A Note on Generalized <i>q</i>-Difference Equations and Their Applications Involving <i>q</i>-Hypergeometric Functions

Abstract

Our investigation is motivated essentially by the demonstrated applications of the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, in many diverse areas. Here, in this paper, we use two q-operators T(a,b,c,d,e,yDx) and E(a,b,c,d,e,yθx) 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.

Keywords

• <i>q</i>-difference operator
• <i>q</i>-binomial theorem
• <i>q</i>-hypergeometric functions
• <i>q</i>-Chu-Vandermonde summation formula
• Andrews-Askey integral
• <i>q</i>-series and <i>q</i>-integral identities