Some Evaluation of Quadratic Euler Sums

TL;DR

This paper derives new formulas for quadratic Euler sums involving harmonic numbers, providing closed-form evaluations using zeta values, polylogarithms, and exploring their relationships with integrals of polylogarithms.

Contribution

It introduces novel formulas for quadratic Euler sums with harmonic numbers and connects these sums to zeta values, polylogarithms, and integral representations.

Findings

New closed-form formulas for quadratic Euler sums

Relationships established between Euler sums and polylogarithm integrals

Expressions involving Riemann zeta values and polylogarithms

Abstract

In this paper, we obtain some formulas for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. By using these formulas, we give new closed form sums of several quadratic Euler series through Riemann zeta values, polylogarithm functions and linear sums. Furthermore, some relationships between Euler sums and integrals of polylogarithm functions are established.