location:  Publications → journals → CJM
Abstract view

# Characterizations of Continuous and Discrete $q$-Ultraspherical Polynomials

Published:2010-11-06
Printed: Feb 2011
Department of Mathematics, King Saud University, Riyadh, Saudi Arabia
• Josef Obermaier,
Helmholtz Zentrum München, German Research Center for Environmental Health, Institute of Biomathematics and Biometry, Ingolstädter Landstr. 1, 85764 Neuherberg, Germany
 Format: HTML LaTeX MathJax PDF

## Abstract

We characterize the continuous $q$-ultraspherical polynomials in terms of the special form of the coefficients in the expansion $\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$ being the Askey--Wilson divided difference operator. The polynomials are assumed to be symmetric, and the connection coefficients are multiples of the reciprocal of the square of the $L^2$ norm of the polynomials. A similar characterization is given for the discrete $q$-ultraspherical polynomials. A new proof of the evaluation of the connection coefficients for big $q$-Jacobi polynomials is given.
 Keywords: continuous $q$-ultraspherical polynomials, big $q$-Jacobi polynomials, discrete $q$-ultra\-spherical polynomials, Askey--Wilson operator, $q$-difference operator, recursion coefficients
 MSC Classifications: 33D45 - Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 42C05 - Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]

 top of page | contact us | privacy | site map |