Evaluations of nonlinear Euler sums of weight ten

TL;DR

This paper derives new identities for nonlinear Euler sums of weight ten, providing closed forms and relations to multiple zeta values, enhancing understanding of complex series in mathematical analysis.

Contribution

It introduces a new family of identities for Euler sums, derives closed forms for quadratic sums of weight ten, and relates them to multiple zeta values.

Findings

Closed forms for all quadratic Euler sums of weight ten.

Relations established between multiple zeta (star) values and nonlinear Euler sums.

Numerical evaluations of several Euler sums of weight ten.

Abstract

In this paper we present a new family of identities for Euler sums and integrals of polylogarithms by using the methods of generating function and integral representations of series. Then we apply it to obtain the closed forms of all quadratic Euler sums of weight equal to ten. Furthermore, we also establish some relations between multiple zeta (star) values and nonlinear Euler sums. As applications of these relations, we give new closed form representations of several cubic Euler sums through single zeta values and linear sums. Finally, with the help of numerical computations of Mathematica or Maple, we evaluate several other Euler sums of weight ten.