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

## TL;DR

This paper introduces new homogeneous $q$-difference operators and uses them to analyze generalized Hahn polynomials, deriving various $q$-identities and formulas.

## Contribution

It constructs specific homogeneous $q$-shift and $q$-difference operators and applies them to represent and study generalized Hahn polynomials.

## Key findings

- Derived generating functions for generalized Hahn polynomials
- Established Mehler's and Roger's formulas in the $q$-context
- Presented new $q$-identities and extended generating functions

## Abstract

In this paper, we first construct the homogeneous $q$-shift operator $\widetilde{E}(a,b;D_{q})$ and the homogeneous $q$-difference operator $\widetilde{L}(a,b; \theta_{xy})$. 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.