Hermite Polynomials and Gaussian distribution Links

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Construct Hermite Polynomials from Gaussian distribution functions (see copy below)

Hermite polynomials from Wiki

Addition on Hermite polynomials


Construct Hermite Polynomials from Gaussian distribution functions, copied from link above:

Let us look at the Hermite polynomials in somewhat more detail. Consider a Gaussian function, .

.

We write

.

This defines the Hermite polynomials.

.

The parity of the Hermite polynomials is (-1)ⁿ.

H_n(z) is a nth degree polynomial in z.

Proof:

This statement holds for n = 1 and n = 2.

Let . Then

.

If H_n-1(z) is a polynomial of degree n-1, the H_n(z) is a polynomial of degree n. The statement therefore holds for all n.

The generating function

Consider F(z+l) = exp(-(z+l)²).

A Taylor series expansion, , yields

.

Let l’ = -l. Then

.

. (Taylor series expansion)

is called the generating function for the Hermite polynomials H_n(z).

Recurrence relations

We have already found

.

Other recurrence relations exist.

and combining the first two relations

.

[ is obtained by differentiating with respect to z,

, and equating terms of equal power of l.

is obtained by differentiating with respect to l.]

Summary

The Hermite polynomials H_n(z) are defined by

or .

The generating function of the Hermite polynomials is

.

The recurrence relations are

, ,

and

.

The differential equation satisfied by the Hermite polynomials is

.

The normalization and orthogonality condition are