Characterizations of Continuous and Discrete q-Ultraspherical Polynomials

Ismail, Mourad E. H. ; Obermaier, Josef

Canadian journal of mathematics, Tome 63 (2011) no. 1, pp. 181-199

Résumé

We characterize the continuous q-ultraspherical polynomials in terms of the special form of the coefficients in the expansion ${{\mathcal{D}}_{q}}{{P}_{n}}\left( x \right)$ in the basis $\left\{ {{P}_{n}}\left( x \right) \right\},{{\mathcal{D}}_{q}}$ 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.

Détail

DOI : 10.4153/CJM-2010-080-0

Mots-clés : 33D45, 42C05, continuous q-ultraspherical polynomials, big q-Jacobi polynomials, discrete q-ultraspherical polynomials, Askey–Wilson operator, q-difference operator, recursion coefficients

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