Multiple zeta values and Euler sums

TL;DR

This paper explores relationships between multiple zeta values, Euler sums, and harmonic numbers, providing new formulas and reductions to zeta polynomials, enriching the understanding of these special functions.

Contribution

It introduces new relationships between multiple zeta values and star values, and provides explicit recurrence formulas for certain combined sums, advancing the theoretical framework.

Findings

Closed form representations of Euler sums in terms of Riemann zeta values

Reduction of combined sums to polynomials in zeta values

Explicit recurrence formulas for sums involving multiple zeta values

Abstract

In this paper, we establish some expressions of series involving harmonic numbers and Stirling numbers of the first kind in terms of multiple zeta values, and present some new relationships between multiple zeta values and multiple zeta star values. The relationships obtained allow us to find some nice closed form representations of nonlinear Euler sums through Riemann zeta values and linear sums. Furthermore, we show that the combined sums \[H\left( {a,b;m,p} \right) := \sum\limits_{a + b = m - 1} {\zeta \left( {{{\left\{ p \right\}}_a},p + 1,{{\left\{ p \right\}}_b}} \right)}\quad (m\in \N,p>1) \] and \[{H^ \star }\left( {a,b;m,p} \right) := \sum\limits_{a + b = m - 1} {{\zeta ^ \star }\left( {{{\left\{ p \right\}}_a},p + 1,{{\left\{ p \right\}}_b}} \right)}\quad (m\in \N,p>1) \] are reducible to polynomials in zeta values, and give explicit recurrence formulas. Some interesting (known or new) consequences and illustrative examples are considered.