  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
${}_2\phi_1$ and ${}_8\phi_7$.
