PlanetMath: Hermite polynomials

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Hermite polynomials

(Definition)

The polynomial solutions of the Hermite differential equation, with a non-negative integer, are usually normed so that the highest degree term is and called the Hermite polynomials . The Hermite polynomials may be defined explicitly by

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

(1)

since this is a polynomial having the highest degree term

and satisfying the Hermite equation. The first six Hermite polynomials are

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

and the general polynomial form is

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

Differentiating this termwise gives

$H'_n(z) = 2n\left[(2z)^{n-1}-\frac{(n-1)(n-2)}{1!}(2z)^{n-3}+ \frac{(n-1)(n-2)(n-3)(n-4)}{2!}(2z)^{n-5}-+\cdots\right],$

i.e.

$\displaystyle H'_n(z) = 2nH_{n-1}(z).$

(2)

We shall now show that the Hermite polynomials form an orthogonal set on the interval $(-\infty,\,\infty)$ with the weight factor $e^{-x^2}$ . Let ; using (1) and integrating by parts we get

$\displaystyle (-1)^n\int_{-\infty}^\infty H_m(x)H_n(x)e^{-x^2}\,dx = \int_{-\infty}^\infty H_m(x)\frac{d^ne^{-x^2}}{dx^n}\,dx =$

$\displaystyle = \operatornamewithlimits{\Big/}_{\!\!\!-\infty}^{\,\quad\infty}H... ...{dx^{n-1}} -\int_{-\infty}^\infty H'_m(x)\frac{d^{n-1}e^{-x^2}}{dx^{n-1}}\, dx.$

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

$\displaystyle \int_{-\infty}^\infty H_m(x)H_n(x)e^{-x^2}\,dx = 2(-1)^{1+n}m\int_{-\infty}^\infty H_{m-1}(x)\frac{d^{n-1}e^{-x^2}}{dx^{n-1}}\,dx.$

Repeating the integration by parts gives the result

$\displaystyle \int_{-\infty}^\infty H_m(x)H_n(x)e^{-x^2}\,dx = 2^m(-1)^{m+n}m!\int_{-\infty}^\infty H_0(x)\frac{d^{n-m}e^{-x^2}}{dx^{n-m}}\,dx =$

$\displaystyle = 2^m(-1)^{m+n}m!\operatornamewithlimits{\Big/}_{\!\!\!-\infty}^{\,\quad\infty}\frac{d^{n-m-1}e^{-x^2}}{dx^{n-m-1}} = 0,$

whereas in the case

the result

$\displaystyle \int_{-\infty}^\infty (H_n(x))^2e^{-x^2}\,dx = 2^n(-1)^{2n}n!\int_{-\infty}^\infty e^{-x^2}\,dx = 2^nn!\sqrt{\pi}$

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

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

$\displaystyle \xi \mapsto \Psi_n(\xi) = C_nH_n(\xi)e^{-\frac{\xi^2}{2}}.$

"Hermite polynomials" is owned by pahio.

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See Also: substitution notation

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Cross-references: wave functions, harmonic oscillator, orthonormal set, functions, area under Gaussian curve, integration by parts, vanish, derivatives, interval, integer, Hermite differential equation, solutions, polynomial
There are 6 references to this entry.

This is version 14 of Hermite polynomials, born on 2005-05-17, modified 2006-10-06.
Object id is 7061, canonical name is HermitePolynomials.
Accessed 7321 times total.

Classification:

AMS MSC:	12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
	26A09 (Real functions :: Functions of one variable :: Elementary functions)
	26C05 (Real functions :: Polynomials, rational functions :: Polynomials: analytic properties, etc.)
	33B99 (Special functions :: Elementary classical functions :: Miscellaneous)
	33E30 (Special functions :: Other special functions :: Other functions coming from differential, difference and integral equations)

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