  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.
