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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, n1,··· ,nN (a1·n+c1)(a2·n+c2)···(aP ·n+cP ) (b1·n+d1)(b2·n+d2)···(bQ·n+dQ) xn1 1 · · · xnN N with ai ·n = Nj =1 ai jn j , etc., in a small parameter around rational values of ci,di’s. Type I sum corresponds to the case where, in the limit →0, the summand reduces to a rational function of nj’s times xn1 1 · · · xnN 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 →0 and xi =1; we consider some specific cases where atmost six functions remain in the limit →0. The algorithms enable evaluations of arbitrary expansion coefficients in 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 aMathematica package, in which these algorithms are implemented.
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