Identities for -ultraspherical polynomials and Jacobi functions

Author:
H. T. Koelink

Journal:
Proc. Amer. Math. Soc. **123** (1995), 2479-2487

MSC:
Primary 33C45; Secondary 33D55

DOI:
https://doi.org/10.1090/S0002-9939-1995-1273504-8

MathSciNet review:
1273504

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Abstract | References | Similar Articles | Additional Information

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