Hermite polynomials

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In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, 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.

[edit] Definition

The Hermite polynomials are defined either by

$H_n(x)=(-1)^n e^{x^2/2}\frac{d^n}{dx^n}e^{-x^2/2}\,\!$

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

$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}\,\!$

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

$H_n^\mathrm{phys}(x) = 2^{n/2}H_n^\mathrm{prob}(\sqrt{2}\,x)\,\!$ .

These are Hermite polynomial sequences of different variances; see the material on variances below.

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

$\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$

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

The first five (probabilists') Hermite polynomials.

The first ten probabilists' Hermite polynomials are:

$H_0(x)=1\,$

$H_1(x)=x\,$

$H_2(x)=x^2-1\,$

$H_3(x)=x^3-3x\,$

$H_4(x)=x^4-6x^2+3\,$

$H_5(x)=x^5-10x^3+15x\,$

$H_6(x)=x^6-15x^4+45x^2-15\,$

$H_7(x)=x^7-21x^5+105x^3-105x\,$

$H_8(x)=x^8-28x^6+210x^4-420x^2+105\,$

$H_9(x)=x^9-36x^7+378x^5-1260x^3+945x\,$

$H_{10}(x)=x^{10}-45x^8+630x^6-3150x^4+4725x^2-945\,$

and the first ten physicists' Hermite polynomials are:

$H_0(x)=1\,$

$H_1(x)=2x\,$

$H_2(x)=4x^2-2\,$

$H_3(x)=8x^3-12x\,$

$H_4(x)=16x^4-48x^2+12\,$

$H_5(x)=32x^5-160x^3+120x\,$

$H_6(x)=64x^6-480x^4+720x^2-120\,$

$H_7(x)=128x^7-1344x^5+3360x^3-1680x\,$

$H_8(x)=256x^8-3584x^6+13440x^4-13440x^2+1680\,$

$H_9(x)=512x^9-9216x^7+48384x^5-80640x^3+30240x\,$

$H_{10}(x)=1024x^{10}-23040x^8+161280x^6-403200x^4+302400x^2-30240\,$

[edit] Properties

H_n is a polynomial of degree n. The probabilists' version has leading coefficient 1, while the physicists' version has leading coefficient 2ⁿ.

[edit] Orthogonality

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

$e^{-x^2/2}\,\!$ (probabilist)

$e^{-x^2}\,\!$ (physicist)

i.e., we have

$\int_{-\infty}^\infty H_n(x)H_m(x)\,e^{-x^2/2}\,dx=n!\sqrt{2\pi}~\delta_{\mathit{nm}}$ (probabilist)

$\int_{-\infty}^\infty H_n(x)H_m(x)\,e^{-x^2}\,dx={n!2^n}{\sqrt{\pi}}~\delta_{\mathit{nm}}$ (physicist)

where δ_ij is the Kronecker delta, which equals unity when n = m and zero otherwise. The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function.

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

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

in which the inner product is given by the integral including a gaussian function

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

[edit] Hermite's differential equation

The nth Hermite polynomial satisfies Hermite's differential equation:

$H_n''(x)-xH_n'(x)+nH_n(x)=0.\,\!$ (probabilist)

$H_n''(x)-2xH_n'(x)+2nH_n(x)=0.\,\!$ (physicist)

[edit] Recursion relation

The sequence of Hermite polynomials also satisfies the recursion

$H_{n+1}(x)=xH_n(x)-H_n'(x).\,\!$ (probabilist)

$H_{n+1}(x)=2 xH_n(x)-H_n'(x).\,\!$ (physicist)

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

$H_n'(x)=nH_{n-1}(x),\,\!$ (probabilist)

$H_n'(x)=2nH_{n-1}(x),\,\!$ (physicist)

or equivalently,

$H_n(x+y)=\sum_{k=0}^n{n \choose k}x^k H_{n-k}(y)$ (probabilist)

$H_n(x+y)=\sum_{k=0}^n{n \choose k}H_{k}(x) (2y)^{(n-k)}$ (physicist)

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

It follows that the Hermite polynomials also satisfy the recurrence relation

$H_{n+1}(x)=xH_n(x)-nH_{n-1}(x),\,\!$ (probabilist)

$H_{n+1}(x)=2xH_n(x)-2nH_{n-1}(x).\,\!$ (physicist)

These last relations, together with the initial polynomials H₀(x) and H₁(x), can be used in practice to compute the polynomials quickly.

