# M1: Hermite Polynomials

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:

$$H_0(x)=1\,$$

$$H_1(x)=2x\,$$

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

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

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

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

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

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

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

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

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