Taiwanese Journal of Mathematics

ON VECTOR GENERALIZATIONS OF VANDERMONDE'S CONVOLUTION

Yao Lin Ong

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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

Permanent link to this document
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

Subjects
Primary: 40A25: Approximation to limiting values (summation of series, etc.) {For the Euler-Maclaurin summation formula, see 65B15} 40B05: Multiple sequences and series (should also be assigned at least one other classification number in this section)
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11M06: $\zeta (s)$ and $L(s, \chi)$ 33E20: Other functions defined by series and integrals

Keywords
Vandermonde's convolution multiple zeta value shuffle product

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


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