ON VECTOR GENERALIZATIONS OF VANDERMONDE'S CONVOLUTION

Abstract

In this paper, we introduce the vector generalizations of the well-known Vandermonde's convolution such as \begin{align*} & \sum_{\boldsymbol{j} = \boldsymbol{\alpha_{1}} + \cdots + \boldsymbol{\alpha_{k}}} \sum_{\boldsymbol{\ell} = \boldsymbol{\beta_{1}} + \cdots + \boldsymbol{\beta_{k}}} \prod_{i=1}^{k} \binom{|\boldsymbol{\alpha_{i}}| + |\boldsymbol{\beta_{i}}|}{|\boldsymbol{\alpha_{i}}|} M(\boldsymbol{\alpha_{i}}) M(\boldsymbol{\beta_{i}}) \\ &= M(\boldsymbol{j}) M(\boldsymbol{\ell}) \binom{|\boldsymbol{j}| + |\boldsymbol{\ell}| + k -1}{|\boldsymbol{j}|, |\boldsymbol{\ell}|, k - 1}, \end{align*} where $\boldsymbol{j}$, $\boldsymbol{\ell}$, $\boldsymbol{\alpha_{i}}$ and $\boldsymbol{\beta_{i}}$ are the vectors with nonnegative integer components, and $M(\boldsymbol{j})$ is the multinomial coefficient defined by $\binom {j_1+j_2+ \cdots +j_m}{j_1,\ldots, j_m}$ with the value $\frac {|j|!}{j_1! \cdots j_m!}$. The main interest in such generalization comes from the number of multiple zeta values in relations produced from the shuffle product of two sets of multiple zeta values in their iterated integral representations over simplices. Several generalizations of Vandermonde's convolution type are given as well.