A Note on Generalized $q$-Difference Equations and Their Applications Involving $q$-Hypergeometric Functions

TL;DR

This paper introduces new generalized $q$-difference equations and operators, extending classical $q$-series identities and integrals, with potential applications in special functions and combinatorics.

Contribution

It develops novel $q$-operator-based generalizations of key $q$-series identities and integrals, expanding the theoretical framework of $q$-difference equations.

Findings

New $q$-operator-based generalizations of the $q$-binomial theorem

Extensions of the $q$-Chu-Vandermonde summation formula

Generalizations of the Andrews-Askey integral

Abstract

In this paper, we use two $q$-operators $\mathbb{T}(a,b,c,d,e,yD_x)$ and $\mathbb{E}(a,b,c,d,e,y\theta_x)$ to derive two potentially useful generalizations of the $q$-binomial theorem, a set of two extensions of the $q$-Chu-Vandermonde summation formula and two new generalizations of the Andrews-Askey integral by means of the $q$-difference equations. We also briefly describe relevant connections of various special cases and consequences of our main results with a number of known results.