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:

$H_0 = 1\,$

$H_1 = 0\,$

$H_2 = -2\,$

$H_3 = 0\,$

$H_4 = +12\,$

$H_5 = 0\,$

$H_6 = -120\,$

$H_7 = 0\,$

$H_8 = +1680\,$

$H_9 =0\,$

$H_{10} = -30240\,$

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

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

Reference ^[1] 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.)

Notes

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