Algorithms to evaluate multiple sums for loop computations

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, with aⁱ.n=∑^N(_j=1) a(_ij_)nj, 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 Xⁿ¹₁...X^nN_N; 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.