Hermite number

In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. Typically they are defined for physicists' Hermite polynomials.

Formal definition
The numbers Hn = Hn(0), where Hn(x) is a Hermite polynomial of order n, may be called Hermite numbers.

The first Hermite numbers are:
 * $$H_0 = 1\,$$
 * $$H_1 = 0\,$$
 * $$H_2 = -2\,$$
 * $$H_3 = 0\,$$
 * $$H_4 = +12\,$$
 * $$H_5 = 0\,$$
 * $$H_6 = -120\,$$
 * $$H_7 = 0\,$$
 * $$H_8 = +1680\,$$
 * $$H_9 =0\,$$
 * $$H_{10} = -30240\,$$

Recursion relations
Are obtained from recursion relations of Hermitian polynomials for x = 0:


 * $$H_{n} = -2(n-1)H_{n-2}.\,\!$$

Since H0 = 1 and H1 = 0 one can construct a closed formula for Hn:


 * $$H_n =

\begin{cases} 0, & \mbox{if }n\mbox{ is odd} \\ (-1)^{n/2} 2^{n/2} (n-1)!! , & \mbox{if }n\mbox{ is even} \end{cases} $$

where (n - 1)!! = 1 &times; 3 &times; ... &times; (n - 1).

Usage
From the generating function of Hermitian polynomials it follows that


 * $$\exp (-t^2 + 2tx) = \sum_{n=0}^\infty H_n (x) \frac {t^n}{n!}\,\!$$

Reference gives a formal power series:


 * $$H_n (x) = (H+2x)^n\,\!$$

where formally the n-th power of H, Hn, is the n-th Hermite number, Hn.  (See Umbral calculus.)