Hermite polynomial

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In mathematics and physics, Hermite polynomials form a well-known class of orthogonal polynomials. In quantum mechanics they appear as eigenfunctions of the harmonic oscillator and in numerical analysis they play a role in Gauss-Hermite quadrature. The functions are named after the French mathematician Charles Hermite (1822–1901).

[edit] Orthonormality

The Hermite polynomials Hn(x) are orthogonal in the sense of the following inner product:

 \left(H_{n'}, H_{n}\right)\equiv \int_{-\infty}^\infty H_{n'}(x)H_n(x)\; e^{-x^2}\, \mathrm{d}x = \delta_{n'n}\, h_n.

That is, the polynomials are defined on the full real axis and have weight w(x) = exp(−x²). Their orthogonality is expressed by the appearance of the Kronecker delta δn'n. The normalization constant is given by

 N_n \equiv \sqrt{\frac{1}{h_n}}  = \left(\frac{1}{\pi}\right)^{1/4}\, \frac{1}{\sqrt{2^n\,n!}}.

Normalization is to unity

 N_n^2\; \int_{-\infty}^\infty H_{n}(x)H_n(x)\; e^{-x^2}\, \mathrm{d}x = 1.

The polynomials NnHn(x) are orthonormal, which means that they are orthogonal and normalized to unity.

[edit] Explicit expression

 H_n(x) = n!\, \sum_{m=0}^{\lfloor N/2\rfloor} \; (-1)^{m} \frac{1}{m!(n-2m)!} \, (2x)^{n-2m}

here \scriptstyle \lfloor N/2\rfloor = N/2 if N even and \scriptstyle \lfloor N/2\rfloor = (N-1)/2 if N odd.

[edit] Recursion relation

Orthogonal polynomials can be constructed recursively by means of a Gram-Schmidt orthogonalization procedure. This procedure yields the following relation

 H_{n+1}(x)=2xH_n(x)-2nH_{n-1}(x) \;\quad\hbox{with}\quad H_0(x) = 1.

The first few follow immediately from this relation,

 H_1 = 2x, \quad H_2 = 4x^2 - 2, \quad H_3 = 8x^3 -12x, \quad \ldots

[edit] Differential equation

The polynomials Hn(x) satisfy the Hermite differential equation

 \frac{d^2 H}{dx^2}-2x\,\frac{dH}{dx}+ 2\alpha H=0,\quad \alpha\in\mathbb{R}.

for the special case \alpha\in\mathbb{Z}^+, i.e., for natural positive α.

[edit] Symmetry

 H_n(-x) = (-1)^n H_n(x)\;,

the functions of even n are symmetric under x → −x and those of odd n are antisymmetric under this substitution.

[edit] Rodrigues' formula

  H_n(x)= (-1)^n\, e^{x^2}\frac{d^n}{dx^n}\, e^{-x^2}.

[edit] Generating function

 e^{2xt}\,e^{-t^2}=\sum_{n=0}^\infty\;H_n(x)\; \frac{t^n}{n!}

First few terms

 \left(1 +2xt +\frac{1}{2} (2x)^2 t^2 + \frac{1}{6} (2x)^3 t^3+\cdots\right)\left(1 -t^2 +\cdots\right)= 1 + 2x\; t + (4x^2 -2)\;\frac{t^2}{2} +  (8 x^3 -12 x)\; \frac{t^3}{6} + \cdots

so that

 H_0 =1, \quad H_1(x) = 2x,\quad H_2(x) = 4x^2-2, \quad H_3(x) = 8x^3-12x.

[edit] Differential relation

 \frac{dH_n(x)}{dx} = 2n H_{n-1}(x).

[edit] Sum formula

 H_n(x+y) = \left(\frac{1}{\sqrt{2}}\right)^n \sum_{k=0}^n \binom{n}{k} \; H_k(\sqrt{2}\,x) \; H_{n-k}(\sqrt{2}\,y),

where \binom{n}{k} is a binomial coefficient.

[edit] References

M. Abramowitz and I.A. Stegun (Eds), Handbook of Mathematical Functions, Dover, New York (1972). Chapter 22

Abramowitz and Stegun online

Eric W. Weisstein, Hermite Polynomial

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