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

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.