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Identities for $ q$-ultraspherical polynomials and Jacobi functions


Author: H. T. Koelink
Journal: Proc. Amer. Math. Soc. 123 (1995), 2479-2487
MSC: Primary 33C45; Secondary 33D55
MathSciNet review: 1273504
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Abstract: A q-analogue of a result by Badertscher and Koornwinder [Canad. J. Math. 44 (1992), 750-773] relating the action of a Hahn polynomial of differential operator argument on ultraspherical polynomials to an ultraspherical polynomial of shifted order and degree is derived. The q-analogue involves q-Hahn polynomials, continuous q-ultraspherical polynomials, and a shift operator. Another limit as q tends to 1 yields an identity for Jacobi functions. Combination with another result of Badertscher and Koornwinder gives a curious formula for Jacobi functions.


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DOI: https://doi.org/10.1090/S0002-9939-1995-1273504-8
Keywords: Continuous q-ultraspherical polynomial, q-Hahn polynomial, Jacobi function, ultraspherical polynomial, continuous symmetric Hahn polynomial
Article copyright: © Copyright 1995 American Mathematical Society