Hermite polynomials

The polynomial solutions of the Hermite differential equation, 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 polynomials $H_n(z)$.  The Hermite polynomials may be defined explicitly by

$H_n(z):={(-1)}^n{e}^{z^2}{\displaystyle \frac{d^n}{d{z}^n}}{e}^{-z^2},$ (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

$$H_n(x)={(-1)}^n\sum _{m_1+2m_2=n}\frac{n!}{m_1!m_2!}{(-1)}^{m_1+m_2}{(2x)}^{m_1}.$$

The first six Hermite polynomials are

$H_0(z)\equiv \mathrm{\hspace{0.33em}1},$
$H_1(z)\equiv \mathrm{\hspace{0.33em}2}z,$
$H_2(z)\equiv \mathrm{\hspace{0.33em}4}{z}^2-2,$
$H_3(z)\equiv \mathrm{\hspace{0.33em}8}{z}^3-12z,$
$H_4(z)\equiv \mathrm{\hspace{0.33em}16}{z}^4-48z^2+12,$
$H_5(z)\equiv \mathrm{\hspace{0.33em}32}{z}^5-160z^3+120z,$

and the general is

$$H_n(z)\equiv {(2z)}^n-\frac{n(n-1)}{1!}{(2z)}^{n-2}+\frac{n(n-1)(n-2)(n-3)}{2!}{(2z)}^{n-4}-+\mathrm{\dots }$$

Differentiating this termwise gives

$$H_n^{\prime }(z)=\mathrm{\hspace{0.33em}2}n[{(2z)}^{n-1}-\frac{(n-1)(n-2)}{1!}{(2z)}^{n-3}+\frac{(n-1)(n-2)(n-3)(n-4)}{2!}{(2z)}^{n-5}-+\mathrm{\dots }],$$

i.e.

$H_n^{\prime }(z)=\mathrm{\hspace{0.33em}2}n{H}_{n-1}(z).$ (2)

The Hermite polynomials are sometimes scaled to such ones ${\mathrm{He}}_{\mathrm{n}}$ which obey the differentiation rule

${\mathrm{He}}_n^{\prime }(z)=n{\mathrm{He}}_{n-1}(z).$ (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  $(-\mathrm{\infty },\mathrm{\infty })$  with the weight factor (http://planetmath.org/OrthogonalPolynomials) $e^{-x^2}$.  Let  ;  using (1) and integrating by parts (http://planetmath.org/IntegrationByParts) we get

${(-1)}^n{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{H}_m(x)H_n(x)e^{-x^2}𝑑x$ $\mathrm{\hspace{0.33em}}={\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{H}_m(x){\displaystyle \frac{d^n{e}^{-x^2}}{d{x}^n}}𝑑x$

$\mathrm{\hspace{0.33em}}=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{/}}{H}_m(x){\displaystyle \frac{d^{n-1}{e}^{-x^2}}{d{x}^{n-1}}}-{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{H}_m^{\prime }(x){\displaystyle \frac{d^{n-1}{e}^{-x^2}}{d{x}^{n-1}}}𝑑x.$

The substitution portion here equals to zero because $e^{-x^2}$ and its derivatives vanish at $\pm \mathrm{\infty }$.  Using then (2) we obtain

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{H}_m(x)H_n(x)e^{-x^2}𝑑x=\mathrm{\hspace{0.33em}2}{(-1)}^{1+n}m{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{H}_{m-1}(x)\frac{d^{n-1}{e}^{-x^2}}{d{x}^{n-1}}𝑑x.$$

Repeating the integration by parts gives the result

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{H}_m(x)H_n(x)e^{-x^2}𝑑x$ $\mathrm{\hspace{0.33em}}={\mathrm{\hspace{0.33em}2}}^m{(-1)}^{m+n}m!{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}{H}_0(x){\displaystyle \frac{d^{n-m}{e}^{-x^2}}{d{x}^{n-m}}}𝑑x$

$\mathrm{\hspace{0.33em}}={\mathrm{\hspace{0.33em}2}}^m{(-1)}^{m+n}m!\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{/}}{\displaystyle \frac{d^{n-m-1}{e}^{-x^2}}{d{x}^{n-m-1}}}=\mathrm{\hspace{0.33em}0},$

whereas in the case  $m=n$  the result

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{(H_n(x))}^2{e}^{-x^2}𝑑x={\mathrm{\hspace{0.33em}2}}^n{(-1)}^{2n}n!{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}𝑑x={\mathrm{\hspace{0.33em}2}}^{n}n!\sqrt{\pi }$$

(see area under Gaussian curve). The results that the functions  $x\mapsto \frac{H_n(x)}{\sqrt{2^{n}n!\sqrt{\pi }}}{e}^{-\frac{x^2}{2}}$  form an orthonormal set on  $(-\mathrm{\infty },\mathrm{\infty })$.

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

$$\xi \mapsto {\mathrm{\Psi }}_n(\xi )=C_n{H}_n(\xi )e^{-\frac{{\xi }^2}{2}}.$$

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