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

Abstract

In this paper, we first construct the homogeneous $q$-shift operator $\tilde{E}(a,b;D_q)$ and the homogeneous $q$-difference operator $\tilde{L}(a,b;{\theta }_{xy})$. We then apply these operators in order to represent and investigate generalized Cauchy and a general form of Hahn polynomials. We derive some $q$-identities such as: generating functions, extended generating functions, Mehler's formula and Roger's formula for these $q$-polynomials.

15 pages