Evaluation of eight different families of cubic Euler sums

TL;DR

This paper explicitly calculates nonlinear cubic Euler sums of degrees four to six across eight families, expressing results in terms of zeta and polylogarithmic values, advancing understanding of these complex sums.

Contribution

It provides explicit formulas for nonlinear cubic Euler sums of degrees four to six in terms of zeta and polylogarithmic values for eight families.

Findings

Explicit formulas for Euler sums in terms of zeta and polylogarithmic values.

Complete calculation of nonlinear Euler sums for three types of denominators.

Results applicable to degrees four, five, and six Euler sums.

Abstract

We present a study on cubic Euler sums of degree four, five and six, where three different types of denominators $1/k^n$, $1/((2k-1)^n)$ and $1/(k(2k-1))$ will be considered We demonstrate that for all three orders the complete variety of corresponding nonlinear Euler sums belonging to the eight different families can be explicitly calculated in terms of zeta values and polylogarithmic values $Li_4(1/2)$, $Li_5(1/2)$, $Li_6(1/2)$, $Li_6(-1/2)$ and $Li_6(-1/8)$.