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

Abstract

In this paper, we use two $q$-operators $\mathbb{T}(a,b,c,d,e,y{D}_x)$ 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.

17 pages

References in corpus (3)

• A note on generalized $q$-difference equations for general Al-Salam--Carlitz polynomials
• Generating Functions for Some Families of the Generalized Al-Salam-Carlitz $q$-Polynomials
• $q$-difference equations for homogeneous $q$-difference operators and their applications

Cited by in corpus (2)

• A General Family of $q$-Hypergeometric Polynomials and Associated Generating Functions
• A note on generalized $q$-difference equations for general Al-Salam--Carlitz polynomials