Applications of Generalized q-Difference Equations for General q-Polynomials

Zeya Jia,Qiuxia Hu,Dawei Niu,Bilal Khan

doi:10.3390/sym13071222

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.

Highlights

Introduction and MotivationAndrews [1,2] established and found a nice relationship of the fifth mock theta function with the q-Jacobi polynomials
If we take different choices for f, we obtain a variety of alternating parity questions connected with classical partition identities of Euler, Rogers, Ramanujan and Gordan
Liu and Zeng [32] further studied the relations between the q-difference equations and Rogers–Szegö polynomials

Summary

Introduction and Motivation

Andrews [1,2] established and found a nice relationship of the fifth mock theta function with the q-Jacobi polynomials. While using the homogeneous q-difference equations Cao [26] gave the generating functions for q-hypergeometric polynomials. Liu [20] obtained several important results on Rogers–Szegö polynomials by q-difference equations with two variables as describe in Proposition 1. Liu and Zeng [32] further studied the relations between the q-difference equations and Rogers–Szegö polynomials. They studied if an analytic function in several variables satisfies a system of q-difference equations, it can be expanded in terms of the product of some polynomials.

Generating Function for the Generalized q-Polynomials

Concluding Remarks and Observations