Connection formulas for Askey--Wilson polynomials and related expansions

Abstract

We derive and study expansions of and over the Askey--Wilson polynomials. We study these expansions and examine some limits to the continuous dual $q$-Hahn, Al-Salam--Chihara, continuous big $q$-Hermite and continuous $q$-Hermite polynomials and their $q^{-1}$-analogues. The Poisson kernel for the infinite discrete orthogonality relation for the $q^{-1}$-Al-Salam--Chihara polynomials is derived which in a special case reduces to the Gupta--Masson biorthogonal rational ${}_4\phi_3$-functions. This Poisson kernel implies new infinite series connection relations for the Askey--Wilson polynomials involving these rational ${}_4\phi_3$-functions. We also consider various interesting limits.