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 $H_{2n}$ 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
\[\sum_{n = 1}^\infty \left (\frac{H_n}{n} \right )^2 = \frac{17
  \pi^4}{360},\]
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.