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.^[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 H_n:

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

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

Usage

From the generating function of Hermitian polynomials it follows that

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

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, H_n. (See Umbral calculus.)