Transformation of generalized multiple Riemann zeta type sums with repeated arguments

The aim of this paper is the study of a transformation dealing with the general K-fold infinite series of the form∑n1≥⋯≥nK≥1∏j=1Kanj, especially those, where an=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 ζ⋆({s}K) when an=1/ns. Our result reduces ∑∏anj to a special kind of one-fold infinite series. We apply the main theorem to the rational function R(n)=1/((n+a)s+bs) in case of which the resulting K-fold sum is called the generalized multiple Hurwitz zeta-star function ζ⋆(a,b;{s}K). We construct an effective algorithm enabling the complete evaluation of ζ⋆(a,b;{2s}K) with a∈{0,−1/2}, b∈R∖{0}, (K,s)∈N2, 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 ζ⋆({2s}K) and ζ⋆({3}K) corresponding to a=b=0.