Bilinear Generating Functions for Orthogonal Polynomials |
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Authors: | H. T. Koelink J. Van der Jeugt |
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Affiliation: | (1) Werkeenheid Algemene Wiskunde ITS-TWI-AW Technische Universiteit Delft Postbus 5031 2600 GA Delft The Netherlands koelink@twi.tudelft.nl, NL;(2) Vakgroep Toegepaste Wiskunde en Informatica Universiteit Gent Krijgslaan 281-S9 B-9000 Gent Belgium Joris.VanderJeugt@rug.ac.be, BE |
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Abstract: | Using realizations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner—Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner—Pollaczek polynomials; this result is also known as the Burchnall—Chaundy formula. For the positive discrete series representations of the quantized universal enveloping algebra U q (su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey—Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials. July 6, 1997. Date accepted: September 23, 1998. |
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Keywords: | . Orthogonal polynomials Bilinear generating functions Poisson kernels Lie algebra Quantum algebra. AMS Classification. 33C80 33D80 33C25 33D25 17B20 17B37. |
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