Multinomial Sum Formulas of Multiple Zeta Values

Abstract

For a pair of positive integers $n,k$ with $n\geq 2$, in this paper we prove that $$ \sum_{r=1}^k\sum_{|\bf\alpha|=k}{k\choose\bf\alpha} \zeta(n\bf\alpha)=\zeta(n)^k =\sum^k_{r=1}\sum_{|\bf\alpha|=k} {k\choose\bf\alpha}(-1)^{k-r}\zeta^\star(n\bf\alpha), $$ where $\bf\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_r)$ is a $r$-tuple of positive integers. Moreover, we give an application to combinatorics and get the following identity: $$ \sum^{2k}_{r=1}r!{2k\brace r}=\sum^k_{p=1}\sum^k_{q=1}{k\brace p}{k\brace q} p!q!D(p,q), $$ where ${k\brace p}$ is the Stirling numbers of the second kind and $D(p,q)$ is the Delannoy number.