, information\nloss on both the model and the domain is possible. Second, \nrelationships and dependencies expressed in the program should be\nlargely preserved. And third, abstractions should be (semi-)\nautomatically computable. We address this challenge with the\nfollowing contributions.\n\n\\begin{itemize}\n\\item We introduce a method to abstract ASP programs in order to\n  obtain an over-approximation of the answer sets of a program\n  $\\Pi$. That is, a program $\\Pi'$ is constructed such that each answer\n  set $I$ of $\\Pi$ is abstracted to some answer set $I'$ of $\\Pi'$; \n  While this abstraction is many to one, {\\em spurious} answer sets of\n  $\\Pi'$ not corresponding to any answer set of $\\Pi$ may exist. \n \n\n\n\\item For abstraction, we consider omission of literals and also domain abstraction,\n  where domain elements are merged. Note that omitting is different\n  from forgetting literals (see \\cite{DBLP:conf\/lpnmr\/Leite17} for an overview), as the latter aims at\n  preserving information.\n \n  The abstraction types \n  can be combined and in principle iterated to build\n  hierarchical abstractions.\n\n\\item The method largely preserves the structure of the rules and\n  works modularly for non-ground programs. Thus, it is particularly attractive\n  for abstraction of parameterized problems, as e.g.,\\ in the search\n  scenario (grid size $n$). Furthermore, it respects built-in\n  predicates such as equality ($=$), comparisons ($<,\\leq$) etc., and\n  can be readily implemented, with little information on the underlying\n  abstraction.\n\\item We illustrate the use of the abstraction method for reasoning about\n   actions, in particular to find counterexamples to an \n   agent policy. Here, it can be particularly useful to\n   identify  and explain ``essential'' aspects of failure.\n\\end{itemize}\n\nWhile abstraction for ASP programs is motivated by applications in reasoning\nabout actions, the approach is domain independent and can be\nutilized in other contexts as well.\n\n\\section{Preliminaries}\n\n\\leanparagraph{ASP} A logic program $\\Pi$ is a set of rules $r$ of the form\n\n\\smallskip\n\n\\centerline{$\\alpha_0 \\leftarrow \\alpha_1,\\dots,\\alpha_m,\\mi{not}\\\n\\alpha_{m+1},\\dots,\\mi{not}\\ \\alpha_n,\\ \\ 0\\,{\\leq}\\, m \\,{\\leq}\\, n,$}\n\n\\smallskip\n\n\\noindent where each $\\alpha_i$ is a first-order (function-free) atom and \n$\\mi{not}$\nis default negation; $r$ is a \\emph{constraint}\nif $\\alpha_0$ is falsity ($\\bot$, then omitted) and a \\emph{fact}  if\n$n\\,{=}\\,0$. We also write $\\alpha_0 \\leftarrow B^+(r),\\mi{not}\\ B^-(r)$,\nwhere $B^+(r)$ (positive body) is the set $\\{\\alpha_1, \\dots,\n\\alpha_m\\}$ and $B^-(r)$ (negative body) the set\n$\\{\\alpha_{m+1},$ $\\dots,\\alpha_n\\}$, \nor $\\alpha_0 {\\leftarrow} B(r)$.\nRules with variables \nstand for the set of their ground instances. \nSemantically, \n$\\Pi$ induces a set of answer sets\n\\cite{gelfond1991classical}, which\nare Herbrand models (sets $I$ of ground atoms) of $\\Pi$ \njustified\nby the rules, \nin that $I$ is a minimal model of $f\\Pi^I=$ $\\{ r \\in \\Pi\n\\mid I \\models B(r)\\}$\n\\cite{FLP04}.\nThe set of answer sets of a program $\\Pi$ is denoted as $AS(\\Pi)$. \nNegative literals $\\neg \\alpha$ can be \nencoded \nusing atoms $\\mi{neg}\\_\\alpha$ and\nconstraints $\\leftarrow \\alpha,\\mi{neg}\\_\\alpha$.\n\n \n\nCommon syntactic extensions are \\emph{choice rules} of the form\n$\\{\\alpha\\} \\leftarrow B$, which stands for the rules $\\alpha \\leftarrow\nB, \\mi{not}\\ \\alpha'$ and $\\alpha' \\leftarrow B, \\mi{not}\\, \\alpha$, where $\\alpha'$ is a\nnew atom, and cardinality constraints and conditional\nliterals\n\\cite{simons2002extending}; in particular, $i_\\ell\\{\\,a(X)\\,{:}\\,b(X)\\,\\}i_u$ is true whenever at least $i_\\ell$ and at most $i_u$ instances of $a(X)$ subject to $b(X)$ are true.\n\n\\leanparagraph{Describing actions and states} \nASP is used to describe dynamic domains by a ``history\nprogram\" \\cite{lif99c}, \nwhose answer sets represent\npossible evolutions of the system over a \ntime interval. This is\nachieved by adding a time variable to the atoms, and\nintroducing action atoms that may cause changes\nover time. An action is defined by its preconditions and effects over the\natoms. For illustration, the following rule describes a \\emph{direct effect} of the\naction $\\mi{goTo(X,Y)}$ over the robot's location $rAt(X,Y)$\n\\vspace*{-0.25\\baselineskip}\n\\begin{equation} \\begin{array} l\n\\mi{rAt}(X,Y,T{+}1) \\leftarrow \\mi{goTo}(X,Y,T).\n\\end{array} \\eeq {eq:direct}\n\\vspace*{-1.5\\baselineskip}\n\n\\noindent Actions can also have \\emph{indirect effects} over the state (rules not\nmentioning actions); e.g.,\n                             \nthe robot location  is visited:\n\\vspace*{-0.25\\baselineskip}\n\\begin{equation} \\begin{array} l\n\\mi{visited}(X,Y,T) \\leftarrow \\mi{rAt}(X,Y,T).\n\\end{array} \\eeq {eq:indirect}\n\\noindent Inertia laws (unaffectedness) can be elegantly expressed, e.g.\\\n\\begin{eqnarray}\n \\mi{rAt}(X,Y,T{+}1) \\! \\leftarrow \\!\\!\\!\\!\\! & \\mi{rAt}(X,Y,T),  \\mi{not}\\, \\neg \\mi{rAt}(X,Y,T{+}1). \\nonumber\n\n\\end{eqnarray}\n\n\\noindent says that the robot location remains by default the same.\n\nOne can also give further restrictions on the state, e.g., the robot and\nan obstacle can never be\nin the same %\ncell.\n\\vspace*{-0.25\\baselineskip}\n\\begin{equation} \\begin{array} l\n\\bot \\leftarrow \\mi{rAt}(X,Y,T), \\mi{obsAt}(X,Y,T).\n\\end{array} \\eeq {eq:obs}\n\n\\noindent Constraints can also define \\emph{preconditions} of an action, e.g.,\n\\vspace*{-0.25\\baselineskip}\n\\begin{equation} \\begin{array} l\n\\bot \\leftarrow \\mi{goTo}(X,Y,T), \\mi{obsAt}(X,Y,T).\n\\end{array} \\eeq {eq:precond}\nDedicated action languages carry this idea further with special \nsyntax for such axioms \\cite{gelfondaction98}, and can be translated to ASP \\cite{giunchiglia2004nct}.