The Generalized Hermite Polynomials: Application to the Heat Equation

Abstract

We consider the family of Hermite Polynomials (HP) and we introduce the γ-order Generalized Hermite Polynomials (GHP) H_r,γ(w), r ∈ N, γ = γ(m) = m/(m−1) ,m > 1, m ∈ N. Briefly, we prove the following points:

(i) The Differential of the γ-order Generalized Normal distribution is related to Hr,γ(w)

(ii) The Generalized form of the Heat Equation (GHE) is an application of the new extension H_r,γ(w)

(iii) Due to our definitions with γ = 2 we are reduced to the classical case

(iv) the introduced GHE covers an existing extension. The provided examples and figures clarify the introduced extensions.