Evaluations of some quadratic Euler sums

TL;DR

This paper introduces a new method using integral computations of polylogarithms to evaluate quadratic Euler sums involving harmonic numbers, establishing relations to linear sums and expressing results in terms of zeta values.

Contribution

It presents a novel approach to evaluate quadratic Euler sums, deriving new closed-form representations in terms of zeta values and linear sums.

Findings

Established relations between quadratic and linear Euler sums.

Derived new closed-form expressions involving zeta values.

Provided a systematic integral-based evaluation method.

Abstract

This paper develops an approach to the evaluation of quadratic Euler sums that involve harmonic numbers. The approach is based on simple integral computations of polyloga- rithms. By using the approach, we establish some relations between quadratic Euler sums and linear sums. Furthermore, we obtain some closed form representations of quadratic sums in terms of zeta values and linear sums. The given representations are new.