\n\n\\leanparagraph{Describing a policy} In addition to defining actions as above, ASP can also be used for further reasoning about the actions by singling out some of them under certain conditions. A policy that singles out the actions to execute from the current state can be described with a set\nof rules,\nwhere rules of\nform $a {\\leftarrow} B$ choose an action $a$ when certain\nconditions $B$ are satisfied in the state. \nFurther rules may \ndescribe auxiliary literals that are used by $B$. \n\nThe rules below make the agent move towards some farthest point on the grid, unless the person is seen or caught. \nIn the latter case, the agent moves towards the person's location. \n\\begin{equation}\n \\begin{split}\n&1\\,\\{\\mi{goTo}(X1,Y1,T) : \\mi{farthest}(X,Y,X1,Y1,T)\\}\\,1\\\\%[-.5ex]\n& \\leftarrow \\mi{rAt}(X,Y,T), \\mi{not\\ seen}(T), \\mi{not\\ caught}(T).\\\\%[-.5ex]\n&\\mi{goTo}(X,Y,T) \\leftarrow \\mi{seen}(T), \\mi{not\\ caught}(T), \\mi{pAt}(X,Y,T). \n\\end{split}\n\\label{eq:pol_formula}\n\\raisetag{24pt}\n\\end{equation} \n\\vspace*{-0.25\\baselineskip}\n\n\\noindent The farthest point is determined by the agent's location and the\ncells considered at that state;\nit is thus an indirect effect of\nthe previous move.\nThis also applies to $\\mi{seen}$ and $\\mi{caught}$:\n\\vspace*{-0.25\\baselineskip}\n\\begin{equation} \\begin{split}\n&\\mi{caught}(T) \\leftarrow \\mi{rAt}(X,Y,T), \\mi{pAt}(X,Y,T).\\\\%[-.5ex]\n&\\mi{seen}(T) \\leftarrow \\mi{seeReachable}(X,Y,T), \\mi{pAt}(X,Y,T).\n\\end{split}\n\\raisetag{26pt}\n \\eeq {eq:pol_formula2}\nNotice that above is on choosing single actions. For policies that choose a sequence of actions, the policy rules will be more involved, as the stages of the plan might have to be considered.\n\n\n\\section{Constructing an Abstract ASP Program}\\label{sec:auto_abs}\n\nOur aim is to over-approximate a given program through constructing a\nsimpler program by reducing the vocabulary and preserving the behavior\nof the original program (i.e., the\nresults of reasoning on the original program are not lost), \nat the cost of obtaining spurious\nsolutions. \n\n\n\\begin{defn}\nGiven two programs $\\Pi$ and $\\Pi'$ with $|{\\bf L}|{\\geq}|{\\bf L}'|$, $\\Pi'$ is an \\emph{abstraction} of $\\Pi$ if there exists a mapping $m : {\\bf L} \\rightarrow {\\bf L'}$ such that for $I\\in AS(\\Pi)$, $I'=\\{m(l) ~|~ l \\in I\\}$ is an answer set of $\\Pi'$.\n\\end{defn}\n\n\nWe consider two important base cases for an abstraction mapping $m$. Literal\nomission is about omitting certain literals from the program, while\ndomain abstraction is on clustering different constants in the domain\nand treating them as equal. \n\n\n\\begin{defn}\nGiven a program $\\Pi$ and its abstraction $\\Pi'$,\n\\begin{myenumerate}\n\\item $\\Pi'$ is a \\emph{literal omission abstraction of $\\Pi$} if a\n  set $L \\subseteq {\\bf L}$ of literals is omitted and the rest is\n  kept, i.e.,\n  ${\\bf L}' = {\\bf L}\n  \\setminus L$ and $m(l)=\\emptyset$ if $l\\in L$ and $m(l)=l$ otherwise.\n\\item $\\Pi'$ is a \\emph{domain abstraction of $\\Pi$} if there is a function $m_{d}\\,{:}\\,D\n\\,{\\rightarrow}\\, \\widehat{D}$ for a Herbrand domain $D$ and its abstraction $\\widehat{D}$, such that for $l\\,{=}\\,p(v_1,\\dots,v_n)$ we have $m(l)\\,{=}\\,p(m_{d}(v_1),$ \\ldots, $m_{d}(v_n))$.\n\\end{myenumerate}\n\\end{defn}\n\nIn the following sections, we show a systematic way of building an abstraction of a given ASP program.\nWhen constructing an abstract program for a given mapping, the aim is\nto ensure that every original answer set $I$ is mapped to some\nabstract answer set, while (unavoidably) some spurious abstract answer\nsets may be introduced. Thus, an over-approximation of the original program is achieved.\nThe abstraction types can be composed to obtain further abstractions.\n\nNotice that literal omission is different\nthan forgetting (see \\cite{DBLP:conf\/lpnmr\/Leite17} for an overview),\nas it ensures the over-approximation of the original program by making\nsure that all of the original answer sets are preserved in the\nabstract program, without resorting to language extensions such as\nnested logic programs that otherwise might be necessary. \n\n\\vspace*{-0.5\\baselineskip}\n\n\\subsection{Literal omission} \n\n\nGiven $L$, we build from $\\Pi$ a program $\\Pi_{\\overline{L}}^m$\nas follows. For every literal $l \\in ({\\bf L} \\setminus L) \\cup \\{\\bot \\}$ and rule\n$r: l \\leftarrow B(r)$ in $\\Pi$, \n\\begin{enumerate}[(1)]\n\\itemsep=2pt\n\\item if $ B(r) \\subseteq {\\bf L} \\setminus L$, we include $m(l)\n  \\leftarrow m(B(r))$;\n\\item otherwise,  if $l\\,{\\neq}\\,\\bot$ we include for every $l' \\in B(r) \\cap\n  L$ the rule $0 \\{ m(l) \\} 1 \\leftarrow m(B(r)\\setminus \\{l'\\}).$\n\\end{enumerate}\nNotice that constraints are omitted in the constructed program if the body contains an omitted literal. If instead, the constraint gets shrunk, then for some interpretation $\\widehat{I}$, the body may fire in $\\Pi_{\\overline{L}}^m$, while it was not the case in $\\Pi$ for any $I \\in AS(\\Pi)$ s.t. $m(I)=\\widehat{I}$. Thus \n$I$ cannot be mapped to an abstract answer set of $\\Pi_{\\overline{L}}^m$, i.e., $\\Pi_{\\overline{L}}^m$ is not an over-approximation of $\\Pi$.\n\nOmitting non-ground literals means omitting\nall occurrences of the predicate.\nIf in a rule $r$, the omitted non-ground literal $p(V_1,\\dots,V_n)$ shares some arguments, $V_i$, with the head $l$, then $l$ is conditioned over $\\mi{dom}(V_i)$ (a special predicate to represent the Herbrand domain) in the constructed rule, so that all values of $V_i$ are considered.\n\n\\begin{exmp}\\label{ex:toy}\nConsider the following simple program $\\Pi$:\n\\vspace*{-.25\\baselineskip}\n\\begin{align}\n  a(X_1,X_2) &\\leftarrow c(X_1), b(X_2). \\label{eq:1}\\\\\n  d(X_1,X_2) &\\leftarrow a(X_1,X_2), X_1{\\leq}X_2.