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 hypergeometric-type sums, $\sum_{n_1,...,n_N} [Gamma(a1.n+c1) Gamma(a2.n}+c2) ... Gamma(aM.n+cM)] / [Gamma(b1.n+d1) Gamma(b2.n+d2) ... Gamma(bM.n+dM)] x1^n1...xN^nN $ with $ai.n=\sum_{j=1}^N a_{ij}nj$, etc., in a small parameter epsilon around rational values of ci,di's. Type I sum corresponds to the case where, in the limit epsilon -> 0, the summand reduces to a rational function of nj's times x1^n1...xN^nN; 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 epsilon -> 0 and xi=1; we consider some specific cases where at most six Gamma functions remain in the limit epsilon -> 0. The algorithms enable evaluations of arbitrary expansion coefficients in epsilon 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.