The operator G(a; b;Dq) for the polynomials Wn(x; y; a; b; q)

Abstract

We give an identity which can be regarded as a basic result for this paper.  We give special  value to the parameters in this identity to get some welknown identities such as Euler identity and Cauchy identity.  Inspired by this indentity  we introduce an operator G(a, b;D_q). The exponential operator R(bD_q) defined by Saad and Sukhi [11] can be considered as a special case of the operator G(a, b;Dq) for a = 0.  Also  we  introduce a polynomials W_n(x, y, a, b; q). Al-Salam-Carlitz polynomials U_n(x, y, b; q) [4] is a special case of Wn(x, y, a, b; q) for a = 0. So all the identities for the polynomials W_n(x, y, a, b; q) are extensions of formulas for the Al-Salam-Carlitz polynomials  U_n(x, y, a; q). We give an operator proof for the generating function, the Rogers formula and the Mehlers formula for W_n(x, y, a, b; q). Rogers formula leads to the inverse linearization formula. We give another Rogers-type formula for the polynomials W_n(x, y, a, b; q).