On 𝐵𝐶 type basic hypergeometric orthogonal polynomials

Stokman, Jasper

doi:10.1090/S0002-9947-99-02551-9

On $BC$ type basic hypergeometric orthogonal polynomials
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by Jasper V. Stokman

Trans. Amer. Math. Soc. 352 (2000), 1527-1579

DOI: https://doi.org/10.1090/S0002-9947-99-02551-9

Published electronically: November 17, 1999

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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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