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Characterizations of Continuous and Discrete $q$-Ultraspherical Polynomials

  Published:2010-11-06
 Printed: Feb 2011
  • Mourad E. H. Ismail,
    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
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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 continuous $q$-ultraspherical polynomials, big $q$-Jacobi polynomials, discrete $q$-ultra\-spherical polynomials, Askey--Wilson operator, $q$-difference operator, recursion coefficients
MSC Classifications: 33D45, 42C05 show english descriptions Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]
33D45 - Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
42C05 - Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]
 

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