\\label{eq:2}\n\\end{align} \n\\vspace*{-1.25\\baselineskip}\n\nIn omitting $c(X)$, while rule \\eqref{eq:2} remains the same, rule \\eqref{eq:1} changes to \n$0\\{a(X_1,X_2)\\,{:}\\,\\mi{dom}(X_1)\\}1 \\leftarrow b(X_2)$.\nFrom\n$\\Pi$\nand the facts $c(1),b(2)$, we get the answer set $\\{c(1)$, $b(2)$,\n$a(1,2)$, $d(1,2)\\}$, and with $c(2),b(2)$ we get\n$\\{c(2)$, $b(2),$ $a(2,2)$, $d(2,2)\\}$.\n After omitting $c(X)$, the abstract answer\nsets with fact $b(2)$ become $\\{b(2),$ $a(1,2),$ $d(1,2)\\}$ and\n$\\{b(2)$, $a(2,2),$ $d(2,2)\\}$, which cover the original answers, so that all original answer sets can be mapped to\nsome abstract answer set.\n\\end{exmp}\n\nFor a semantical more fine-grained removal, e.g., removing $c(X)$ for $X{<}3$, rules may be split in cases, e.g., (\\ref{eq:1}) into $X_1{<}3$ and $X_1{\\geq} 3$, and\ntreated after renaming separately.\n\nThe following result shows that $\\Pi_{\\overline{L}}^m$ can be seen as an over-approximation of $\\Pi$\n\n\\begin{thm}\nFor every $I \\in AS(\\Pi)$ and set $L$ of literals,\n$I_{|\\overline{L}} \\in AS(\\Pi_{\\overline{L}}^m)$ where $I_{|\\overline{L}} =I\\setminus L$.\n\\end{thm}\n\n\\noindent By introducing choice rules for any rule that contains the omitted literal, all possible cases that could have been achieved by having the omitted literal in the rule are covered. Thus, the abstract answer sets cover the original answer sets.\n\n\\vspace*{-0.25\\baselineskip}\n\n\\subsection{Domain abstraction} \n\nAbstraction on the domain, $D$, divides it into equivalence classes, $\\widehat{D}=\\{\\hat{d}_1,\\dots,\\hat{d}_k\\}$, where some values of the variables are seen as equal. \nSuch an abstraction can be constructed by keeping the structure of the\nliterals, and having abstract rules similar to the original ones. The\noriginal rule may rely on certain built-in relations between the\nliterals' variables, e.g., $=,\\neq,<,\\leq$, such as \\eqref{eq:2};\nwe can automatically lift \nthem to the abstraction\n(discussed below), and \naim to use\n\\smallskip\n\n\\centerline{$d(\\widehat{X}_1,\\widehat{X}_2) \\leftarrow a(\\widehat{X}_1,\\widehat{X}_2), \\widehat{X}_1{\\leq}\\widehat{X}_2.$}\n\n\\smallskip\n\n\\noindent where $\\widehat{X}_1,\\widehat{X}_2$ are variables ranging over $\\widehat{D}$.\nHowever, due to the mapping, the lifted relations may create\nuncertainties which must be dealt with. \nE.g.\\\nfor a mapping $m_d(\\{1,2,3\\})=k$, the atom $a(k,k)$ can be true in\nthe abstract state because $a(3,2)$ is true in the\noriginal state. \n The original program can have answer sets $I$ that contain (i) $a(3,2),\\mi{not}\\ d(3,2)$, or (ii) $a(2,2),d(2,2)$. If we keep the structure of the original rule, in any abstract answer set $d(k,k)$ must hold if $a(k,k)$ holds; hence, no $I$ with (i) can be mapped to an abstract answer set. This would result in losing a possible answer set. \nWe can avoid this\nby using an altered rule\n\\smallskip\n\n\\centerline{$0\\{d(\\widehat{X}_1,\\widehat{X}_2)\\}1 \\leftarrow a(\\widehat{X}_1,\\widehat{X}_2), \\widehat{X}_1{\\leq}\\widehat{X}_2.$}\n\n\\smallskip\n\nA naive approach would \nabstract all rules by modifying the heads\nto choice rules. However,\nnegation in rule bodies may cause\na loss of\noriginal answer sets in the abstraction. Say we have a rule with negation in the body, $d(X,X) \\leftarrow \\mi{not}\\ a(X,X)$. If it is only changed to a choice rule in the abstract program, when $a(k,k)$ holds we will not have $d(k,k)$, while originally we can have $\\{d(2,2),a(3,2)\\}$. Such rules must be treated specially to catch the cases of obtaining $d(k,k)$ while $a(k,k)$ holds.\n\nFor a finer-grained and systematic approach,\nwe focus on rules of form $r: l \\leftarrow B(r), \\Gamma_{\\mi{rel}}(r)$\nwhere the variables \nin $B(r)$ are standardized apart\nand $\\Gamma_{\\mi{rel}}$ consists\nof built-in relation literals that\nimpose restrictions \non the variables in $B(r)$.\n\n\\begin{exmp}\n\\label{ex:standardize}\nThe rules  (\\ref{eq:1}) and (\\ref{eq:2}) are standardized apart and they\nhave  $\\Gamma_{\\mi{rel}}(r)=\\top$ (or a dummy $X\\,{=}\\,X$) and\n$\\Gamma_{\\mi{rel}}(r)=X_1\\leq X_2$, respectively. The rule $\\mi{c} \\leftarrow\n\\mi{r}(X,Y), \\mi{p}(X,Y)$ is rewritten to the rule $\\mi{c} \\leftarrow\n\\mi{r}(X_1,Y_1), \\mi{p}(X_2,Y_2),\\Gamma_{rel}$ with $\\Gamma_{rel}\\,{=}\\, (X_1\\,{=}\\,X_2,Y_1\\,{=}\\,Y_2)$.\n\\end{exmp}\nThe basic idea is as follows:\nwhen\nconstructing the abstract program, we either (i) just abstract each\nliteral in a rule, or (ii) in case of uncertainty due to abstraction,\nwe guess the rule head to catch possible cases. The uncertainty may\noccur \ndue to having relation restrictions over non-singleton equivalence\nclasses\n(i.e. $|m_d^{-1}(\\hat{d}_i)|>1$), \nor having negative literals that are mapped to non-singleton abstract literals.\n\n\nTo the best of our knowledge, this is the first such approach of abstracting ASP programs.\n\n\n\\leanparagraph{Abstracting the relations} For simplicity, we first\nfocus on binary relations, e.g., $=,<,\\leq,\\neq$, \nand $\\Gamma_{rel}(r)$ \nof the form $\\mi{rel}(X,c)$ or $\\mi{rel}(X,Y)$.\n\nIt is necessary to\nreason about the cases that can occur for the truth values of ${\\mi{rel}}(\\hat{d}_1,\\hat{d}_2)$, for $\\hat{d}_1,\\hat{d}_2 \\in \\widehat{D}$, in order to obtain minimal abstract models that cover the original answer sets. There are four cases to consider:\n\n\n\n\n\\noindent\\begin{tabular}{r@{}l}\n I &  $\\phantom{\\neg} \\mi{rel}(\\hat{d}_1,\\hat{d}_2) \\wedge \\forall x_1\\,{\\in}\\, \\hat{d}_1,\\forall x_2\\,{\\in}\\, \\hat{d}_2. \\mi{rel}(x_1,x_2)$ \\\\%[1ex]\n II & $\\neg \\mi{rel}(\\hat{d}_1,\\hat{d}_2) \\wedge \\forall x_1\\,{\\in}\\,  \\hat{d}_1,\\forall x_2\\,{\\in}\\, \\hat{d}_2. \\neg \\mi{rel}(x_1,x_2)$ \n     \\\\%[1ex]\n III  &  $\\phantom{\\neg} \\mi{rel}(\\hat{d}_1,\\hat{d}_2) \\wedge \\exists x_1\\,{\\in}\\, \\hat{d}_1,\\exists x_2\\,{\\in}\\, \\hat{d}_2. \\neg \\mi{rel}(x_1,x_2)$\\\\%[1ex]\n  IV &        $\\neg \\mi{rel}(\\hat{d}_1,\\hat{d}_2) \\wedge \\exists x_1\\,{\\in}\\, \\hat{d}_1,\\exists x_2\\,{\\in}\\, \\hat{d}_2. \\mi{rel}(x_1,x_2)$\n\\end{tabular}\n\n\n\\noindent For $\\mi{rel}(\\hat{d}_1,\\hat{d}_2){=} \\top$, Case III is\nmore common in domain abstractions, while case I occurs e.g.,\\ for singleton\nmappings (i.e., $|\\hat{d}_1|=|\\hat{d}_2|=1$) or for negative relations such as $\\neq$. For $\\mi{rel}(\\hat{d}_1,\\hat{d}_2) {=}\\bot$, Case II is the common case, e.g., $=,\\leq$, whereas case IV may occur for negative relations or $<$.