[edit] Generating function

The Hermite polynomials are given by the exponential generating function

$\exp (xt-t^2/2) = \sum_{n=0}^\infty H_n(x) \frac {t^n}{n!}\,\!$ (probabilist)

$\exp (2xt-t^2) = \sum_{n=0}^\infty H_n(x) \frac {t^n}{n!}\,\!$ (physicist)

[edit] Expected value

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

$E(H_n(X))=\mu^n.\,\!$ (probabilist)

[edit] Relations to other functions

[edit] Laguerre polynomials

The Hermite polynomials can be expressed as a special case of the Laguerre polynomials.

$H_{2n}(x) = (-4)^{n}\,n!\,L_{n}^{(-1/2)}(x^2)\,\!$ (physicist)

$H_{2n+1}(x) = 2(-4)^{n}\,n!\,x\,L_{n}^{(1/2)}(x^2)\,\!$ (physicist)

[edit] Relation to confluent hypergeometric functions

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

$H_{n}(x) = 2^{n}\,U\left(\frac{1-n}{2},\frac{3}{2};x^2\right)$ (physicist)

where $U (a, b; z)$ is Whittaker's confluent hypergeometric function. Similarly,

$H_{2n}(x) = (-1)^{n}\,\frac{(2n)!}{n!} \,_1F_1\left(-n,\frac{1}{2};x^2\right)$ (physicist)

$H_{2n+1}(x) = (-1)^{n}\,\frac{(2n+1)!}{n!}\,2x \,_1F_1\left(-n,\frac{3}{2};x^2\right)$ (physicist)

where $\,_1F_1(a,b;z)=M(a,b;z)$ is Kummer's confluent hypergeometric function.

[edit] Differential operator representation

The probabilists' Hermite polynomials satisfy the identity

$H_n(x)=e^{-D^2/2}x^n\,\!$

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.

Since the power series coefficients of the exponential are well known, and higher order derivatives of the monomial xⁿ can be written down explicitly, this differential operator representation gives rise to a concrete formula for the coefficients of H_n that can be used to quickly compute these polynomials.

Since the formal expression for the Weierstrass transform W is e^D², we see that the Weierstrass transform of (√2)ⁿH_n(x/√2) is xⁿ. Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding Maclaurin series.

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.

[edit] Contour integral representation

The Hermite polynomials have a representation in terms of a contour integral, as

$H_n(x)=\frac{n!}{2\pi i}\oint\frac{e^{tx-t^2/2}}{t^{n+1}}\,dt$ (probabilist)

$H_n(x)=\frac{n!}{2\pi i}\oint\frac{e^{2tx-t^2}}{t^{n+1}}\,dt$ (physicist)

with the contour encircling the origin.

[edit] Generalization

The (probabilists') Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is

$\frac{1}{\sqrt{2\pi}}e^{-x^2/2}\,\!$

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

$H_n^{[\alpha]}(x)\,\!$

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

$(2\pi\alpha)^{-1/2}e^{-x^2/(2\alpha)}.\,\!$

They are given by

$H_n^{[\alpha]}(x) = \alpha^{-n/2}H_n^{[1]}\left(\frac{x}{\sqrt{\alpha}}\right)=e^{-\alpha D^2/2}x^n.\,\!$

In particular, the physicists' Hermite polynomials are

$H_n^{[1/2]}(x).\,\!$

$H_n^{[\alpha]}(x)=\sum_{k=0}^n h^{[\alpha]}_{n,k}x^k\,\!$

then the polynomial sequence whose nth term is

$\left(H_n^{[\alpha]}\circ H^{[\beta]}\right)(x)=\sum_{k=0}^n h^{[\alpha]}_{n,k}\,H_k^{[\beta]}(x)\,\!$

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

$\left(H_n^{[\alpha]}\circ H^{[\beta]}\right)(x)=H_n^{[\alpha+\beta]}(x)\,\!$

and

$H_n^{[\alpha+\beta]}(x+y)=\sum_{k=0}^n{n\choose k}H_k^{[\alpha]}(x) H_{n-k}^{[\beta]}(y).\,\!$

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

[edit] "Negative variance"

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

$H_n^{[-\alpha]}(x)\,\!$

the sequence that is inverse to the one similarly denoted but without the minus sign, and thus speak of Hermite polynomials of negative variance. For α > 0, the coefficients of H_n^[−α](x) are just the absolute values of the corresponding coefficients of H_n^[α](x).

