Multiple zeta values and Euler sums

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 \[H\left( {a,b;m,p} \right) := \sum\limits_{a + b = m - 1} {ζ\left( {{\left\{ p \right\}_a},p + 1,{\left\{ p \right\}_b}} \right)}\quad (m\in \N,p>1) \] and \[{H^ \star }\left( {a,b;m,p} \right) := \sum\limits_{a + b = m - 1} {{ζ^ \star }\left( {{\left\{ p \right\}_a},p + 1,{\left\{ p \right\}_b}} \right)}\quad (m\in \N,p>1) \] are reducible to polynomials in zeta values, and give explicit recurrence formulas. Some interesting (known or new) consequences and illustrative examples are considered.