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Transformation of generalized multiple Riemann zeta type sums with repeated arguments

Marian Genčev
Journal / Anthology

Journal of Mathematical Analysis and Applications
Year: 2017
Volume: 449
Page range: 490-513

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 K j=1 anj , especially those, where an=R(n)is a rational function satisfying certain simple conditions. These sums represent the direct generalization of the well-known multi-ple Riemann zeta-star function with repeated arguments ζ({s}K)when an=1/ns. Our result reduces anjto 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 evalua-tion of ζ(a, b; {2s}K)with a ∈{0,−1/2}, b ∈R \{0}, (K, s) ∈N2, by means of adifferential 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.

*Applied Mathematics