On a generalization of the Rogers generating function

Abstract

We derive a generalized Rogers generating function and corresponding definite integral, for the continuous q-ultraspherical polynomials by applying its connection relation and utilizing orthogonality. Using a recent generalization of the Rogers generating function by Ismail & Simeonov expanded in terms of Askey-Wilson polynomials, we derive corresponding generalized expansions for the continuous q-Jacobi, and Wilson polynomials with two and four free parameters respectively. Comparing the coefficients of the Askey-Wilson expansion to our continuous q-ultraspherical/Rogers expansion, we derive a new quadratic transformation for basic hypergeometric series connecting 2 φ 1 and 8 φ 7 .

Some results for sums of products of Chebyshev and Legendre polynomials

“…For example, the case λ = 1/2 and λ = 1 in (1.1) gives the Legendre polynomials P n (x) and the Chebyshev polynomials of the second kind U n (x), respectively. The Chebyshev polynomials of the first kind T n (x) can be expressed by the Gegenbauer polynomials, as follows (see, e.g., [1,3]):…”