  We derive generalized generating functions for basic hypergeometric
orthogonal polynomials by applying connection relations with one free parameter
to them. In particular, we generalize generating functions for the
Askey-Wilson, continuous $q$-ultraspherical/Rogers, little $q$-Laguerre/Wall,
and $q$-Laguerre polynomials. Depending on what type of orthogonality these
polynomials satisfy, we derive corresponding definite integrals, infinite
series, bilateral infinite series, and $q$-integrals.

Classical Analysis and ODEs