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In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. Typically they are defined for physicists' Hermite polynomials.

Formal definition

he 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

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