The Geometry of Hermite Polynomials

On the left is a three-dimensional plot of a Hermite polynomial in two variables and , and on the right is a 2D plot of the surface cut by a plane perpendicular to the axis.


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The two-variable Hermite polynomial
has been shown to be the solution of the heat equation
with boundary condition
The solution written in an operational form reads
which can be exploited to infer a kind of geometrical understanding of the Hermite polynomials in 3D.
The geometrical content of this operational identity is shown in -- space. The exponential operator transforms an ordinary monomial into a special 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. It is worth stressing that only for negative values of do the polynomials exhibit zeros (snapshots 3 and 4), in accordance with the fact that in this region they realize an orthogonal set.
[1] P. Appell and Kampé de Fériét, 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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