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, as the eigenstates of the quantum harmonic oscillator. They are named in honor of Charles Hermite.

Definition

The Hermite polynomials are defined either by

[H_n(x)=(-1)^n e^\frace^]

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

[H_n(x)=(-1)^n e^\frace^]

(the "physicists' Hermite polynomials"). These two definitions are not exactly equivalent; either is a trivial rescaling of the other. 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}e^]

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

The first several 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\,]

in probabilists' notation, or

[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\,]

in physicists' notation.

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 function (measure)

[e^\,] (probabilist)

or

[e^\,] (physicist)

i.e., we have

[\int_^\infty H_n(x)H_m(x)\,e^\,dx=n!\sqrt~\delta_] (probabilist)

or

[\int_^\infty H_n(x)H_m(x)\,e^\,dx=}~\delta_] (physicist)

where δ_ij is the Kronecker delta, which equals unity when n = m and zero otherwise. This is the same as saying they are orthogonal with respect to the normal probability distribution. They form an orthogonal basis of the Hilbert space of functions satisfying

[\int_^\infty\left|f(x)\right|^2\,e^\,dx<\infty,]

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

[\langle f,g\rangle=\int_^\infty f(x)\overline\,e^\,dx.]

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)

Recursion relation

The sequence of Hermite polynomials also satisfies the recursion

[H_(x)=xH_n(x)-H_n'(x).\,] (probabilist)

[H_(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_(x),\,] (probabilist)

[H_n'(x)=2nH_(x),\,] (physicist)

or equivalently,

[H_n(x+y)=\sum_^nx^k H_(y)] (probabilist)

(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_(x)=xH_n(x)-nH_(x)\,] (probabilist)

[H_(x)=2xH_n(x)-2nH_(x).\,] (physicist)

Generating function

The Hermite polynomials are given by the exponential generating function

[\exp (xt-t^2/2) = \sum_^\infty H_n(x) \frac ] (probabilist)

[\exp (2xt-t^2) = \sum_^\infty H_n(x) \frac ] (physicist)

Contour integral

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

[H_n(x)=\frac\oint\frac}}\,dt] (physicist)

with the contour encircling the origin.

Operator identity

The Hermite polynomials satisfy the identity

[H_n(x)=e^x^n] (probabilist)

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. 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.

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)

Generalization

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

[(2\pi)^e^\,dx]

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

[H_n^(x)]

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

[(2\pi\alpha)^e^\,dx.]

They are given by

[H_n^(x)=e^x^n.]

If

[H_n^(x)=\sum_^n h^_x^k]

then the polynomial sequence whose nth term is

[\left(H_n^\circ H^\right)(x)=\sum_^n h^_\,H_k^(x)]

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

[\left(H_n^\circ H^\right)(x)=H_n^(x)]

and

[H_n^(x+y)=\sum_^nH_k^(x)

H_^(y).]

The last identity is expressed by saying that this parameterized 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

[H_n^(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^(\mu)]

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

[\sum_^n H_k^(x) H_^(y)=H_n^(x+y)=(x+y)^n.]

Relation to the Laguerre polynomials

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

[H_(x) = (-4)^\,n!\,L_^(x^2)] (physicist)

[H_(x) = 2(-4)^\,n!\,x\,L_^(x^2)] (physicist)

The Hermite Functions

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

[_n(x) = \frac}}\,e^H_n(x).\,]

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

[\int_^\infty \psi_n(x)\psi_m(x)\,dx= \delta_] (physicist)

They 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.

Eigenfunctions of the Fourier transform

The Hermite functions [_n(x)] are eigenfunctions of the Fourier transform, with eigenvalues [-i^n].

Combinatorial interpretation of the 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.

References

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . (See chapter 22).

• Norbert Wiener, The Fourier Integral and Certain of its Applications, (1958) Dover Publications, New York. ISBN 0486-60272-9 .

 

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