Explicit evaluation of quadratic Euler sums

TL;DR

This paper derives explicit formulas for quadratic Euler sums involving harmonic numbers, providing new closed-form representations in terms of the Riemann zeta function and linear sums, advancing the understanding of these mathematical series.

Contribution

It introduces new explicit formulas for quadratic Euler sums involving harmonic and alternating harmonic numbers, and offers novel closed-form representations using the Riemann zeta function.

Findings

New explicit formulas for quadratic Euler sums

Closed-form representations involving Riemann zeta function

Advances in understanding Euler sums and their relations

Abstract

In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic Euler sums through Riemann zeta function and linear sums. The given representations are new.