Hermite polynomials

They can then be standardized into the Hermite polynomials .

This fact is equivalent to the corresponding statement for Hermite polynomials (see above).

There are two different ways of standardizing the Hermite polynomials:

This implies Hermite polynomials can be expressed in terms of F as well.

The Hermite polynomials can be expressed as a special case of the parabolic cylinder functions.

Typically they are defined for physicists' Hermite polynomials.

For further details, see Hermite polynomials.

In the case of Hermite polynomials, this reduces to the conventional recursion formula for that sequence.

The Hermite polynomials evaluated at zero argument are called Hermite numbers.

Its solutions are Hermite polynomials.

See also generalized Hermite polynomials.

The Hermite polynomials are orthogonal with respect to the Gaussian distribution with zero mean value.

The functions H are the Hermite polynomials:

The first eleven probabilists' Hermite polynomials are:

The generalized Laguerre polynomials are related to the Hermite polynomials:

(Hermite polynomials are another example.)

Many authors, particularly probabilists, use an alternate definition of the Hermite polynomials, with a weight function of instead of .

He also published an article on Hermite polynomials together with Joseph Gillis, giving him an Erdos number of 2.

Here is a tiny sample of them, relating to the Chebyshev, associated Laguerre, and Hermite polynomials:

Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding Maclaurin series.

(the "physicists' Hermite polynomials").

Considering the problem in cylindrical coordinates, one can write higher-order modes using Laguerre polynomials instead of Hermite polynomials.

In the case of the harmonic potential, the wave functions solutions of the one dimensional quantum harmonic oscillator are known as Hermite polynomials.

This example shows how the Hermite polynomials and Laguerre polynomials are interrelated through the Wigner-Weyl transform.

PC was first introduced by Wiener where Hermite polynomials were used to model stochastic processes with Gaussian random variables.