2-variable generalized Hermite matrix polynomials and Lie algebra representation
type: Item
instance of: scholarly article
title: 2-variable generalized Hermite matrix polynomials and Lie algebra representation
zbMATH Open document ID: 1229.33018
DOI: 10.1016/S0034-4877(10)00024-8
author: Subuhi Khan
author: Nusrat Raza
published in: Reports on Mathematical Physics
publication date: 2011-06-03T00:00:00Z
review text: Motivated by the definition of 2-variable Hermite Kampé de Fériet polynomials \(H_n(x,y)\) via their generating function, \[ \exp(xt + y t^2) =\sum_{n=0}^\infty H_n(x,y) \frac{t^n}{n!}, \] and by several extensions of classical orthogonal polynomials to matrix parameters, the authors introduce the set of 2-variable generalized Hermite polynomials \(H^\lambda\) via \[ \exp\left(\lambda\left(xt\sqrt{ A/2} + y t^2 I\right)\right) =\sum_{n=0}^\infty H^\lambda_n(x,y;A) \frac{t^n}{n!}, \] where \(A\) is a matrix in \(\mathbb C^{m\times m}\), and \(\lambda\) a complex parameter. In particular, \[ H^1_n(x,y;A)=H_n\left(x\sqrt{A/2},y\right). \] Several algebraic properties of \(H^\lambda\) (such as their explicit representation or the differential equation) are obtained. Also, the authors use the representation theory of the harmonic oscillator Lie algebra \(\mathcal G(0,1)\) in order to derive further generating identities for the 2-variable generalized Hermite polynomials \(H^\lambda\).
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 33C50
Mathematics Subject Classification ID: 43A65
zbMATH DE Number: 5904004
zbMATH Keywords: Multivariate Hermite polynomials
zbMATH Keywords: matrix polynomials
zbMATH Keywords: Lie groups
zbMATH Keywords: Lie algebra
zbMATH Keywords: representation theory
zbMATH Keywords: generating relation
reviewed by: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
Wikidata QID: Q115339927
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full work available at URL: s0034-4877(10)00024-8
OpenAlex ID: W1995201278
description: scientific article; zbMATH DE number 5904004