Hermite Calculus

Abstract

We develop a new method of umbral nature to treat blocks of Hermite and of Hermite like polynomials as independent algebraic quantities. The Calculus we propose allows the formulation of a number of “practical rules” yielding significant simplifications in computational problems involving integrals and partial differential equations as well. The procedure we adopt is particularly useful to enter more deeply in the algebraic structure of Hermite polynomials. It provides indeed a tool allowing a generalization of the recently introduced geometrical point of view to the interplay between ordinary monomials and Hermite polynomials