Hermite Polynomial
Let $n_1,n_2$ be non-negative integers and $x$ a real number, this application evaluates the Hermite polynomials $H_{n_1}(x),\ldots,H_{n_2}(x)$, where the Hermite polynomial of order $n$ at x is given by \[ H_n(x)=(-1)^n\mathrm{e}^{x^2}\frac{d^n}{dx^n}\mathrm{e}^{-x^2}.\]
Note that this is the physicist's Hermite polynomial. The probabilist's Hermite polynomial is given by \[ \mathbf{H}_n(x)=(-1)^n\mathrm{e}^{\frac{x^2}{2}}\frac{d^n}{dx^n}\mathrm{e}^{-\frac{x^2}{2}}.\] As the latter can also be calculated from the former using the relation \[ \mathbf{H}_n(x)=2^{-\frac{n}{2}}H_n\left(\frac{x}{\sqrt{2}}\right),\] it is not evaluated in this application.