On a generalization of the Rogers generating function

J Math Anal Appl. 2019 Jul;475(2):10.1016/j.jmaa.2019.01.068. doi: 10.1016/j.jmaa.2019.01.068.


We derive a generalization of the Rogers generating function for the continuous q-ultraspherical/Rogers polynomials whose coefficient is a 2 ϕ 1. From that expansion, we derive corresponding specialization and limit transition expansions for the continuous q-Hermite, continuous q-Legendre, Laguerre, and Chebyshev polynomials of the first kind. Using a generalized expansion of the Rogers generating function in terms of the Askey-Wilson polynomials by Ismail & Simeonov whose coefficient is a 8 ϕ 7, we derive corresponding generalized expansions for the Wilson, continuous q-Jacobi, and Jacobi polynomials. By comparing the coefficients of the Askey-Wilson expansion to our continuous q-ultraspherical/Rogers expansion, we derive a new quadratic transformation for basic hypergeometric functions which relates an 8 ϕ 7 to a 2 ϕ 1. We also obtain several definite integral representations which correspond to the above mentioned expansions through the use of orthogonality.

Keywords: Basic hypergeometric orthogonal; Basic hypergeometric series; Connection coefficients; Definite integrals; Eigenfunction expansions; Generating functions; polynomials.