On 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 | References | Similar Articles | Additional Information

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 -Racah polynomials and multivariable big and little -Jacobi polynomials are proved by taking suitable limits in the orthogonality relations for the Koornwinder polynomials. In particular new proofs of several well-known -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