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:

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 Hn:

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, Hn. (See Umbral calculus.)

Notes

  1. ↑ 1.0 1.1 Weisstein, Eric W. "Hermite Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HermiteNumber.html