Connection formulas for Askey--Wilson polynomials and related expansions

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.