\n\n\\begin{exmp}\nConsider $\\mi{rel}(X,Y)=X\\leq Y$ and a mapping $m_d(\\{1\\})=\\hat{d}_1, m_d(\\{2,3\\})=\\hat{d}_k$ with an order $\\hat{d}_1 < \\hat{d}_k$ on the abstract values. Notice that case I occurs for $\\hat{d}_1 \\leq \\hat{d}_k$ and $\\hat{d}_1 \\leq \\hat{d}_1$, while case III occurs for $\\hat{d}_k \\leq \\hat{d}_k$. The latter is due to the possibility of having $3 \\leq 2$ which is false.\n\\end{exmp}\n\nThe cases that the equivalence classes have for a binary $\\mi{rel}$\ncan be computed by simple queries and represented by facts of form $\\mi{type}_{\\mi{rel}}^{\\mi{case}}(\\hat{d}_1,\\hat{d}_2)$\nfor each equivalence classes $\\hat{d}_1,\\hat{d}_2$.\n\n\\leanparagraph{Program abstraction} We start with a procedure for programs\nwith rules $r: l \\leftarrow B(r), \\mi{rel}(t_1,t_1')$\nwhere $|B^-(r)|{\\leq} 1$. \n\nFor any rule $r$ and $*{\\in} \\{+,-\\}$, let the set\n$S^{*}_{\\mi{rel}}(r)=\\{l_j \\in B^{*}(r) \\mid arg(l_j) \\cap\n\\{t_1,t_1'\\} \\neq \\emptyset\\}$ be the positive and negative literals, respectively, that share an argument with $\\mi{rel}(t_1,t_1')$.\nWe assume for simplicity that $B^-(r) \\subseteq S_{rel}(r)$\nand discuss how to handle rules not meeting this assumption  later.\n\nWe build a program $\\Pi_{dom}^m$ according to the mapping $m$\nas follows. \nFor any rule $r: l \\leftarrow B(r), \\mi{rel}(t_1,t_1')$ in $\\Pi$, we\nadd:\n\\vspace*{-.125\\baselineskip}\n\n\\begin{enumerate}[\\hspace{1cm}\\quad(1)]\n\n\\item[(0)] If $B^+(r)\\setminus S^+_{\\mi{rel}}(r)\\neq \\emptyset$:\n\\vspace*{-.25\\baselineskip}\n\\begin{enumerate} [$~\\hspace{-1.5em}$(a)]\n\\item If $\\mi{rel}(t_1,t_1'){=} \\top:$~ $m(l) \\leftarrow m(B(r))$. \n\\end{enumerate}\n\\vspace*{-.25\\baselineskip}\n\n\\item If $S_{\\mi{rel}}^+(r)\\neq \\emptyset$:\n\\vspace*{-.35\\baselineskip}\n\\begin{enumerate} [$~\\hspace{-1.5em}$(a)]\n\\itemsep=3pt\n\\item $m(l) \\leftarrow m(B(r)), {rel}(\\hat{t}_i,\\hat{t}_j),\\mi{type}^{\\textup{I}}_{\\mi{rel}}(\\hat{t}_i,\\hat{t}_j)$.\n\\item $0\\{m(l)\\}1 \\leftarrow m(B(r)), {rel}(\\hat{t}_i,\\hat{t}_j),\\mi{type}^{\\textup{III}}_{\\mi{rel}}(\\hat{t}_i,\\hat{t}_j)$.\n\\item $0\\{m(l)\\}1 \\leftarrow m(B(r)), \\neg {rel}(\\hat{t}_i,\\hat{t}_j),\\mi{type}^{\\textup{IV}}_{\\mi{rel}}(\\hat{t}_i,\\hat{t}_j)$.\n\\end{enumerate}\n\\vspace*{-.25\\baselineskip}\n\\item If $l_i {\\in} S_{\\mi{rel}}^-(r)$:\n\\vspace*{-.2\\baselineskip}\n\\begin{enumerate} [$~\\hspace{-2em}$(a$'$)] \n\\itemsep=3pt\n\\item $m(l) {\\leftarrow} m(B(r)), {rel}(\\hat{t}_i,\\hat{t}_j).$\n\\item[(b$'$)] $0\\{m(l)\\}1 {\\leftarrow} m(B^{\\mi{shift}}_{l_i}(r)), {rel}(\\hat{t}_i,\\hat{t}_j), \\mi{type}^{\\textup{III}}_{\\mi{rel}}(\\hat{t}_i,\\hat{t}_j).$  \n\\item[(c$'$)] same as (c), if $S_{\\mi{rel}}^+(r){=} \\emptyset$;\\\\[1pt]\n$ 0\\{m(l)\\}1 {\\leftarrow} m(B^{\\mi{shift}}_{l_i}(r)), \\neg {rel}(\\hat{t}_i,\\hat{t}_j),\\mi{type}^{\\textup{IV}}_{\\mi{rel}}(\\hat{t}_i,\\hat{t}_j).$ \\\\[1pt]\n$~0\\{m(l)\\}1 {\\leftarrow} m(B^{\\mi{shift}}_{l_i}(r)), {rel}(\\hat{t}_i,\\hat{t}_j), \\mi{type}^{\\textup{IV}}_{\\mi{rel}}(\\hat{t}_i,\\hat{t}_j).$ \n\\end{enumerate}\n\\vspace*{-.3\\baselineskip}\n\\end{enumerate}\n}\n\\noindent where $B^{\\mi{shift}}_{l_i}(r){=}B^+(r)\\cup \\{l_i\\},\\mi{not}\\ B^-(r){\\setminus} \\{l_i\\}$.\n\nCase (0) is the special case of having positive literals that do not share arguments with $\\mi{rel}$. If $\\mi{rel}{=}\\top$, then it will not be processed by next steps. Thus, the abstraction of $r$ is added. The assumption on $B^-(r)$ about being included in $S_{rel}(r)$ prohibits the case $B^-(r){\\setminus} S^-_{\\mi{rel}}(r){\\neq} \\emptyset$. \n\n\nIf $\\mi{rel}(t_1,t'_1)$ shares arguments with a positive body literal, we add\nrules to grasp the possible cases resulting from the relation type. In\ncase of uncertainty, \nthe head is made a choice,\nand for case IV, we flip the relation, $\\neg\\mi{rel}$, to catch the\ncase of the relation holding true. If $\\mi{rel}(t_1,t'_1)$ shares arguments with\na negative body literal, we need to \ngrasp the\nuncertainty arising\nfrom negation. We do this by adding rules in which we shift the\nrelated literal to \nthe positive body, via $B^{\\mi{shift}}_{l_i}(r)$.\n\n(2-c$'$) deals with the special case of\na type IV relation and a negative literal, e.g., $b(X_1) \\leftarrow\n\\mi{not}\\ a(X_1,X_2), X_1 {\\neq} X_2$. If\n$r$ \nis abstracted only\nby keeping the same structure,\n$m(B(r))$ \nmight not be\nsatisfied by abstract literals that actually have corresponding\nliterals which satisfy\n$B(r)$. \nE.g., $a(2,3){=}\\bot$\nsatisfies $r$; this can only be reflected in the abstraction by\n$a(k,k){=}\\bot$ which actually\ndoes \nnot satisfy\n$m(B(r))$. \nThus, \nwhen building the abstract rules, rules \nfor all combinations of shifting the literal and flipping the relation need to be added.\n\nNotably, the construction of $\\Pi^m_{dom}$ is modular, rule by\nrule;\nfacts $p(\\vec{t})$ are simply lifted to abstract facts $p(m(\\vec{t}))$. \n\n\n\\begin{exmp}\\label{ex:toy_dom}\nConsider the rules from Example~\\ref{ex:toy} plus \n\\begin{align}\n  e(X_1) &\\leftarrow \\mi{not}\\ a(X_1,X_2), X_1{=}X_2.\\footnotemark  \\label{eq:3}\n\\end{align} \n\\footnotetext{In order to ensure safety, these rules can be extended with special built-in domain predicates which do not require to be standardized apart.