M1: Hermite Polynomials

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Nov 3, 2020
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- Mathematical Functions
- M2: Legendre Polynomials

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The "physicists' Hermite polynomials" are given by

$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}=\left (2x-\frac{d}{dx} \right )^n \cdot 1 .$

These are Hermite polynomial sequences of different variances; see the material on variances below. the first eleven physicists' Hermite polynomials are:

$H_0(x)=1\,$

$H_1(x)=2x\,$

$H_2(x)=4x^2-2\,$

$H_3(x)=8x^3-12x\,$

$H_4(x)=16x^4-48x^2+12\,$

$H_5(x)=32x^5-160x^3+120x\,$

$H_6(x)=64x^6-480x^4+720x^2-120\,$

$H_7(x)=128x^7-1344x^5+3360x^3-1680x\,$

$H_8(x)=256x^8-3584x^6+13440x^4-13440x^2+1680\,$

$H_9(x)=512x^9-9216x^7+48384x^5-80640x^3+30240x\,$

$H_{10}(x)=1024x^{10}-23040x^8+161280x^6-403200x^4+302400x^2-30240\,$

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