On a generalization of the Rogers generating function

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.