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Bull. Korean Math. Soc. 2024; 61(3): 671-697

Online first article December 6, 2023      Printed May 31, 2024

https://doi.org/10.4134/BKMS.b230299

Copyright © The Korean Mathematical Society.

Some evaluations of infinite series involving Dirichlet type parametric harmonic numbers

Hongyuan Rui, Ce Xu, Xiaobin Yin

Anhui Normal University; Anhui Normal University; Anhui Normal University

Abstract

In this paper, we formally introduce the notion of a general parametric digamma function $\Psi (-s; A, a)$ and we find the Laurent expansion of $\Psi (-s; A, a)$ at the integers and poles. Considering the contour integrations involving $\Psi (-s; A, a)$, we present some new identities for infinite series involving Dirichlet type parametric harmonic numbers by using the method of residue computation. Then applying these formulas obtained, we establish some explicit relations of parametric linear Euler sums and some special functions (e.g.~trigonometric functions, digamma functions, Hurwitz zeta functions etc.). Moreover, some illustrative special cases as well as immediate consequences of the main results are also considered.

Keywords: General parametric digamma function, parametric linear Euler sums, contour integrations, residue computations, parametric harmonic numbers, Hurwitz zeta functions

MSC numbers: Primary 11M32

Supported by: Ce Xu is supported by the National Natural Science Foundation of China (Grant No. 12101008), the Natural Science Foundation of Anhui Province (Grant No. 2108085QA01) and the University Natural Science Research Project of Anhui Province (Grant No. KJ2020A0057).

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