Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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SOME EVALUATIONS OF INFINITE SERIES INVOLVING DIRICHLET TYPE PARAMETRIC HARMONIC NUMBERS

  • Hongyuan Rui (School of Mathematics and Statistics Anhui Normal University) ;
  • Ce Xu (School of Mathematics and Statistics Anhui Normal University) ;
  • Xiaobin Yin (School of Mathematics and Statistics Anhui Normal University)
  • Received : 2023.05.18
  • Accepted : 2023.08.14
  • Published : 2024.05.31
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Abstract

In this paper, we formally introduce the notion of a general parametric digamma function Ψ(−s; A, a) and we find the Laurent expansion of Ψ(−s; A, a) at the integers and poles. Considering the contour integrations involving Ψ(−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.

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Acknowledgement

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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