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. Inspiredby this identity we introduce an operator G(a, b;Dq). The exponential operator R(bDq)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 Wn(x, y, a, b; q). Al-Salam-Carlitz polynomialsUn(x, y, b; q) [4] is a special case of Wn(x, y, a, b; q) for a = 0. So all the identities for thepolynomials Wn(x, y, a, b; q) are extensions of formulas for the Al-Salam-Carlitz polynomialsUn(x, y, a; q). We give an operator proof for the generating function, the Rogers formula andthe Mehlers formula for Wn(x, y, a, b; q). Rogers formula leads to the inverse linearizationformula. We give another Rogers-type formula for the polynomials Wn(x, y, a, b; q).