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Characterizations of Continuous and Discrete $q$-Ultraspherical Polynomials |
Canad. J. Math. 63(2011), 181-199
Published:2010-11-06
Printed: Feb 2011
Printed: Feb 2011
- Mourad E. H. Ismail,
- Josef Obermaier,
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] |