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 H₀ = 1 and H₁ = 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, Hⁿ, is the n-th Hermite number, H_n. (See Umbral calculus.)