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

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