}\n\\noindent  over the domain $D=\\{1,2,3\\}$.\nSuppose $\\widehat{D}{=}\\{\\hat{d}_1,\\hat{d}_k\\}$ with mapping $m_{d}(1){=}\\hat{d}_1$, $m_{d}(\\{2,3\\}){=}\\hat{d}_k$. The abstract program constructed is as follows, in simplified form:\n\\vspace*{-.25\\baselineskip}\n\\begin{align}\n  a(\\widehat{X}_1,\\widehat{X}_2) &\\leftarrow c(\\widehat{X}_1), b(\\widehat{X}_2) \\label{eq:11}\\\\%[-.6ex]\n  d(\\widehat{X}_1,\\widehat{X}_2) &\\leftarrow a(\\widehat{X}_1,\\widehat{X}_2), \\widehat{X}_1 {\\leq} \\widehat{X}_2, \\widehat{X}_1 {=} \\hat{d}_1,\\widehat{X}_2 {=} \\hat{d}_k \\!\\!\\label{eq:21}\\\\%[-.6ex]\n  \\hspace{-0.45cm} 0\\{d(\\widehat{X}_1,\\widehat{X}_2)\\}1 & \\leftarrow a(\\widehat{X}_1,\\widehat{X}_2), \\widehat{X}_1 {\\leq} \\widehat{X}_2,\\widehat{X}_1{=} \\hat{d}_k,\\widehat{X}_2 {=} \\hat{d}_k \\label{eq:22}\\\\%[-.6ex]\n  e(\\widehat{X}_1) &\\leftarrow \\mi{not}\\ a(\\widehat{X}_1,\\widehat{X}_2), \\widehat{X}_1{=}\\widehat{X}_2 \\!\\!\\!\\!\\label{eq:31}\\\\%[-.6ex]\n0\\{e(\\widehat{X}_1)\\}1 &\\leftarrow a(\\widehat{X}_1,\\widehat{X}_2), \\widehat{X}_1{=}\\widehat{X}_2, \\widehat{X}_1{=} \\hat{d}_k,\\widehat{X}_2 {=} \\hat{d}_k \\label{eq:33}\n\\end{align} \n\\vspace*{-1\\baselineskip}\n\n\\noindent Here the $\\mi{type}^{case}_{\\mi{rel}}$ literals have been evaluated, and\nredundant rules are omitted.\nObserve that \\eqref{eq:11} is same as \\eqref{eq:1} as it has\n$\\mi{rel}{=}\\top$. \nFrom \\eqref{eq:2}, we get \\eqref{eq:21} for $\\hat{d}_1,\\hat{d}_k$ which have case I for $\\leq$, and \\eqref{eq:22} for $\\hat{d}_k,\\hat{d}_k$ that have case III. From \\eqref{eq:3}, we get \\eqref{eq:31} and \\eqref{eq:33} with shifting for case III.\n\nFor given facts $c(3),b(2)$,\n$I =\\{a(3,2),e(1),e(2),e(3),c(3),b(2)\\} \\in AS(\\Pi)$.\nAfter applying m,\nthe facts become $c(\\hat{d}_k),b(\\hat{d}_k)$ and\n$\\{a(\\hat{d}_k,\\hat{d}_k),e(1),e(\\hat{d}_k),c(\\hat{d}_k),b(\\hat{d}_k)\\} \\in AS(\\Pi^m_{dom})$, which covers\n$I$. Note that the choice rule \\eqref{eq:33} ensures\nthat $e(\\hat{d}_k)$ can still be obtained even \nw\\textbf{}hen $a(\\hat{d}_k,\\hat{d}_k)$ holds. It likewise covers\n$\\{c(2), b(3), a(2,3), d(2,3), e(1), e(2), e(3)\\} \\in AS(\\Pi)$ for the facts $c(2),b(3)$.\n\n\\end{exmp}\n\nWe prove that the abstraction procedure constructs a system $\\Pi_{dom}^m$ that over-approximates $\\Pi$.\n\n\\begin{thm}\nLet $m$ be a domain abstraction over $\\Pi$.\nThen for every\n$I \\in AS(\\Pi)$, $m(I) \\in AS(\\Pi_{dom}^m)$.\n\\end{thm}\n\n\\begin{proof}[Proof (sketch)]\nWith the \nrules (0a), (1a-1b), and (2a$'$), we ensure that\n$\\widehat{I}$ is a\nmodel\nof\n$\\Pi^m_{dom}$, as we either keep the structure\nof a rule $r$ or change it to a choice rule. The rules added in steps\n(1b-1c) and (2b$'$-2c$'$) \nserve to catch the cases that may\nviolate the minimality of the model due to a negative literal or a\nrelation over non-singleton equivalence classes. The rules (1b,2b$'$) deal with\nhaving a literal (resp.\\ relation literal) that is false in\n$I$ but thought to be true in the abstract model $\\hat{I}$, and (1c,2c$'$)\ndeal with \na literal (resp.\\ relation literal) that is thought to be false in \n$\\hat{I}$ but true in $I$.\n\\end{proof}\n\\vspace*{-0.4\\baselineskip}\n\n\n\\leanparagraph{General case} The construction can be applied to more general programs by focusing on two aspects: 1) $|B^-(r)|{>}1$: For multiple negative literals in the rule, the shifting must be applied to each negative literal. 2) $|\\Gamma_{\\mi{rel}}|{>}1$: To handle multiple relation literals, a straightforward approach is to view $\\Gamma_{\\mi{rel}} = \\mi{rel}(t_1,t'_1),$ \\ldots, $\\mi{rel}(t_k,t'_k)$ as \na literal of an $n$-ary relation $\\mi{rel}'(X_1,X'_1,\\ldots,X_k,X'_k)$, $n\\,{=}\\,2k$.\nThe abstract version of such built-ins $\\mi{rel}'$ and the type cases I-IV are readily lifted\n\nLet $\\Pi_{dom}^{*\\ m}$ be the program obtained from a program $\\Pi$\nwith the generalized abstraction procedure.\nThen we \nobtain:\n\\begin{thm}\nFor every $I \\in AS(\\Pi)\n, $m(I) \\in AS(\\Pi_{dom}^{*\\ m})$.\n\\end{thm}\n\n\nFor constraints, the\nsteps creating choice rules can be skipped \nas we cannot guess over $\\bot$. \nFurther simplifications and optimizations\ncan help to \navoid introducing too many spurious answer sets.\nSyntactic extensions can also be addressed. Rules with choice and cardinality constraints can be lifted with the same structure. For conditional literals with conditions over negative literals, additional rules with shifting will be necessary; otherwise, the condition can be lifted the same.\n\n\\section{Using Abstraction for Policy Refutation}\\label{abs_policy}\n\nAs an application case, we are interested in the problem of defining declarative policies for reactive agents and reasoning about their behavior, especially in non-deterministic environments with uncertainty in the initial state. In such environments, searching for a plan that reaches the main goal easily becomes troublesome. Therefore, we focus on defining policies that choose a sequence of actions from the current state with the current observations, in order to achieve some subgoal, and then checking the overall behavior of these policies. More details of such policies can be found in \\cite{zgs16jelia}.\n\n\\leanparagraph{Background}\nFormally, a \\emph{system} \n$\\ensuremath{A}\\,{=}\\,\\langle \\mathcal{S},\\mathcal{S}_0,\\mathcal{A},\\Phi\\rangle$\nconsists of a finite set $\\mathcal{S}$ of states, a set\n$\\mathcal{S}_0 \\,{\\subseteq}\\, \\mathcal{S}$ of initial states, a finite set\n$\\mathcal{A}$ of actions, and a non-deterministic transition relation\n$\\Phi: \\mathcal{S}\\,{\\times}\\,\\mathcal{A} \\rightarrow 2^{\\mathcal{S}}$. \n\n\nA sequence $\\sigma=a_1,a_2,\\dots,a_n$ of actions is \\emph{executable}, if\n\n\\smallskip\n\n\\centerline{$\\exists s_0,\\dots,s_n \\in \\mathcal{S}_0\\, \\forall\\, 0\\leq i<n:\ns_{i+1} \\,{\\in}\\, \\Phi(s_i,a_{i+1})$}\n\n\\smallskip\n\n\\noindent holds. We denote such \\emph{(potential) plans} by $\\Sigma$. By\n$\\Sigma(s)$, resp.