On the convolutions of sums of multiple zeta(-star) values of height one

Abstract

In this paper, we investigate the sums of mutliple zeta(-star) values of height one: $Z_{\pm }(n)={\sum }_{a+b=n}(\pm 1{)}^b\zeta (\{1{\}}^a,b+2)$, $Z_{\pm }^{\star }(n)={\sum }_{a+b=n}(\pm 1{)}^b{\zeta }^{\star }(\{1{\}}^a,b+2)$. In particular, we prove that the weighted sum ${\sum }_{\begin{array}{c}0\le m\le p\\ m:\mathrm{even}\end{array}}{\sum }_{\mid \mathbf{\alpha }\mid =p+3}{2}^{{\alpha }_{m+1}+1}\zeta ({\alpha }_0,{\alpha }_1,\dots ,{\alpha }_m,{\alpha }_{m+1}+1)$ can be evaluated through the convolution of $Z_{-}(m)$ and $Z_{+}(n)$ with $m+n=p$.

Any comments are welcome