Geometric Properties of Generalized Hermite Polynomials

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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.

Contributed by: Marcello Artioli and Giuseppe Dattoli (April 2015)
Open content licensed under CC BY-NC-SA


Snapshots


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.

References

[1] P. Appell and Kampé de Fériet, Fonctions hypergéométriques et hypersphériques polynômes d'Hermite, Paris: Gautier-Villars, 1926.

[2] 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.



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