Published online by Cambridge University Press: 20 November 2018
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.