Multinomial Sum Formulas of Multiple Zeta Values

TL;DR

This paper establishes new sum formulas for multiple zeta values involving multinomial coefficients, providing identities that connect these sums to combinatorial numbers like Stirling and Delannoy numbers.

Contribution

The paper introduces novel multinomial sum formulas for multiple zeta values and applies these to derive combinatorial identities involving Stirling and Delannoy numbers.

Findings

Proved sum formulas relating multiple zeta values and their star versions.

Derived a new combinatorial identity involving Stirling and Delannoy numbers.

Connected multiple zeta value identities with classical combinatorial quantities.

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.