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Characterizations of Continuous and Discrete q-Ultraspherical Polynomials
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- Journal:
- Canadian Journal of Mathematics / Volume 63 / Issue 1 / 01 February 2011
- Published online by Cambridge University Press:
- 20 November 2018, pp. 181-199
- Print publication:
- 01 February 2011
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- Article
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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.