\\ $\\Phi_\\Sigma(s,\\sigma)$, we denote those \nexecutable from $s$, \nresp.\\ the set of states $s_n$ reached by them. The latter\ninduces the transition function\n$\\Phi_\\Sigma: \\mathcal{S} \\times \\Sigma \\rightarrow 2^\\mathcal{S}$ of\nthe system\n$\\ensuremath{A}_\\Sigma{=}\\langle \\mathcal{S},\\mathcal{S}_0,\\Sigma,\\Phi_\\Sigma\\rangle$.\n\nWe consider policies for a \ngoal $\\mu$,\nthat guide the agent with action sequences \ncomputed according to the knowledge base $KB$,\nwhich is a formal world model in a transition system view. \n\n\\begin{defn}[Policy]\nGiven a system $\\ensuremath{A}=\\langle \\mathcal{S},\\mathcal{S}_0,\\mathcal{A},\\Phi\\rangle$ and a set $\\Sigma$ of plans with actions of $\\mathcal{A}$, a \\emph{policy} is \na function $P_{\\mu, KB}:\\mathcal{S} {\\rightarrow} 2^{\\Sigma}$\ns.t. $P_{\\mu, KB}(s) \\,{\\subseteq}\\, \\Sigma(s)$. \n\\end{defn}\n\\noindent \nAssuming that $\\mu$ and $KB$ are fixed, we omit subscripts of $P$.\nInformally, the agent executes at state $s$ any plan $\\sigma\\in P_{\\mu, KB}(s)$ to achieve a (hidden) subgoal, and continues at state $s'\\in\n\\Phi_\\Sigma(s,\\sigma)$ by executing any plan $\\sigma'\\in P_{\\mu,\nKB}(s')$ for the next subgoal, etc, until the goal $\\mu$ is established.\n\n\nWe focus on trajectories in\n$\\ensuremath{A}_\\Sigma$\nfollowed by the policy,\nand consider goals $\\mu$ expressed as propositional formulas \nover the states, which are consistent sets of propositional literals. The aim is to reach some state \nsatisfying $\\mu$, from all possible initial states and through all\npolicy trajectories;\ni.e., \nthe policy works if $\\forall s_0\\,{\\in}\\,\\mathcal{S}_0 :\ns_0\\,{\\models}\\, {\\bf AF}\\mu$\nholds,\nwhere {\\bf A} ranges over all trajectories under $P\\!$ starting at some\ninitial state $s_0$ in $A_\\Sigma$.\n\n\\begin{exmp}\nIn the missing person scenario, the goal $\\mu$ can be expressed as\n$\\mu=\\bigvee_{i,j=1}^n \\mi{rAt}(i,j)\\land \\mi{pAt}(i,j)$, \nwhere $\\mi{rAt}(X,Y)$ (resp.\\ $\\mi{pAt}(X,Y)$)  states that the \nrobot (resp.\\ person) is at position $X,Y$, \nor by using a designated atom $\\mi{caught}$ that is defined by this formula.\n\n\\end{exmp}\n\n\n\n\n\\subsection{Abstraction over the policy behavior}\n\n\nIn order to abstract the policy behavior, we consider abstraction on the states and also on the actions; Figure~\\ref{fig:abs_fig}(a) illustrates our abstraction approach. The abstract system is then built straightforwardly over the transitions \nof the policy.\n\n\\leanparagraph{State abstraction}We consider a set\n$\\Psi$ of abstract literals\nfor the set\n$\\hat{\\mathcal{S}} \\,{\\subseteq}\\, 2^{\\Psi}$ of abstract states, and\nan abstraction function\n$h_{st}\\,{:}\\,\\mathcal{S} \\,{\\rightarrow}\\, \\hat{\\mathcal{S}}$\non states, on which further conditions might be imposed.\nFor convenience, we denote\n$h_{st}(s)$\nby $\\hat{s}$, and identify\n$\\hat{s}$\n\n with the states $\\{ s\\,{\\in}\\, \\mathcal{S}\\ {\\mid}\\\n h_{st}(s)\\,{=}\\, \\hat{s}\\}$\nabstracted to it.\n\n\\begin{figure}[t]\n\\caption{State and action abstraction}\n\\vspace{0.5em}\n\\label{fig:abs_fig}\n\\centering\n\\subfigure[][Abstraction]{\\begin{tikzpicture}\n   \\path (0,0) node(x) {$s$}\n(0,1.5) node(y) {$\\hat{s}$};\n\\draw[->] (x) -- node[midway,left]{$h_{st}$} (y);\n\\path (2,1.5) node(z) {$\\hat{s}'$}\n(2,0) node(t) {$s'$};\n\\draw[->] (z) -- node[midway,right]{$h^{-1}_{st}$} (t);\n\\draw[->] (y) -- node[midway,above]{$\\hat{a}$} (z);\n\\draw[->] (x) -- node[midway,below]{$\\sigma$} (t);\n\\node(k) at (1,0.75) {$h_{act}$};\n\\draw[dashed] (1,0) -- (k);\n\\draw[->,dashed] (k) -- (1,1.5);\n\\end{tikzpicture}}~~~\n\\subfigure[][Domain abstraction]{\\includegraphics[scale=.6]{abs.pdf}}\n\\vspace{-1.5em}\n\\end{figure}\n\n\n\\leanparagraph{Action abstraction} \nWe consider a set $\\hat{\\mathcal{A}}$ of abstract actions\nfor $\\Psi$, and an abstraction function\n$h_{act}{:} \\Sigma {\\rightarrow} \\hat{\\mathcal{A}}$ which maps \naction sequences $\\sigma$ to abstract actions $\\hat{a}$.\nSimilarly, \nwe denote $h_{act}(\\sigma)$ by $\\hat{\\sigma}$ and identify\n$\\hat{a}$ with $\\{ \\sigma \\,{\\in}\\, \\Sigma \\mid h_{act}(\\sigma)\\,{=}\\,\\hat{a}\\}$.\n\n\n\\begin{exmp}[ctd]\\label{ex:regions}\nFigure~\\ref{fig:abs_fig}(b) shows an example of a domain abstraction that maps the large domain into a smaller sized domain, with $h_{st|D}(\\mi{rAt}(X,Y)){=}\\mi{rAt}(\\lceil \\frac{X}{n\/2}\\rceil,\\lceil\\frac{Y}{n\/2}\\rceil){=}\\mi{rAt}(Rx,Ry),$ $Rx,Ry {\\in} \\{1,2\\}$. Similarly, $\\mi{goTo}(X,Y)$ is mapped to some $\\mi{goTo}(Rx,Ry)\n. For simplicity, we will refer to the abstract cells as regions $\\{nw,ne,sw,se\\}$.\n\\end{exmp}\n\\vspace*{-0.5\\baselineskip}\n\n\\leanparagraph{Abstract system with an abstracted policy}\nThe transitions of\n$\\ensuremath{\\widehat{A}}$ are defined over those \nin $\\ensuremath{A}$ \nchosen by the policy.\n\n\\begin{defn}\nFor a system\n$\\ensuremath{A}=\\langle \\mathcal{S},\\mathcal{S}_0,\\mathcal{A},\\Phi\\rangle$, a set\n$\\Sigma$ of plans \nover $\\mathcal{A}$, a transition function\n$\\Phi_\\Sigma$, and a policy $P$, an abstract system $\\widehat{A}=\\langle \\hat{\\mathcal{S}}, \\hat{\\mathcal{S}}_0, \\hat{\\mathcal{A}}, \\widehat{\\Phi}_P \\rangle$\nis \\emph{generated} by a state abstraction $h_{st}$ and \naction abstraction $h_{act}$, if \n\n\\vspace{1pt}\n\n\\begin{compactitem}[--]\n\\item $\\hat{\\mathcal{S}}_0 = \\{\\hat{s}_0 \\mid s_0 \\in \\mathcal{S}_0\\}$ are the initial abstract states, and \n\\item $\\widehat{\\Phi}_{P}: \\widehat{S} \\times  \\hat{\\mathcal{A}} \\rightarrow \\widehat{S}$ is the abstract transition function according to the policy $P$, defined as\n\\smallskip\n\n\\centerline{$\n\\widehat{\\Phi}_{P}(\\hat{s},\\hat{a}) {=} \n\\{\n\\hat{s}' \\mid  \\exists s'' \\in\n\\hat{s}, \\sigma \\in P(s'')\\cap \\hat{a}:\ns' \\in \\Phi_\\Sigma(s'',\\sigma)\\}.