Hermite polynomials

Definition

In mathematics, the Hermite polynomials, named in honor of Charles Hermite[?] (pronounced "air MEET"), compose a polynomial sequence defined either by

<math>H_n(x)=(-1)^n e^{x^2/2}\frac{d^n}{dx^n}e^{-x^2/2}</math>

or sometimes by

<math>H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}.</math>

Below, we follow the first convention. That convention is sometimes preferred by probabilists because

<math>\frac{1}{\sqrt{2\pi}}e^{-x^2/2}</math>

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

The first several Hermite polynomials are:

<math>H_0(x)=1</math>

<math>H_1(x)=x</math>

<math>H_2(x)=x^2-1</math>

<math>H_3(x)=x^3-3x</math>

<math>H_4(x)=x^4-6x^2+3</math>

Orthogonality

The nth function in this list is an nth-degree polynomial for n = 0, 1, 2, 3, .... These polynomials are orthogonal with respect to the weight

<math>e^{-x^2/2}\,dx,</math>

i.e., we have

<math>\int_{-\infty}^\infty H_n(x)H_m(x)\,e^{-x^2/2}\,dx=0{\rm\ whenever\ }n\neq m.</math>

They form an orthogonal basis of the Hilbert space of functions satisfying

<math>\int_{-\infty}^\infty\left|f(x)\right|^2\,e^{-x^2/2}\,dx<\infty,</math>

in which the inner product is given by

<math>\langle f,g\rangle=\int_{-\infty}^\infty f(x)\overline{g(x)}\,e^{-x^2/2}\,dx.</math>

Various properties

The nth Hermite polynomial satisfies Hermite's differential equation:

<math>H_n(x)-xH_n'(x)+nH_n(x)=0.</math>

The sequence of Hermite polynomials also satisfies the recursion

<math>H_{n+1}(x)=xH_n(x)-H_n'(x).</math>

The Hermite polynomials constitute an Appell sequence, i.e., they are a polynomial sequence satisfying the identity

<math>H_n'(x)=nH_{n-1}(x),</math>

or equivalently,

<math>H_n(x+y)=\sum_{k=0}^n{n \choose k}x^k H_{n-k}(y)</math>

(the equivalence of these last two identities may not be obvious, but its proof is a routine exercise). The Hermite polynomials satisfy the identity

<math>H_n(x)=e^{-D^2/2}x^n</math>

where D represents differentiation with respect to x, and the exponential is interpreted by expanding it as a power series. There are no delicate questions of convergence of this series when it operates on polynomials, since all but finitely many terms vanish. Indeed, the existence of some formal power series g(D), with nonzero constant coefficient, such that H_n(x) = g(D)xⁿ is another equivalent to the statement that these polynomials form an Appell sequence. Since they are an Appell sequence they are a fortiori a Sheffer sequence.

If X is a random variable with a normal distribution with standard deviation 1 and expected value μ then

<math>E(H_n(X))=\mu^n.</math>

Generalization

The Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution

<math>(2\pi)^{-1/2}e^{-x^2/2}\,dx</math>

which has expected value 0 and variance 1. One may speak of Hermite polynomials

<math>H_n^{[\alpha]}(x)</math>

of variance α, where α is any positive number. These are orthogonal with respect to the normal probability distribution

<math>(2\pi\alpha)^{-1/2}e^{-x^2/(2\alpha)}\,dx.</math>

They are given by

<math>H_n^{[\alpha]}(x)=e^{-\alpha D^2/2}x^n.</math>

If

<math>H_n^{[\alpha]}(x)=\sum_{k=0}^n c_{n,k}x^k</math>

then the polynomial sequence whose nth term is

<math>\left(H_n^{[\alpha]}\circ H^{[\beta]}\right)(x)=\sum_{k=0}^n c_{n,k}H_k^{[\beta]}(x)</math>

is the umbral composition of the two polynomial sequences, and it can be shown to satisfy the identities

<math>\left(H_n^{[\alpha]}\circ H^{[\beta]}\right)(x)=H_n^{[\alpha+\beta]}(x)</math>

and

<math>H_n^{[\alpha+\beta]}(x+y)=\sum_{k=0}^n{n\choose k}H_k^{[\alpha]}(x) H_{n-k}^{[\beta]}(y).</math>

The last identity is expressed by saying that this parametrized family of polynomial sequences is a cross-sequence.

"Negative variance"

Since polynomial sequences form a group under the operation of umbral composition, one may denote by

<math>H_n^{[-\alpha]}(x)</math>

the sequence that is inverse to the one similarly denoted but without the minus sign, and thus speak of Hermite polynomials of negative variance. These arise as moments of normal probability distributions: The nth moment of the normal distribution with expected value μ and variance σ² is

<math>E(X^n)=H_n^{[-\sigma^2]}(\mu)</math>

where X is a random variable with the specified normal distribution. A special case of the cross-sequence identity then says that

<math>\sum_{k=0}^n {n\choose k}H_k^{[\alpha]}(x) H_{n-k}^{[-\alpha]}(y)=H_n^{[0]}(x+y)=(x+y)^n.</math>