On a system of $q$-partial differential equations with applications to $q$-series

TL;DR

This paper develops an expansion theorem for functions satisfying $q$-partial differential equations and applies it to extend transformation formulas for $q$-series, enriching the theory of $q$-calculus.

Contribution

It introduces a new expansion theorem for multivariable functions solving $q$-partial differential equations and applies it to extend known $q$-series transformation formulas.

Findings

Established an expansion theorem for $q$-partial differential equations.

Extended Andrews' transformation formula for the $q$-Lauricella function.

Demonstrated applications to $q$-series and $q$-calculus.

Abstract

Using the theory of functions of several variables and $q$-calculus, we prove an expansion theorem for the analytic function in several variables which satisfies a system of $q$-partial differential equations. Some curious applications of this expansion theorem to $q$-series are discussed. In particular, an extension of Andrews' transformation formula for the $q$-Lauricella function is given.