Abstract. In this paper, we study a family of generalized Weyl algebras {A p (λ, µ, K q [s, t])} and their polynomial extensions. We will show that the algebra) S when none of p and q is a root of unity. As an application, we determine all the height-one prime ideals and the center foris cancellative. Then we will determine the automorphism group and solve the isomorphism problem for the generalized Weyl algebras A p (λ, µ, K q [s, t]) and their polynomial extensions in the case where none of p and q is a root of unity. We will establish a quantum analogue of the Dixmier conjecture and compute the automorphism group for the simple localization (A p (1, 1, K q [s, t])) S . Moreover, we will completely determine the automorphism group for the algebra A p (1, 1, K q [s, t]) and its polynomial extension when p = 1 and q = 1.