For a non-negative integer $n$, the HermiteH(n, x) function computes the nth Hermite polynomial.

The Hermite polynomials are orthogonal on the interval $\left(-\mathrm{\infty }\,\mathrm{\infty }\right)$ with respect to the weight function $w\left(x\right)={\ⅇ}^{-x^2}$. They satisfy:

Hermite polynomials satisfy the following recurrence relation:

where HermiteH(0,x) = 1 and HermiteH(1,x) = 2*x.

For n different from a non-negative integer, the analytic extension of the Hermite polynomial is given by:

$\mathrm{HermiteH}\left(3\,x\right)$

$\mathrm{HermiteH}\left(3\,x\right)$

$\mathrm{simplify}\left(\,'\mathrm{HermiteH}'\right)$

$8x^3-12x$

$\mathrm{HermiteH}\left(3.2\,2.1\right)$

$59.58210770$