(untitled) (JOURNAL OF ADVANCES IN MATHEMATICS, 2015)
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;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 polynomials Un(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 Wn(x, y, a, b; q) are extensions of formulas for the Al-Salam-Carlitz polynomials Un(x, y, a; q). We give an operator proof for the generating function, the Rogers formula and the Mehlers formula for Wn(x, y, a, b; q). Rogers formula leads to the inverse linearization formula. We give another Rogers-type formula for the polynomials Wn(x, y, a, b; q).