Applications of Generalized <i>q</i>-Difference Equations for General <i>q</i>-Polynomials

Abstract

Andrews gave a remarkable interpretation of the Rogers–Ramanujan identities with the polynomials ρe(N,y,x,q), and it was noted that ρe(∞,−1,1,q) is the generation of the fifth-order mock theta functions. In the present investigation, several interesting types of generating functions for this q-polynomial using q-difference equations is deduced. Besides that, a generalization of Andrew’s result in form of a multilinear generating function for q-polynomials is also given. Moreover, we build a transformation identity involving the q-polynomials and Bailey transformation. As an application, we give some new Hecke-type identities. We observe that most of the parameters involved in our results are symmetric to each other. Our results are shown to be connected with several earlier works related to the field of our present investigation.

Keywords

• <i>q</i>-difference equations
• <i>q</i>-polynomial
• generating function
• Hecke-type series
• Bailey transformation