\n$}\n\\vspace*{-0.25\\baselineskip}\n\\end{compactitem}\n\\end{defn}\nNote that any abstract transition $\\hat{s},\\hat{a},\\hat{s}'$\nin $\\widehat{A}$\nmust \nstem from a transition $s,\\sigma,s'$ in $\\mathcal{A}$\nvia policy $P$,\ni.e., an abstract transition is introduced only if there is a corresponding original transition. This gives an over-approximation of the policy's behavior on the original system \\cite{clarke03}.\n\n\n\\subsubsection{Constructing an abstract system}\nWe can apply the abstraction method to ASP programs \nwith action\ndescriptions and policy rules, where we focus on \npolicies with single action plans, \nwith some particulars.\n\n\\begin{itemize}\n \\item\n It is possible to have\nthe mapping $m_a{:}{\\cal A}{\\rightarrow} \\hat{\\cal A}$\ncreate abstract\npredicates, i.e., $m_a(a(v_1,\\dots,v_n))=\\hat{a}(m_{d}(v_1),\\dots,$\n$m_{d}(v_n))$ where $\\hat{a} = \\widehat{a(v_1,\\dots,v_n)}$ depends also on the arguments of $a$ in order keep some possibly necessary details of the original actions in the abstract action.  However, with action atoms occurring in transition descriptions\nonly positively in rule bodies, and in policy rules only in rule\nheads, no further treatment of these atoms is necessary.\n\n\\item For policy rules \n$a {\\leftarrow} B$ that select an action $a$, abstract rules\n$0\\{\\hat{a}\\}1 {\\leftarrow} \\widehat{B}$ \n(while correct) are undesirable as they allow to skip the action and would create a spurious trajectory.\nTo have an optimization over the abstraction, \nthis can be avoided by\nliterals $l$ in\n$B$ with singleton mappings, i.e.,\n$|m^{-1}(\\hat{l})|{=}1$, or with a non-singleton mapping\nwhere $l {\\in} B^{+}\\setminus S^+_{\\mi{rel}}$, or with cases \nthat allow for simplification of the choice rules.\n\n\\item Time arguments amount to a special sort, and we do not abstract\nover it (i.e., each time point $t$ is abstracted to itself).\nThus, time variables, terms etc.\\ simply remain unaffected.\n\n\\item For plan abstraction\n$h_{act}{:} \\Sigma {\\rightarrow} \\hat{\\cal A}$, dedicated atoms\n$\\sigma$ can describe plans with their effects, obtained \nfrom unfolding\nthe \neffect rules of the actions in $\\sigma$\nand their preconditions.\n\\end{itemize}\n\n\\begin{exmp}[ctd]\nBy omitting most of the details\nexcept the directly\naffected literals and the literals related with the goal condition,\nthe domain abstraction \nin Ex.\\ref{ex:regions} \nyields the following abstract rules\nfor \\eqref{eq:pol_formula}-\\eqref{eq:pol_formula2}:\n\\begin{align}\n&1\\{\\mi{goTo}(\\widehat{X},\\widehat{Y},T){:} \\widehat{X},\\widehat{Y} {\\in} \\widehat{D}\\}1 {\\leftarrow} \\mi{not}\\ \\mi{seen}(T), \\mi{not}\\ \\mi{caught}(T).\\nonumber\\\\%[-.5ex]\n&\\mi{goTo}(\\widehat{X},\\widehat{Y},T) \\leftarrow \\mi{seen}(T), \\mi{not}\\ \\mi{caught}(T), \\mi{pAt}(\\widehat{X},\\widehat{Y},T).\\nonumber\\\\%[-.5ex]\n&0\\{\\mi{caught}(T)\\}1 \\leftarrow \\mi{rAt}(\\widehat{X},\\widehat{Y},T), \\mi{pAt}(\\widehat{X},\\widehat{Y},T).\\label{eq:abs_rules}\\\\%[-.5ex]\n&0\\{\\mi{seen}(T)\\}1 \\leftarrow \\mi{pAt}(\\widehat{X},\\widehat{Y},T). \\nonumber\n\\end{align}\n\\end{exmp}\n\n\\subsection{Counterexample search} Recall \nour aim of over-approxi\\-mating the problem\nof checking whether \nobeying the policy $P$ always reaches the goal $\\mu$ \n(i.e., all paths starting from ${\\cal S}_0$\nreach a state that satisfies $\\mu$).\nFor policies where all\nstates have outgoing transitions, state abstractions that distinguish\nthe goal conditions can avoid false positives. \nThat is, if no ``bad'' abstract trajectory exists in which\n$\\mu$ is unachieved (a {\\em counterexample}), then no \n``bad'' original trajectory exists;\nand we can check the policy behavior on\nthe abstract system, as in \\cite{clarke03}.\n\nConcretely, we search for an abstract counterexample (cex)\ntrajectory in the abstract system $\\ensuremath{\\widehat{A}}$\nof length at most $n$, where\n$n$ is large enough. As the original space state $\\mathcal{S}$ is\nfinite, any path trajectory longer than $|\\mathcal{S}|+1$\nclearly must loop. If we cannot find such a counterexample trajectory, the policy works, cf.\\ \\cite{ClarkeKOS04}. \nOn the other hand,  if  a cex trajectory\n$\\hat{\\tau}=\\hat{s}_0,\\hat{a}_0,\\dots,\\hat{s}_n$ is found, we need to\ncheck whether $\\hat{\\tau}$ has a corresponding concrete trajectory $\\tau$ in $\\ensuremath{A}_\\Sigma$. \nThe counterexample is \\emph{spurious}, if no such \n$\\tau$ exists.\n\n\\begin{figure}[t]\n\\caption{}\n\\centering\n\\subfigure[][a counterexample trajectory]{\\includegraphics[scale=0.75]{abs_failure_notarget2_vfinal_new_1_new.pdf}}\n\\subfigure[][a corresponding concrete trajectory with failure]{\\includegraphics[scale=0.75]{abs_failure_notarget2_vfinal_new_new.pdf}}\n\\label{fig:failure}\n\n\\vspace*{-1.5\\baselineskip}\n\n\\end{figure}\n\n\n\\begin{exmp}[ctd]\\label{ex:spur}\nA 3-step counterexample (Figure~\\ref{fig:failure}(a)) to\nfinding the person\nis \n$\\hat{\\tau}\\,{=}\\,\\mi{rAt}(nw,0),\\mi{goTo}(\\mi{nw},0),\\mi{rAt}(\\mi{nw}$, $1),\\mi{goTo}(\\mi{ne},1),\\mi{rAt(ne},2),\\mi{goTo}(\\mi{ne},2),\\mi{rAt(ne},3)$.\nFigure~\\ref{fig:failure}(b) shows a corresponding trajectory\nin \n$\\ensuremath{A}_\\Sigma$;\nit fails at step 2 to find an\naction corresponding to\n$\\mi{goTo}(\\mi{ne})$, \nas the\npolicy \nwould move to $\\mi{nw}$. In fact, no corresponding trajectory \nwithout\nfailure can be found; so $\\hat{\\tau}$ is spurious.\n\\end{exmp}\n \nExample~\\ref{ex:spur} shows that, as expected, omitting most of the\ndetails of the problem makes it easy to encounter spurious\ntrajectories. We need to add back some of the details that the policy uses in order to reduce spurious transitions.