On the $q$-partial differential equations and $q$-series

TL;DR

This paper develops a method using $q$-partial differential equations and complex analysis to expand functions in terms of Rogers-Szeg ext{"o} polynomials, leading to new $q$-identities and transformations.

Contribution

It introduces a novel expansion theorem for functions satisfying $q$-partial differential equations, enabling the derivation of new $q$-transformation formulas and identities.

Findings

Derived a general $q$-transformation formula.

Proved an expansion theorem for analytic functions with $q$-partial differential equations.

Presented a multilinear generating function for Rogers-Szeg ext{"o} polynomials.

Abstract

Using the theory of functions of several complex variables, we prove that if an analytic function in several variables satisfies a system of $q$-partial differential equations, then, it can be expanded in terms of the product of the Rogers-Szeg\H{o} polynomials. This expansion theorem allows us to develop a general method for proving $q$-identities. A general $q$-transformation formula is derived, which implies Watson's $q$-analog of Whipple's theorem as a special case. A multilinear generating function for the Rogers-Szeg\H{o} polynomials is given. The theory of $q$-exponential operator is revisited.