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.

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.

References

[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.