# 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.[1]

The first Hermite numbers are:

${\displaystyle H_{0}=1\,}$
${\displaystyle H_{1}=0\,}$
${\displaystyle H_{2}=-2\,}$
${\displaystyle H_{3}=0\,}$
${\displaystyle H_{4}=+12\,}$
${\displaystyle H_{5}=0\,}$
${\displaystyle H_{6}=-120\,}$
${\displaystyle H_{7}=0\,}$
${\displaystyle H_{8}=+1680\,}$
${\displaystyle H_{9}=0\,}$
${\displaystyle H_{10}=-30240\,}$

## Recursion relations

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

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

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

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

where (n - 1)!! = 1 × 3 × ... × (n - 1).

## Usage

From the generating function of Hermitian polynomials it follows that

${\displaystyle \exp(-t^{2})=\sum _{n=0}^{\infty }H_{n}{\frac {t^{n}}{n!}}\,\!}$

Reference [1] gives a formal power series:

${\displaystyle H_{n}(x)=(H+2x)^{n}\,\!}$

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

## Notes

1. ^ a b Weisstein, Eric W. "Hermite Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HermiteNumber.html