Tridiagonalization of the hypergeometric operator and the Racah-Wilson algebra
type: Item
instance of: scholarly article
zbMATH Open document ID: 1351.33010
author: Vincent X. Genest
author: Luc Vinet
author: Alexei Zhedanov
author: Mourad E. H. Ismail
published in: Proceedings of the American Mathematical Society
publication date: 2016-08-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1506.07803
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33C80
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 6620009
zbMATH Keywords: hypergeometric operator
zbMATH Keywords: tridiagonalization
zbMATH Keywords: Wilson polynomials
zbMATH Keywords: Hahn polynomials
MaRDI profile type: Publication
OpenAlex ID: W2963411637
arXiv ID: 1506.07803
cites work: Q4222737
cites work: On liftingq-difference operators in the Askey scheme of basic hypergeometric polynomials
cites work: q-Hermite Polynomials and Classical Orthogonal Polynomials
cites work: Q4432376
cites work: Superintegrability in two dimensions and the Racah-Wilson algebra
cites work: The equitable Racah algebra from three $\mathfrak {su}(1,1)$ algebras
cites work: Mutual integrability, quadratic algebras, and dynamical symmetry
cites work: Q3182020
cites work: The \(J\)-matrix method
cites work: Spectral analysis of certain Schrödinger operators
cites work: Spectral properties of operators using tridiagonalization
cites work: Hypergeometric Orthogonal Polynomials and Their q-Analogues
cites work: Q3721151
cites work: ``Hidden symmetry'' of Askey-Wilson polynomials
DOI: 10.1090/PROC/13082
title: Tridiagonalization of the hypergeometric operator and the Racah-Wilson algebra
review text: This contribution develops the tridiagonalization of the hypergeometric operator NEWLINE\[NEWLINEL=x(1-x)\partial^2_x+(\alpha+1-(\alpha+\beta+2)x) \partial_xNEWLINE\]NEWLINE where \(\alpha,\beta\) are real constants such that \(\alpha,\beta>-1\), provided by a set of eigenfunctions of an operator \(M\) of the form NEWLINE\[NEWLINEM=\tau_1XL+\tau_2LX+\tau_3X+\tau_0,NEWLINE\]NEWLINE where \(Xf(x)=xf(x)\) and \(\tau_0,\tau_1,\tau_2,\tau_3\) are constants fulfilling \(\tau_1+\tau_2\neq 0\). The authors explore a connection between the tridiagonalization and the generic Wilson polynomials and uncover ``an embedding of the Racah algebra into the Jacobi algebra in their standard realization on the space of polynomials.'' The tridiagonalization in the case for which \(\tau_1+\tau_2=0\) is also treated being enlightened in this situation a connection with the generic Hahn polynomials.
reviewed by: Teresa A. Mesquita
description: scientific article; zbMATH DE number 6620009