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Continuous Hahn Polynomials of Differential Operator Argument and Analysis on Riemannian Symmetric Spaces of Constant Curvature

Published online by Cambridge University Press:  20 November 2018

Erich Badertscher  and
Tom H. Koornwinder
Erich Badertscher
Affiliation:
Mathematisches Institut, Sidlerstrasse 5, CH-3012 Bern, Switzerland email: badertscher@mai.unibe.ch
Tom H. Koornwinder
Affiliation:
CWI, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands
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Abstract

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For the three types of simply connected Riemannian spaces of constant curvature it is shown that the associated spherical functions can be obtained from the corresponding (zonal) spherical functions by application of a differential operator of the form p(i d/dt), where p belongs to a system of orthogonal polynomials: Gegenbauer polynomials, Hahn polynomials or continuous symmetric Hahn polynomials. We give a group theoretic explanation of this phenomenon and relate the properties of the polynomials p to the properties of the corresponding representation. The method is extended to the case of intertwining functions.

Keywords

43A9043A8533C8033C45spaces of constant curvaturespherical functionsassociated spherical functionsBessel functionsJacobi functionsGegenbauer polynomialsHahn polynomialscontinuous symmetric Hahn polynomials.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 4 , 01 August 1992 , pp. 750 - 773
Copyright
Copyright © Canadian Mathematical Society 1992

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