A Lie-theoretic interpretation of multivariate hypergeometric polynomials
type: Item
instance of: scholarly article
title: A Lie-theoretic interpretation of multivariate hypergeometric polynomials
zbMATH Open document ID: 1248.33028
DOI: 10.1112/S0010437X11007421
author: Plamen Iliev
published in: Compositio Mathematica
publication date: 2012-06-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1101.1683
Mathematics Subject Classification ID: 33C80
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 39A14
zbMATH DE Number: 6051034
zbMATH Keywords: multivariate Krawtchouk polynomials
zbMATH Keywords: Lie algebra \(sl_{d+1}\)
zbMATH Keywords: partial difference operators
zbMATH Keywords: bispectral problem
MaRDI profile type: Publication
arXiv ID: 1101.1683
cites work: Orthogonal Polynomials on the Multinomial Distribution
cites work: Bispectrality of multivariable Racah-Wilson polynomials
cites work: Discrete orthogonal polynomials and difference equations of several variables
cites work: Two linear transformations each tridiagonal with respect to an eigenbasis of the other
cites work: (𝑛+1,𝑚+1)-hypergeometric functions associated to character algebras
cites work: A Multi-dimensional Linear Growth Birth and Death Process
cites work: Differential equations in the spectral parameter
review text: In 1971, Griffiths used a generating function to define polynomials in \(d\) variables orthogonal with respect to the multinomial distribution. The polynomials possess a duality between the discrete variables and the degree indices. In 2004, \textit{H. Mizukawa} and \textit{H. Tanaka} [Proc. Am. Math. Soc. 132, No. 9, 2613--2618 (2004; Zbl 1059.33020)] related these polynomials to character algebras and the Gelfand hypergeometric series. Using this approach, they clarified the duality and obtained a new proof of the orthogonality.NEWLINENEWLINEIn the present paper, these polynomials are interpreted within the context of the Lie algebra \(sl_{d+1}(\mathbb{C})\). This approach yields yet another proof of the orthogonality. Moreover, it shows that the polynomials satisfy \(d\) independent recurrence relations each involving \(d^2+d+1\) terms. This, combined with the duality, establishes their bispectrality. Finally, the results are illustrated with several explicit examples.
reviewed by: Nele De Schepper
description: scientific article; zbMATH DE number 6051034