On a system of partial differential equations and the bivariate Hermite polynomials

Using the theory of analytic functions of several complex variables, we prove that if an analytic function in several variables satisfies a system of partial differential equations, then, it can be expanded in terms of the product of the bivariate Hermite polynomials. This expansion theorem allows us to develop a systematic method to prove the identities involving the bivariate Hermite polynomials. With this expansion theorem, we can easily derive, among others, the Mehler formula, Nielsen's formulas, Doetsch's formula, the addition formula, Weisner's formulas, Carlitz's formulas for the bivariate Hermite polynomials.