Hermite Polynomials

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Hermite

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Hermite Polynomials

DEFINITION

The Hermite polynomials are defined either by

:

H_n(x)=(-1)^n e^{x^2/2}rac{d^n}{dx^n}e^{-x^2/2}

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

:

H_n(x)=(-1)^n e^{x^2}rac{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. 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

:

rac{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 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 ''n''th function in this list is an ''n''th-degree polynomial for ''n'' = 0, 1, 2, 3, .... These Polynomials Are Orthogonal with respect to the ''weight function'' ( Measure )

:

e^{-x^2/2}\,

(probabilist)

or

:

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_{nm}

(probabilist)

or

:

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

(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

with the contour encircling the origin.

OPERATOR IDENTITY

The Hermite polynomials satisfy the identity

:

H_n(x)=e^{-D^2/2}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)^{-1/2}e^{-x^2/2}\,dx

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

:

H_n^{

{Link without Title} }(x)

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

:

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

They are given by

:

H_n^{

{Link without Title} }(x)=e^{-\alpha D^2/2}x^n.

If

:

}_{n,k}x^k

then the polynomial sequence whose ''n''th term is

:

} ight)(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

:

} ight)(x)=H_n^{[\alpha+\beta]}(x)

and

:

H_n^{k}H_k^{[\alpha}(x) 

H_{n-k}^{

{Link without Title} }(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^{

{Link without Title} }(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''^{are just the absolute values of the corresponding coefficients of ''H''_''n''^[α}(''x'').

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

:

E(X^n)=H_n^{

{Link without Title} }(\mu)

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

:

}(y)=H_n^{[0]}(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_{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)

THE HERMITE FUNCTIONS

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

:

{\psi}_n(x) = rac{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 othonormal:

:

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

(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

{\psi}_n(x)

are eigenfunctions of the Fourier Transform , with Eigenvalue s

-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 .