Geometric Properties of Generalized Hermite Polynomials
On the left is a three-dimensional plot of a Hermite-Appell–Kampé de Fériet polynomial in two variables and , and on the right is a 2D plot of the surface cut by a plane perpendicular to the axis. These generalizations of Hermite polynomials are defined in the Details.
The two-variable Hermite–Appell–Kampé de Fériet polynomials are a generalization of the well-known Hermite polynomials. They can be defined by the expansion
The integer specifies the order, while is the degree. The ordinary Hermite polynomials belong to the order (snapshots 1 and 2).
The generalized Hermite polynomials have been shown to be the solution of the generalized heat equation
with boundary condition
The solution, written in operational form, is
The exponential operator provides the transformation of the ordinary monomial , on which it acts, into these special functions. Such a transition can be understood in geometrical terms, which may allow further insight into the nature of these polynomials.
The geometrical content of this operational identity is shown in -- space. The exponential operator transforms an ordinary monomial into a polynomial of the Hermite type. The monomial-polynomial evolution is shown by moving the cutting plane orthogonal to the axis. For a specific value of the polynomial degree , the polynomials lie on the cutting plane, as shown in the snapshots.
 P. Appell and Kampé de Fériet, Fonctions hypergéométriques et hypersphériques polynômes d'Hermite, Paris: Gautier-Villars, 1926.
 G. Dattoli, "Generalized Polynomials, Operational Identities and Their Applications," Journal of Computational and Applied Mathematics, 118(1–2), 2000 pp. 19–28. doi:10.1016/S0377-0427(00)00283-1.