Hermite number

Formal definition

The numbers Hn = Hn(0), where Hn(x) is a Hermite polynomial of order n, may be called Hermite numbers.Weisstein, Eric W. "Hermite Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HermiteNumber.html

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 + 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, H**n, is the n-th Hermite number, H**n. (See Umbral calculus.)

Notes