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 H 2 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:munderover> <m:mo>∑</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>∞</m:mo> </m:munderover> <m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mfrac> <m:mrow> <m:msub> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mi>n</m:mi> </m:msub> </m:mrow> <m:mi>n</m:mi> </m:mfrac> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>17</m:mn> <m:msup> <m:mrow> <m:mi>π</m:mi> </m:mrow> <m:mn>4</m:mn> </m:msup> </m:mrow> <m:mrow> <m:mn>360</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:mrow> </m:math> ∑n = 1^∞ {{{( {{{{H_n}} n}} )}^2} = {{17{π ^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.