Some homogeneous q-difference operators and the associated generalized Hahn polynomials

Abstract

In this paper, we first construct the homogeneous q-shift operator E(a, b; D q ) and the homogeneous q-difference operator L(a, b; x,y ). We then apply these operators in order to represent and investigate a family of generalized Cauchy polynomials and a general form of the q-Hahn polynomials. We derive some q-identities such as generating functions, extended generating functions, Mehler's formula and Rogers' formula for these q-polynomials. Relevant connections of the q-identities presented here with a number of known or new results associated with various specialized families of q-polynomials are also considered.