Operational results on bi-orthogonal Hermite functions

By starting from the concept of the orthogonality property related to the ordinary and generalized two-variable Hermite polynomials, we present some interesting results on the class of bi-orthogonal Hermite functions. The structure of these bi-orthogonal functions is based on the family of the two-index, two-variable Hermite polynomials of type H{m,n}(x,y) and their adjoint G{m,n}(x,y). We will also introduce a dierential representation of the operators acting on the above bi-orthogonal Hermite functions and we will derive some operational identities.