 | This article relies largely or entirely on a single source .(October 2021) |
In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. Typically they are defined for physicists' Hermite polynomials.
| | This article relies largely or entirely on a single source .(October 2021) |
In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. Typically they are defined for physicists' Hermite polynomials.
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:
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).
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.)
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