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, where Hn is a Hermite polynomial of order n, may be called Hermite numbers.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 !! = 1 × 3 ×... ×.
Usage
From the generating function of Hermitian polynomials it follows thatReference gives a formal power series:
where formally the n-th power of H, Hn, is the n-th Hermite number, Hn.