The general form of a Hermite polynomial is give by the following expression: $H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}$ Hermite polynomials are solutions to the [[Hermite equation]]. # Properties 1. $\int_{-\infty}^{\infty}dx H_m(x)H_n(x) \sqrt{\pi}2^n n!\delta_{nm}\;\;\;$ (Orthonormality) 2. $H_{n+1}(x)=2xH_n(x)-2nH_{n-1}(x)\;\;\;$ (recurrence relation) # List of Hermite Polynomials The first 6 Hermite polynomials are listed below $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$ #MathematicalFoundations/Analysis/Functions