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

Abstract

In this paper, we investigate the Euler sums $$G_{n+2}(p,q)=\sum _{1\le k_1<k_2<\cdots <k_{p+1}}\frac{1}{k_1{k}_2\cdots k_p{k}_{p+1}^{n+2}}\sum _{1\le {\ell }_1\le {\ell }_2\le \cdots \le {\ell }_q\le k_{p+1}}\frac{1}{{\ell }_1{\ell }_2\cdots {\ell }_q}.$$ 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.