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On $BC$ type basic hypergeometric orthogonal polynomials


Author: Jasper V. Stokman
Journal: Trans. Amer. Math. Soc. 352 (2000), 1527-1579
MSC (2000): Primary 33D52; Secondary 33D45, 33D80
DOI: https://doi.org/10.1090/S0002-9947-99-02551-9
Published electronically: November 17, 1999
MathSciNet review: 1694379
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Abstract: The five parameter family of Koornwinder's multivariable analogues of the Askey-Wilson polynomials is studied with four parameters generically complex. The Koornwinder polynomials form an orthogonal system with respect to an explicit (in general complex) measure. A partly discrete orthogonality measure is obtained by shifting the contour to the torus while picking up residues. A parameter domain is given for which the partly discrete orthogonality measure is positive. The orthogonality relations and norm evaluations for multivariable $q$-Racah polynomials and multivariable big and little $q$-Jacobi polynomials are proved by taking suitable limits in the orthogonality relations for the Koornwinder polynomials. In particular new proofs of several well-known $q$-analogues of the Selberg integral are obtained.


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Additional Information

Jasper V. Stokman
Affiliation: KdV Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
Address at time of publication: Centre de Mathématiques de Jussieu, Université Paris 6 Pierre et Marie Curie, 4 Place Jussieu, Paris 75252 Cedex 05, France
Email: stokman@math.jussieu.fr

DOI: https://doi.org/10.1090/S0002-9947-99-02551-9
Keywords: Multivariable basic hypergeometric orthogonal polynomials, Koornwinder polynomials, multivariable $q$-Racah polynomials, multivariable big and little $q$-Jacobi polynomials, $q$-Selberg type integrals, residue calculus
Received by editor(s): July 7, 1997
Published electronically: November 17, 1999
Additional Notes: The author was supported by a NISSAN-fellowship of the Netherlands Organization of Scientific Research (NWO)
Article copyright: © Copyright 2000 American Mathematical Society

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