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Hermite form : ウィキペディア英語版
Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The polynomials arise in:
* probability, such as the Edgeworth series;
* in combinatorics, as an example of an Appell sequence, obeying the umbral calculus;
* in numerical analysis as Gaussian quadrature;
* in finite element methods as shape functions for beams;
* in physics, where they give rise to the eigenstates of the quantum harmonic oscillator;
* in systems theory in connection with nonlinear operations on Gaussian noise.
Hermite polynomials were defined by 〔P.S. Laplace: ''Théorie analytique des probabilitte és'' 1812 livre 2, 321-323; Oeuvres VII〕 though in scarcely recognizable form, and studied in detail by Chebyshev (1859).〔P.L.Chebyshev: ''Sur le développement des fonctions à une seule variable'' Bull. Acad. Sci. St. Petersb. I 1859 193-200; Oeuvres I 501-508.〕 Chebyshev's work was overlooked and they were named later after Charles Hermite who wrote on the polynomials in 1864 describing them as new.〔C. Hermite: ''Sur un nouveau développement en série de fonctions'' C. R Acad. Sci. Paris 58 1864 93-100; Oeuvres II 293-303〕 They were consequently not new although in later 1865 papers Hermite was the first to define the multidimensional polynomials.
==Definition==
There are two different ways of standardizing the Hermite polynomials:
* The "probabilists' Hermite polynomials" are given by
::\mathit_n(x)=(-1)^n e^}\frace^}=\left (x-\frac \right )^n \cdot 1 ,
* while the "physicists' Hermite polynomials" are given by
::H_n(x)=(-1)^n e^\frace^=\left (2x-\frac \right )^n \cdot 1 .
These two definitions are not exactly identical; each one is a rescaling of the other,
:H_n(x)=2^} \,x), \qquad \mathit_n(x)=2^}H_n\left(\frac x}
is the probability density function for the normal distribution with expected value 0 and standard deviation 1.
The first eleven probabilists' Hermite polynomials are:
:}_1(x)=x\,
:}_3(x)=x^3-3x\,
:}_5(x)=x^5-10x^3+15x\,
:}_7(x)=x^7-21x^5+105x^3-105x\,
:}_9(x)=x^9-36x^7+378x^5-1260x^3+945x\,
:(x)=x^-45x^8+630x^6-3150x^4+4725x^2-945\,
and the first eleven physicists' Hermite polynomials are:
: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\,
:H_7(x)=128x^7-1344x^5+3360x^3-1680x\,
:H_8(x)=256x^8-3584x^6+13440x^4-13440x^2+1680\,
:H_9(x)=512x^9-9216x^7+48384x^5-80640x^3+30240x\,
:H_(x)=1024x^-23040x^8+161280x^6-403200x^4+302400x^2-30240\,


抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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