Hermite polynomials

Remarks

The first values are $H_2 = 4x^2 - 2$, $H_3 = 8x^3 - 12x$, and $\operatorname{He}_2 = x^2 - 1$, $\operatorname{He}_3 = x^3 - 3x$. Check for $H_2$ with the recursion: $2x \cdot 2x - 2 \cdot 1 \cdot 1 = 4x^2 - 2$. The polynomial $H_n$ has degree $n$ and leading coefficient $2^n$, while $\operatorname{He}_n$ is monic. The two families are related by $H_n(x) = 2^{n/2} \operatorname{He}_n(\sqrt{2} \, x)$. They are orthogonal on $\R$: $\int_{-\infty}^{\infty} H_m(x) H_n(x) e^{-x^2} \, dx = \sqrt{\pi} \, 2^n n! \, \delta_{mn}$. They satisfy $H_n' = 2n \, H_{n-1}$ and $\operatorname{He}_n' = n \operatorname{He}_{n-1}$. In quantum mechanics, the eigenfunctions of the harmonic oscillator are $H_n(x) e^{-x^2/2}$ up to scaling.