Transformation of generalized multiple Riemann zeta type sums with repeated arguments

Abstract

The aim of this paper is the study of a transformation dealing with the general K-fold infinite series of the form

Sigma(n1 >=...>= nK >= 1) Pi(K)(j=1) a(nj),

especially those, where a(n) = R(n) is a rational function satisfying certain simple conditions. These sums represent the direct generalization of the well-known multiple Riemann zeta -star function with repeated arguments zeta*({s}(K)) when a(n) = 1/n(s). Our result reduces Sigma Pi a(nj) to a special kind of one-fold infinite series. We apply the main theorem to the rational function R(n) = 1/((n + a)(s) + b(s)) in case of which the resulting K-fold sum is called the generalized multiple Hurwitz zeta -star function zeta*(a, b; {s}(K)). We construct an effective algorithm enabling the complete evaluation of zeta*(a, b; {2s}(K)) with a is an element of {0, -1/2}, b is an element of R \ {0}, (K, s) is an element of N-2, by means of a differential operator and present a simple 'Mathematica' code that allows their symbolic calculation. We also provide a new transformation of the ordinary multiple Riemann zeta-star values zeta*({2s}(K)) and zeta*({3}(K)) corresponding to a = b = 0.