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:
Recursion relations
Are obtained from recursion relations of Hermitian polynomials for x = 0:
Since H0 = 1 and H1 = 0 one can construct a closed formula for Hn:
where (n - 1)!! = 1 × 3 × ... × (n - 1).
Usage
From the generating function of Hermitian polynomials it follows that
Reference [1] gives a formal power series:
where formally the n-th power of H, Hn, is the n-th Hermite number, Hn. (See Umbral calculus.)
Notes
- 1 2 Weisstein, Eric W. "Hermite Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HermiteNumber.html
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