HERMITE POLYNOMIALS

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.

Contents
Definition
Properties
Orthogonality
Hermite's differential equation
Recursion relation
Generating function
Expected value
Relations to other functions
Laguerre polynomials
Relation to confluent hypergeometric functions
Differential operator representation
Contour integral representation
Generalization
"Negative variance"
Applications
Hermite functions
Combinatorial coefficients
References
External links

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, 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
: rac{1}{sqrt{2pi}}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 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.

Properties


Orthogonality

''H''''n''(''x'') 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{2pi}~delta_{mathit{nm}} (probabilist)
or
: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}^inftyleft|f(x)
ight|^2,e^{-x^2/2},dx
in which the inner product is given by the integral including a gaussian function
:langle f,g
angle=int_{-infty}^infty f(x)overline{g(x)},e^{-x^2/2},dx.
Hermite's differential equation

The ''n''th 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_{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)
Generating function

The Hermite polynomials are given by the exponential generating function
:exp (xt-t^2/2) = sum_{n=0}^infty H_n(x) rac {t^n}{n!},! (probabilist)
:exp (2xt-t^2) = sum_{n=0}^infty H_n(x) rac {t^n}{n!},! (physicist)
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)

Relations to other functions


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)
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},Uleft( rac{1-n}{2}, rac{3}{2};x^2
ight) (physicist)
where U(a,b;z) is Whittaker's confluent hypergeometric function. Similarly,
:H_{2n}(x) = (-1)^{n}, rac{(2n)!}{n!}
,_1F_1left(-n, rac{1}{2};x^2
ight) (physicist)
:H_{2n+1}(x) = (-1)^{n}, rac{(2n+1)!}{n!},2x
,_1F_1left(-n, rac{3}{2};x^2
ight) (physicist)
where ,_1F_1(a,b;z)=M(a,b;z) is Kummer's confluent hypergeometric function.

Differential operator representation


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 ''Hn''(''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.

Contour integral representation


The Hermite polynomials have a representation in terms of a contour integral, as
:H_n(x)= rac{n!}{2pi i}oint rac{e^{2tx-t^2}}{t^{n+1}},dt (physicist)
with the contour encircling the origin.

Generalization


The (probabilists') Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is
: rac{1}{sqrt{2pi}}e^{-x^2/2},!
which has expected value 0 and variance 1. One may speak of Hermite polynomials
:H_n^{[lpha]}(x),!
of variance α, where α is any positive number. These are orthogonal with respect to the normal probability distribution whose density function is
:(2pilpha)^{-1/2}e^{-x^2/(2lpha)}.,!
They are given by
:H_n^{[lpha]}(x)=e^{-lpha D^2/2}x^n.,!
If
:H_n^{[lpha]}(x)=sum_{k=0}^n h^{[lpha]}_{n,k}x^k,!
then the polynomial sequence whose ''n''th term is
:left(H_n^{[lpha]}circ H^{[eta]}
ight)(x)=sum_{k=0}^n h^{[lpha]}_{n,k},H_k^{[eta]}(x),!
is the 'umbral composition' of the two polynomial sequences, and it can be
shown to satisfy the identities
:left(H_n^{[lpha]}circ H^{[eta]}
ight)(x)=H_n^{[lpha+eta]}(x),!
and
:H_n^{[lpha+eta]}(x+y)=sum_{k=0}^n{nchoose k}H_k^{[lpha]}(x) H_{n-k}^{[eta]}(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^{[-lpha]}(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 ''n''th 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 {nchoose k}H_k^{[lpha]}(x) H_{n-k}^{[-lpha]}(y)=H_n^{[0]}(x+y)=(x+y)^n.,!

Applications


Hermite functions

One can define the 'Hermite functions' from the physicists' polynomials:
:{psi}_n(x) = rac{1}{sqrt{n!2^nsqrt{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)
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.
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).

The Hermite functions {psi}_n(x) are eigenfunctions of the continuous Fourier transform, with eigenvalues
-i^n.
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.

References



★

★ Norbert Wiener, ''The Fourier Integral and Certain of its Applications'', (1958) Dover Publications, New York. ISBN 0-486-60272-9 .

External links



★

★ Module for Hermite Polynomial Interpolation by John H. Mathews

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