Algorithms to evaluate multiple sums for loop computations

Abstract

We present algorithms to evaluate two types of multiple sums, which appear in higher-order loop computations. We consider expansions of a generalized hyper-geometric-type sums, \documentclass[12pt]{minimal}\begin{document}$\sum _{n_1,\cdots ,n_N} \frac{ \Gamma ({\bm a}_1\cdot {\bm n}+c_1) \Gamma ({\bm a}_2\cdot {\bm n}+c_2) \cdots \Gamma ({\bm a}_P\cdot {\bm n}+c_P) }{ \Gamma ({\bm b}_1\cdot {\bm n}+d_1) \Gamma ({\bm b}_2\cdot {\bm n}+d_2) \cdots \Gamma ({\bm b}_Q\cdot {\bm n}+d_Q) } x_1^{n_1}\cdots x_N^{n_N}$\end{document}∑n1,⋯,nNΓ(a1·n+c1)Γ(a2·n+c2)⋯Γ(aP·n+cP)Γ(b1·n+d1)Γ(b2·n+d2)⋯Γ(bQ·n+dQ)x1n1⋯xNnN with \documentclass[12pt]{minimal}\begin{document}${\bm a}_i\! \cdot \!{\bm n}\break = \sum _{j=1}^N a_{ij}n_j$\end{document}ai·n=∑j=1Naijnj, etc., in a small parameter ε around rational values of ci,di’s. Type I sum corresponds to the case where, in the limit ε → 0, the summand reduces to a rational function of nj’s times \documentclass[12pt]{minimal}\begin{document}$x_1^{n_1}\cdots x_N^{n_N}$\end{document}x1n1⋯xNnN; ci,di’s can depend on an external integer index. Type II sum is a double sum (N = 2), where ci, di’s are half-integers or integers as ε → 0 and xi = 1; we consider some specific cases where at most six Γ functions remain in the limit ε → 0. The algorithms enable evaluations of arbitrary expansion coefficients in ε in terms of Z-sums and multiple polylogarithms (generalized multiple zeta values). We also present applications of these algorithms. In particular, Type I sums can be used to generate a new class of relations among generalized multiple zeta values. We provide a Mathematica package, in which these algorithms are implemented.