The Hermite polynomials $H_n(x)$ are a set of orthogonal polynomials that have the following properties:

$$\int\limits_{-\infty}^{\infty}e^{-x^2}H_n^2(x)dx = \sqrt{\pi}2^nn!,$$ $$\int\limits_{-\infty}^{\infty}e^{-x^2}H_n(x)H_m(x)dx = 0 \qquad n\ne m.$$

The first two Hermite polynomials are:

$$H_0 = 1,$$ $$H_1=2x.$$

The rest of the Hermite polynomials can be generated with the recursion relation,

$$H_{n+1}(x) = 2xH_n(x)- 2nH_{n-1}(x).$$

$H_n(x)$ crosses zero $n$ times. The code below will calculate the the value of $H_n(x)$. You can change $n$ and $x$ in the code.

<script> function Hermite(n,x) { if (n==0) {return 1;} if (n==1) {return 2*x;} if (n > 1) { Hn_min_1 = 1; Hn = 2*x; for (i = 2; i<(n+1); i++) { Hn_plus_1 = 2*x*Hn - 2*(i-1)*Hn_min_1; Hn_min_1 = Hn; Hn = Hn_plus_1; } return Hn_plus_1; } } //**** change n and x here ***** n = 3; x = 2; document.write("<i>H</i><sub>"+n+"</sub>("+x+") = "+Hermite(n,x)); </script>

The first few Hermite polynomials are plotted below.

$H_n$
	$x$