Some special Euler sums and $\zeta^\star(r+2,\{2\}^n)$

TL;DR

This paper explores special Euler sums, providing new representations and properties, and applies these findings to explicitly compute certain multiple zeta-star values for specific parameters.

Contribution

It introduces new representations and properties of Euler sums and applies them to evaluate specific multiple zeta-star values.

Findings

Derived two new representations of Euler sums

Established a reflection formula for these sums

Explicitly calculated $ zeta^igstar(r+2, extbf{2}^n)$ for r=0,1,2

Abstract

In this paper, we investigate the Euler sums $$ G_{n+2}(p,q)=\sum_{1\leq k_1<k_2<\cdots<k_{p+1}}\frac1{k_1k_2\cdots k_pk_{p+1}^{n+2}} \sum_{1\leq\ell_1\leq\ell_2\leq\cdots\leq\ell_q\leq k_{p+1}}\frac1{\ell_1\ell_2\cdots\ell_q}. $$ We give another two representations, a reflection formula, and some other properties. Then we use these results to calculate $\zeta^\star(r+2,\{2\}^n)$, for $r=0,1,2$, as our applications.