On Vectorized Weighted Sum Formulas of Multiple Zeta Values

Abstract

In this paper, we introduce the vectorization of shuffle products of two sums of multiple zeta values, which generalizes the weighted sum formula obtained by Ohno and Zudilin. Some interesting weighted sum formulas of vectorized type are obtained, such as\begin{align*} &\quad \sum_{\substack{\boldsymbol{\alpha}+\boldsymbol{\beta}=\boldsymbol{k} |\boldsymbol{\alpha}|: \textrm{even}}} M(\boldsymbol{\alpha}) M(\boldsymbol{\beta}) \sum_{|\boldsymbol{c}|=|\boldsymbol{k}|+r+3} 2^{c_{|\boldsymbol{\alpha}|+1}} \zeta(c_0, c_1, \ldots, c_{|\boldsymbol{\alpha}|}, \ldots, c_{|\boldsymbol{k}|+1}+1) \\ &= \frac{1}{2} (2|\boldsymbol{k}|+r+5) M(\boldsymbol{k}) \zeta(|\boldsymbol{k}|+r+4),\end{align*}where $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$ and $\boldsymbol{k}$ are $n$-tuples of nonnegative integers with $|\boldsymbol{k}| = k_1 + k_2 + \cdots + k_n$ even; $M(\boldsymbol{u})$ is the multinomial coefficient defined by $\binom{u_1 + u_2 + \cdots + u_n}{u_1, u_2, \ldots, u_n}$ with the value $\frac{|\boldsymbol{u}|!}{u_1! u_2! \cdots u_n!}$; and $r$ is a nonnegative integer. Moreover, some newly developed combinatorial identities of vectorized types involving multinomial coefficients by extending the shuffle products of two sums of multiple zeta values in their vectorizations are given as well.