Multinomial Sum Formulas of Multiple Zeta Values

Abstract

For a pair of positive integers $n,k$ with $n\ge 2$, in this paper we prove that $$\sum _{r=1}^k\sum _{|\alpha |=\mathbf{k}}\left(\genfrac{}{}{0ex}{}{k}{\alpha }\right)\zeta (n\alpha )=\zeta (\mathbf{n}{)}^{\mathbf{k}}=\sum _{\mathbf{r}=1}^{\mathbf{k}}\sum _{|\alpha |=\mathbf{k}}(\genfrac{}{}{0ex}{}{\mathbf{k}}{\alpha })(-1{)}^{\mathbf{k}-\mathbf{r}}{\zeta }^{\star }(\mathbf{n}\alpha ),$$ where $\alpha =({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{\mathbf{r}})$ is a $r$-tuple of positive integers. Moreover, we give an application to combinatorics and get the following identity: $$\sum _{r=1}^{2k}r!\left\{\genfrac{}{}{0ex}{}{2k}{r}\right\}=\sum _{p=1}^k\sum _{q=1}^k\left\{\genfrac{}{}{0ex}{}{k}{p}\right\}\left\{\genfrac{}{}{0ex}{}{k}{q}\right\}p!q!D(p,q),$$ where $\left\{\genfrac{}{}{0ex}{}{k}{p}\right\}$ is the Stirling numbers of the second kind and $D(p,q)$ is the Delannoy number.

10 pages. Version 2, add Section 5

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