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=∑j=1Naijnj, 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.Comment: 30 pages, 2 figures; address of Mathematica package in Sec.6; version
to appear in J.Math.Phy
