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,H-n(x, y) and their adjoint G(m,n)(x, y). Many of the results presented in the first two sections are well known in literature, but the scheme used is functional to the presentation of the bi-orthogonal Hermite functions discussed in section III. The exposition of the properties satised by the functions (H) over bar (m,n)(x, y) and (G) over bar (m,n)(x, y) is arranged in a non-ordinary way: in fact, we deduce many relations by using the structure of the two-index, two-variable Hermite polynomials, comparing them with the known properties of the ordinary Hermite. We also discuss a differential representation of the operators acting on the above bi-orthogonal Hermite functions and we derive some operational identities to better clarify the role of these Hermite functions.