Monomiality and a New Family of Hermite Polynomials

Abstract

The monomiality principle is based on an abstract definition of the concept of derivative and multiplicative operators. This allows to treat different families of special polynomials as ordinary monomials. The procedure underlines a generalization of the Heisenberg–Weyl group, along with the relevant technicalities and symmetry properties. In this article, we go deeply into the formulation and meaning of the monomiality principle and employ it to study the properties of a set of polynomials, which, asymptotically, reduce to the ordinary two-variable Kampè dè Fèrièt family. We derive the relevant differential equations and discuss the associated orthogonality properties, along with the relevant generalized forms.

Keywords

• special functions 33C52, 33C65, 33C99, 33B10, 33B15
• Hermite polynomials 33C45
• operators theory 44A99, 47B99, 47A62