Connection formulas for Askey–Wilson polynomials and related expansions

Abstract We derive and investigate expansions involving 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 $$q^{-1}$$ q - 1 -analogues. The Poisson kernel for the infinite discrete orthogonality relation for the $$q^{-1}$$ q - 1 -Al-Salam–Chihara polynomials is derived which in a special case reduces to the Gupta–Masson biorthogonal rational $${\phantom {a}}_4\phi _3$$ a 4 ϕ 3 -functions. This Poisson kernel implies new infinite series connection relations for the Askey–Wilson polynomials involving these rational $${\phantom {a}}_4\phi _3$$ a 4 ϕ 3 -functions. We also consider various interesting limits.