Operational results in bi-orthogonal Hermite functions

Abstract

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.