Hermite Polynomials

The Hermite polynomials which are a classical orthogonal polynomial sequence arises in mathematics in probability, such as the Edgeworth series; in combinatorials, as an example of an Appell sequence, obeying the umbral calculus; and in physics, where they give rise to the eigenstates of the quantum harmonic oscillator. They are named in honor of Charles Hermite.

Definition

Hn(x) = (-1)nex2/2 [`d/dx`(e-x2/2)]n

(the "probabilists' Hermite polynomials"), or sometimes by

Hn(x) =(-1)nex2 [`d/dx`(e-x2)]n

(the "physicists' Hermite polynomials"). The above two definitions are not exactly equivalent; either is a rescaling of the other, to wit

Hnphys(x) = 2n/2Hnprob(`sqrt(2)`x )

Usually follow the first convention is followed. That convention is often preferred by probabilists because

`1/sqrt(2Pi)` e-x2/2

is the probability density function for the normal distribution in which expected value 0 and standard deviation 1.

Properties of Hermite polynomials

Hn is a polynomial of degree n. According to probabilists' definition it has leading coefficient 1, while according to physicists' definition it has leading coefficient 2n.

Orthogonality

Hn(x) is an nth-degree polynomial for n = 0, 1, 2, 3, .... These polynomials show orthogonality with respect to the weight function     (measure)

W(x) = e-x2/2   (probabilist)

or

W(x) = e-x2   (physicist)

Completeness

The Hermite polynomials (probabilist or physicist) form the basis of orthogonality of the Hilbert space of functions satisfying
`int_-oo^oo|f(x)|^2w(x)dx` < `oo`


Hermite's differential equation

The probabilists' version of Hermite polynomials are solutions of the differential equation

(e-x2/2 u')' + `lambda`e-x2/2 u = 0

Recursion relation

Hermite polynomials' sequences also satisfies the recursion

Hn+1(x) = xHn(x) - H'n(x) (probabilist)

Hn+1(x) = 2xHn(x) - H'n(x) (physicist)


Applications of Hermite polynomials

Hermite functions

One can define the Hermite functions from the physicists' version of polynomials:

`psi`n(x)  =   `1/sqrt(n!2^nsqrt(Pi))`  e-x2/2 Hn(x)


Recursion relation

Following recursion relations of Hermite polynomials, the Hermite functions satisfy

`psi`'n(x) = `sqrt(n/2)``psi`n-1(x) - `sqrt((n+1)/2)``psi`n+1(x)

Cramér's inequality

The Hermite functions satisfy the following inequality due to Harald Cramér

|`psi`n(x)|   `<=` K

for x real, where the constant K is less than 1.086435.

Hermite functions act as eigenfunctions of the Fourier transform