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

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.