Explicit evaluation of some quadratic Euler-type sums containing double-index harmonic numbers: Tatra Mt. Math. Publ. Number Theory and Cryptology '20

Year, pages: 2020, 73 - 98

In this paper a number of new explicit expressions for quadratic Euler-type sums containing double-index harmonic numbers $H_2n$ 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

∑_{n = 1}^∞ (((H_n) / (n)) )² = ((17 π⁴) / (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.

Stewart, S. 2020. Explicit evaluation of some quadratic Euler-type sums containing double-index harmonic numbers: Tatra Mt. Math. Publ. Number Theory and Cryptology '20. In Tatra Mountains Mathematical Publications, vol. 77, no.3, pp. 73-98. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2020-0034

Stewart, S. (2020). Explicit evaluation of some quadratic Euler-type sums containing double-index harmonic numbers: Tatra Mt. Math. Publ. Number Theory and Cryptology '20. Tatra Mountains Mathematical Publications, 77(3), 73-98. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2020-0034

Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava

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