Hypergeometric ladders and spherical polynomials for root systems \(BC\) of rank 2
type: Item
instance of: scholarly article
title: Hypergeometric ladders and spherical polynomials for root systems \(BC\) of rank 2
zbMATH Open document ID: 0762.33009
DOI: 10.1007/BF01764130
author: Amédée Debiard
author: Bernard Gaveau
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1992-09-27T00:00:00Z
review text: \textit{T. Koornwinder} and \textit{I. Sprinkhuizen-Kuyper} have studied a family of generalized Jacobi polynomials in two variables, denoted by \(R_{n,k}^{\alpha,\beta,\gamma}(\xi,\eta)\), with domain of definition \[ \Omega=\{(\xi,\eta)\in\mathbb{R}^ 2:\;\eta>0,\;1-\xi+\eta>0,\;\xi^ 2>4\eta,\;0<\xi<2\} \] and orthogonality weight \[ w_{\alpha,\beta,\gamma}(\xi,\eta):=\eta^ \alpha(1-\xi+\eta)^ \beta(\xi^ 2-4\eta) ^ \gamma,\quad\text{for } (\xi,\eta)\in\Omega, \] [see SIAM J. Math. Anal. 9, 457-483 (1978; Zbl 0395.33008)]. In the present article, Deviard and Gaveau consider these polynomials in the context of functions on \(\mathbb{R}^ 2\) periodic modulo the root lattice and invariant under the Weyl group of the root system \(BC_ 2\). This approach leads to the construction of several differential operators useful in the study of the \(R_{n,k}^{\alpha,\beta,\gamma}\) (for example, eigenfunction expansion, transformation to contiguous \(n\) or \(k\) values). The authors are able to give detailed expressions for these polynomials, using a ``hypergeometric ladder'' to increase the \(k\)-index from the known expressions for \(k=0\).
Mathematics Subject Classification ID: 33C80
Mathematics Subject Classification ID: 33C50
zbMATH DE Number: 59863
reviewed by: Charles F. Dunkl
MaRDI profile type: Publication
cites work: Q3222473
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cites work: Analysis on Root Systems
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full work available at URL: https://doi.org/10.1007/bf01764130
OpenAlex ID: W56814276
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description: scientific article; zbMATH DE number 59863