Hermite polynomials

The polynomialPlanetmathPlanetmath solutions of the Hermite differential equationMathworldPlanetmath, with n a non-negative integer, are usually normed so that the highest degree (http://planetmath.org/PolynomialRing) is (2z)n and called the Hermite polynomialsDlmfDlmfDlmfMathworldPlanetmath Hn(z).  The Hermite polynomials may be defined explicitly by

Hn(z):=(-1)nez2dndzne-z2, (1)

since this is a polynomial having the highest (2z)n and satisfying the Hermite equation.  The equation (1) is the Rodrigues’s formula for Hermite polynomials.  Using the Faà di Bruno’s formula, one gets from (1) also


The first six Hermite polynomials are

H0(z) 1,
H1(z) 2z,
H2(z) 4z2-2,
H3(z) 8z3-12z,
H4(z) 16z4-48z2+12,
H5(z) 32z5-160z3+120z,

and the general is


Differentiating this termwise gives

Hn(z)= 2n[(2z)n-1-(n-1)(n-2)1!(2z)n-3+(n-1)(n-2)(n-3)(n-4)2!(2z)n-5-+],


Hn(z)= 2nHn-1(z). (2)

The Hermite polynomials are sometimes scaled to such ones Hen which obey the differentiation rule

Hen(z)=nHen-1(z). (3)

Such Hermite polynomials form an Appell sequence.

We shall now show that the Hermite polynomials form an orthogonal set (http://planetmath.org/OrthogonalPolynomials) on the interval(-,)  with the weight factor (http://planetmath.org/OrthogonalPolynomials) e-x2.  Let  m<n;  using (1) and integrating by parts (http://planetmath.org/IntegrationByParts) we get

(-1)n-Hm(x)Hn(x)e-x2𝑑x =-Hm(x)dne-x2dxn𝑑x

The substitution portion here equals to zero because e-x2 and its derivatives vanish at ±.  Using then (2) we obtain

-Hm(x)Hn(x)e-x2𝑑x= 2(-1)1+nm-Hm-1(x)dn-1e-x2dxn-1𝑑x.

Repeating the integration by parts gives the result

-Hm(x)Hn(x)e-x2𝑑x = 2m(-1)m+nm!-H0(x)dn-me-x2dxn-m𝑑x
= 2m(-1)m+nm!/-dn-m-1e-x2dxn-m-1= 0,

whereas in the case  m=n  the result

-(Hn(x))2e-x2𝑑x= 2n(-1)2nn!-e-x2𝑑x= 2nn!π

(see area under Gaussian curve). The results that the functionsMathworldPlanetmathxHn(x)2nn!πe-x22  form an orthonormal set on  (-,).

The Hermite polynomials are used in the quantum mechanical treatment of a harmonic oscillator, the wave functions of which have the form

Title Hermite polynomials
Canonical name HermitePolynomials
Date of creation 2013-03-22 15:16:25
Last modified on 2013-03-22 15:16:25
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 28
Author pahio (2872)
Entry type Definition
Classification msc 33E30
Classification msc 33B99
Classification msc 26C05
Classification msc 26A09
Classification msc 12D99
Related topic SubstitutionNotation
Related topic AppellSequence
Related topic LaguerrePolynomial