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