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

Stewart, Seán Mark

doi:10.2478/tmmp-2020-0034

Abstract

In this paper a number of new explicit expressions for quadratic Euler-type sums containing double-index harmonic numbers H₂_n are given. These are obtained using ordinary generating functions containing the square of the harmonic numbers H_n. As a by-product of the generating function approach used new proofs for the remarkable quadratic series of Au-Yeung $\sum_{n = 1}^{\infty} {(\frac{H_{n}}{n})}^{2} = \frac{17 π^{4}}{360}$ \sum\limits_{n = 1}^\infty {{{\left( {{{{H_n}} \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.

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

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