The Geometry of Hermite Polynomials

​
bound for x and y
5
bound for z
50
cutting plane position y
0
polynomial degree n
3
H
3
(x, y) = 6
3
x
6
+xy
2
x
6
H
3
(x, 0) =
H
3
(x)
​
​
On the left is a three-dimensional plot of a Hermite polynomial in two variables
x
and
y
, and on the right is a 2D plot of the surface cut by a plane perpendicular to the
y
axis.

Details

The two-variable Hermite polynomial
H
n
(x,y)=n!

n
2

∑
r=0
n−2r
x
r
y
(n−2r)!r!
has been shown to be the solution of the heat equation
∂
∂y
H
n
(x,y)=
2
∂
∂
2
x
H
n
(x,y)
with boundary condition
H
n
(x,0)=
n
x
.
The solution written in an operational form reads
H
n
(x,y)=
y
2
∂
x
e
n
x
,
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
x
-
y
-
z
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
y
axis. For a specific value of the polynomial degree
n
, the polynomials lie on the cutting plane, as shown in the snapshots. It is worth stressing that only for negative values of
y
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.

Permanent Citation

Marcello Artioli, Giuseppe Dattoli
​
​"The Geometry of Hermite Polynomials"​
​http://demonstrations.wolfram.com/TheGeometryOfHermitePolynomials/​
​Wolfram Demonstrations Project​
​Published: March 4, 2015