\n\n\\begin{exmp}[ctd]\nLet us apply domain abstraction on $\\mi{farthest}(X,Y,X1,Y1)$, and\nfurther auxiliary literals such as $\\mi{farthestDist}(X,Y,D)$. We get\n$\\mi{farthest}(Rx,Ry,Rx1,Ry1)$ and $\\mi{farthestDist}(Rx,Ry,RD)$,\nwhere $RD \\in \\{0,1\\}$ tells if the distance is $<n\/2$ or $\\geq\nn\/2$. With the added back information in the abstraction, \nin the refined abstract program $\\hat{\\tau}$ is\nno longer encountered.\nTo avoid nondeterminism,\nwe may use actions of the form $\\mi{goWithDist}$,$\\mi{goToPerson}$\nintroduced by  auxiliary action descriptions in the original system.\n\\end{exmp}\n\n\n\\subsection{Failure analysis and refinement}\n\nFurther studying the cause of failure in spurious cex trajectories $\\hat{\\tau}=\\hat{s}_0,\\hat{a}_0,\\dots,\\hat{s}_n$\nwould give hints on which\ninformation to add back in the\nabstraction in order to eliminate $\\hat{\\tau}$. \nA failure state $\\hat{s}_i$ occurs in\n$\\hat{\\tau}$ \ndue to any state $s\\,{\\in}\\, \\hat{s}_i$ that is reachable from some $s_0\\,{\\in}\\, \\hat{s}_0$ following\n$\\ensuremath{A}_\\Sigma$ but has no transition $\\sigma$\ncorresponding to $\\hat{a}_i$ to some state $s'\\,{\\in}\\,\\hat{s}_{i+1}$. \nThe reason can be that (i)\n$P$ determines no plan $\\sigma\\in P(s)$ that can be abstracted to\n$\\hat{a}_i$, or (ii) some such plan $\\sigma$  has no transition to any\nstate in $\\hat{s}_{i+1}$.\nWe can impose conditions on the abstraction to avoid \nthese failures types and \neliminate spurious cex trajectories. However, \nif the policy uses\ninformation depending on facts that are not affected by the\nactions, a systematic approach for refinement becomes necessary.\n\nASP provides the possibility of reasoning over the failures and\nobtaining an explanation on what is missing in the abstraction.\nConstraints about the policy behavior can be obtained,\ne.g., the agent can not travel a smaller distance than\nbefore. Depending on which constraints are violated in the failure,\none can get hints\nfor refining the abstraction. \nThus,\na CEGAR-like approach \\cite{clarke03} can be\nused, where starting from an\ninitial abstraction, one repeats searching for a counterexample, checks \ncorrectness of the latter, obtains a\nfailure explanation in case of spuriousness, and refines the abstraction\nuntil a concrete counterexample is found.\n\\vspace*{-0.5\\baselineskip}\n\n\\subsection{Discussion on evaluation}\n\nA major difficulty of the policy checking problem is the (huge) number\n        of initial states, indirect effects, choice of plans by the\n        agents and possible nondeterministic effects of\n        actions. Therefore existing planning approaches can not be\n        used off-the-shelf. Our preliminary results on the motivating\n        example show that a standard approach of searching for a\n        counterexample for the policy's behavior in the original\n        program is infeasible (i.e., reaches time or memory outage) \n        for larger dimensions (e.g., for $n=64$). \n        Increasing policy sophistication makes counterexample search harder, and outage is hit earlier.\n\t\tOn the other hand, with an abstraction, we are able\n        to easily get some candidate solution. Checking whether\n        the candidate solution is concrete is faster than applying the standard approach. \n        However, in the worst case one\n        needs to go through all of the introduced spurious solutions\n        before reaching a correct one. Ongoing study is on dealing\n        with the encountered spurious abstract solutions and to refine\n        the abstraction to get rid of them.\n\nMoreover, as a side note, even for problems such as numbering each\n        node in an $n\\times n$ grid with its position in a loop-free\n        path (similar to the challenge planning problem VisitAll), the naive\n        approach of guessing a numbering that matches the requirements\n        is known not to scale, e.g., for $n=10$ no solution can be found\n        in 5 hours. On the other hand, building an abstraction,\n        computing and checking the abstract\n        solution and (for now) manually refining the abstraction by\n        hints obtained from the checking is faster to reach a concrete\n        solution.\n\n\\section{Conclusion}\n\nWe introduced a    \nmethod for abstracting ASP programs by\nover-approximating the answer sets, motivated by applications in\nreasoning about actions. It keeps the structure and can be easily implemented, and further optimized.\n\n\\leanparagraph{Related Work}\nIn addition to\nabstraction in planning, \nabstraction has\nbeen studied\nfor agent verification \nin situation\ncalculus action theory \\cite{banihashemi2017abstraction}\nand of\nmulti-agent systems against specifications \ndefined in epistemic\nlogic \\cite{lomuscio2016verification} and alternating time temporal logic\n\\cite{belardinelli2016abstraction}. \n\\citeNselfBYB{Banihashemi et al.}{banihashemi2017abstraction}\nconsider high and low-level agent specifications \nto obtain sound and complete abstraction via bisimulation.\n\\citeNselfBYB{Lomuscio et al.}{lomuscio2016verification} present an automated\npredicate abstraction method in three-valued\nsemantics,\nand refinement based on Craig's interpolants \\cite{belardinelli2016agent}.\nAbstraction has been studied in logic\nprogramming \\cite{COUSOT1992103}, but stable semantics \nwas not addressed.  \n\n\\leanparagraph{Outlook} \nOur work is a necessary starting point for\nreasoning about ASP-based agent policy behavior in large environments. To\novercome\nspurious answer sets, ongoing work covers\nobtaining hints for a refinement \nfrom the constraint violations in\n\nspurious answers or by debugging \nthe inconsistency caused by the spurious\nanswers approach, and a refinement methodology.\n\n\\section*{Acknowledgements}\n\nThis work has been supported by Austrian Science Fund (FWF) project W1255-N23.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}