TY - JOUR

T1 - Algorithms to evaluate multiple sums for loop computations

AU - Anzai, C.

AU - Sumino, Y.

N1 - Funding Information:
We are grateful to Y. Kiyo for fruitful discussion. We also thank J. Blümlein and C. Schneider, and V. A. Smirnov and M. Steinhauser, respectively, for providing information on their computations. The works of C.A. and Y.S. are supported in part by the JSPS Fellowships for Young Scientists and Grant-in-Aid for scientific research (Grant No. 23540281) from MEXT, Japan, respectively.

PY - 2013/3/6

Y1 - 2013/3/6

N2 - 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...XnNN; 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.

AB - 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...XnNN; 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.

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U2 - 10.1063/1.4795288

DO - 10.1063/1.4795288

M3 - Article

AN - SCOPUS:84875879653

VL - 54

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 3

M1 - 033514

ER -