Hermite polynomials are a class of orthogonal polynomials that are defined as follows:
- Probabilist's Hermite polynomials:
$$\text{He}_n(x)=(-1)^n e^{\frac{x^2}{2}}\frac{\mathrm{d}^n}{\mathrm{d}x^n} e^{-\frac{x^2}{2}}$$
- Physicist's Hermite polynomials: $$\text{H}_n(x)=(-1)^{n}e^{x^{2}}\frac{\mathrm{d}^n}{\mathrm{d}x^n} e^{-x^{2}}$$
These polynomials are orthogonal with respect to the weight function
$$\int_{-\infty}^{\infty}\text{H}_m(x)H_n(x)\ e^{-x^2}\mathrm{d}x=\sqrt {\pi}\ 2^n n!\ \delta_{nm}$$
$$\int_{-\infty}^{\infty}\text{He}_m(x)\text{He}_n(x)\ e^{-\frac{x^2}{2}}\mathrm{d}x=\sqrt{2\pi}\ n!\ \delta_{nm}$$
