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Algorithms to evaluate multiple sums for loop computations

C. Anzai
Y. Sumino
Journal / Anthology

Year: 2013
Volume: 54

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 (a1n+c1)(a2n+c2)(aP n+cP ) (b1n+d1)(b2n+d2)(bQn+dQ) xn1 1 xnN N with ai n = Nj =1 ai jn j , etc., in a small parameter  around rational values of ci,dis. Type I sum corresponds to the case where, in the limit →0, the summand reduces to a rational function of njs times xn1 1 xnN N ; ci,dis can depend on an external integer index. Type II sum is a double sum (N = 2), where ci, dis 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.

*Science > Physics