Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers

Seán Mark Stewart

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper a number of new explicit expressions for quadratic Euler-type sums containing double-index harmonic numbers H2n are given. These are obtained using ordinary generating functions containing the square of the harmonic numbers Hn. As a by-product of the generating function approach used new proofs for the remarkable quadratic series of Au-Yeung n=1∞(Hnn)2=17π4360 \sum\limits {n = 1}infty {left({Hn}} (over n}} right)}2} = {{17{pi 4}} over {360} together with its closely related alternating cousin are given. New proofs for other closely related quadratic Euler-type sums that are known in the literature are also obtained.

Original languageEnglish
Pages (from-to)73-98
Number of pages26
JournalTatra Mountains Mathematical Publications
Volume77
Issue number1
DOIs
Publication statusPublished - Dec 1 2020

Keywords

  • Euler sums
  • Generating function
  • Harmonic number
  • Polylogarithm function
  • Riemann zeta function

ASJC Scopus subject areas

  • Mathematics(all)

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