Hermite number
# 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 $H_n$ = $H_n(0),$ where $H_n(x)$ 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 $H_0$ = 1 and $H_1$ = 0 one can construct a closed formula for $H_n:$

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

where (n - 1)!! = 1 × 3 × ... × (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, $H^n,$ is the n-th Hermite number, $H_n.$ (See Umbral calculus.)