These arise as moments of normal probability distributions: The nth moment of the normal distribution with expected value μ and variance σ² is

$E(X^n)=H_n^{[-\sigma^2]}(\mu)\,\!$

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

$\sum_{k=0}^n {n\choose k}H_k^{[\alpha]}(x) H_{n-k}^{[-\alpha]}(y)=H_n^{[0]}(x+y)=(x+y)^n.\,\!$

[edit] Applications

[edit] Hermite functions

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

${\psi}_n(x) = \frac{1}{\sqrt{n!2^n\sqrt{\pi}}}\,e^{-x^2/2}H_n(x).\,\!$

Since these functions contain the square root of the weight function, and have been scaled appropriately, they are orthonormal:

$\int_{-\infty}^\infty \psi_n(x)\psi_m(x)\,dx= \delta_{\mathit{nm}}\,\!$ (physicist)

The Hermite functions are closely related to the Whittaker function (Whittaker and Watson, 1962) $D n (z)$ :

$D_n(z)=(n!\sqrt{\pi})^{1/2}\psi_n(z/\sqrt{2})$

and thereby to other parabolic cylinder functions. The Hermite functions satisfy the differential equation:

$\psi_n''(x)+(2n+1-x^2)\psi_n(x)=0.\,\!$

This equation is equivalent to the Schrödinger equation for a harmonic oscillator in quantum mechanics, so these functions are the eigenfunctions.

Hermite functions 0 (black), 1 (red), 2 (blue), 3 (yellow), 4 (green), and 5 (magenta).

Hermite functions 0 (black), 2 (blue), 4 (green), and 50 (magenta).

[edit] Hermite Functions as Eigenfunctions of the Fourier Transform

The Hermite functions $ψ n (x)$ are the eigenfunctions of the continuous Fourier transform. To see this, take the physicist's version of the generating function and multiply by $exp( - x 2 / 2)$ . This gives

$\exp (-x^2/2 + 2xt-t^2) = \sum_{n=0}^\infty \exp (-x^2/2) H_n(x) \frac {t^n}{n!}\,\!$

Choosing the unitary representation of the Fourier Transform, the Fourier Transform of the left hand side is given by

$\mathcal{F} \{ \exp (-x^2/2 + 2xt-t^2)\} \,$	$= \quad \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^\infty \exp (-ixk)\exp (-x^2/2 + 2xt-t^2)dx \,$
	$= \quad \exp (-k^2/2 - 2kit+t^2) \,$
	$= \quad \sum_{n=0}^\infty \exp (-k^2/2) H_n(k) \frac {(-it)^n}{n!}\,$

The Fourier Transform of the right hand side is given by

$\mathcal{F} \left\{ \sum_{n=0}^\infty \exp (-x^2/2) H_n(x) \frac {t^n}{n!} \right\}\,$

$= \quad \sum_{n=0}^\infty\mathcal{F}\left\{\exp(-x^2/2)H_n(x)\right\}\frac{t^n}{n!} \,$

Equating like powers of $t$ in the transformed versions of the Left and Right Hand Sides gives

$\mathcal{F} \left\{ \exp (-x^2/2) H_n(x) \right\} = (-i)^n \exp (-k^2/2) H_n(k) \,\!$

The Hermite functions $ψ n (x)$ are therefore the orthonormal basis of $\mathcal{L}^2$ which diagonalizes the Fourier Transform operator. In this case, we chose the unitary version of the Fourier Transform, so the eigenvalues are $( - i) n$ .

[edit] Combinatorial coefficients

In the Hermite polynomial H_n(x) of variance 1, the absolute value of the coefficient of x^k is the number of (unordered) partitions of an n-member set into k singletons and (n − k)/2 (unordered) pairs.

[edit] References

Abramowitz, Milton & Stegun, Irene A., eds. (1965), “Chapter 22”, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 .
Wiener, Norbert (1958). The Fourier Integral and Certain of its Applications. New York: Dover Publications. ISBN 0-486-60272-9.
Whittaker, E. T.; Watson, G. N. (1962). A Course of Modern Analysis. London: Cambridge University Press.

[edit] External links