## Taiwanese Journal of Mathematics

### ON VECTOR GENERALIZATIONS OF VANDERMONDE'S CONVOLUTION

Yao Lin Ong

#### 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.

#### Article information

Source
Taiwanese J. Math., Volume 17, Number 5 (2013), 1677-1691.

Dates
First available in Project Euclid: 10 July 2017

https://projecteuclid.org/euclid.twjm/1499706232

Digital Object Identifier
doi:10.11650/tjm.17.2013.2440

Mathematical Reviews number (MathSciNet)
MR3106037

Zentralblatt MATH identifier
1287.40001

#### Citation

Ong, Yao Lin. ON VECTOR GENERALIZATIONS OF VANDERMONDE'S CONVOLUTION. Taiwanese J. Math. 17 (2013), no. 5, 1677--1691. doi:10.11650/tjm.17.2013.2440. https://projecteuclid.org/euclid.twjm/1499706232