New forms of the Cauchy operator and some of their applications

Abstract

In this paper, we first construct the Cauchy q-shift operator T(a, b;Dxy) and the Cauchy q-difference operator L(a, b; θxy). We then apply these operators in order to represent and investigate some new families of q-polynomials which are defined in this paper. We derive some q-identities such as generating functions, symmetry properties and Rogers-type formulas for these q-polynomials. We also give an application for the q-exponential operator R(bDq).