A decomposition of $\zeta(2n+3)$ into sums of multiple zeta values

Abstract

We evaluate the multiple zeta value $\zeta \left(1,{\left\{2\right\}}^{n+1}\right)$ or its dual $\zeta \left({\left\{2\right\}}^n,3\right)$. When $n$ is even, along with stuffle relations already available, it is enough to evaluate all multiple zeta values of the form $\zeta \left({\left\{2\right\}}^a,3,{\left\{2\right\}}^b\right)$ with $a+b=n$. Furthermore, we obtain a decomposition for $2\zeta \left(2n+3\right)$ as

$${\zeta }^{\star }\left(3,{\left\{2\right\}}^n\right)+\zeta \left({\left\{2\right\}}^n,3\right)+\sum _{r=1}^n\sum _{\left|\alpha \right|=n+1}\zeta \left(1,2{\alpha }_1,2{\alpha }_2,\dots ,2{\alpha }_r\right),$$

which also can be used to evaluate $\zeta \left({\left\{2\right\}}^n,3\right)$ when $n$ is even.