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

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 \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.

On a series of Ramanujan, dilogarithm values, and solitons

“…3 (i) matches what is given in the Appendix of[17] by Stewart, which is a good sign our representation of the trilogarithm is indeed useful.…”

“…The polylogarithm function Li p (t) = ∞ n=1 t n n p has important applications in mathematics, especially in evaluating Euler type series and logarithmic integrals [2,3,7,8,9,10,12,15,16,14,17]. The function is continuous on the closed unit disk |t| ≤ 1 and analytic in its interior.…”

“…where ζ(s) is Riemann's zeta function (see [2,3,8,15,16,14,18]). These functions appear in the structure of many generating functions involving the harmonic numbers [3,5,6,9,16,17]…”

Infinite Series Concerning Tails of Riemann Zeta Values

“…Recently, Furdui [5,6], followed by Boyadzhiev [7] and Nguyen [8], evaluated a few series involving the Riemann zeta tails, which are also related to harmonic numbers (cf. [9][10][11][12][13]) and multifold zeta values (cf. [14][15][16][17]).…”

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