In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. Typically they are defined for physicists' Hermite polynomials.
The numbers H<sub>n</sub> = H<sub>n</sub>(0), where H<sub>n</sub>(x) is a Hermite polynomial of order n, may be called Hermite numbers.
The first Hermite numbers are:
Are obtained from recursion relations of Hermitian polynomials for x = 0:
Since H<sub>0</sub> = 1 and H<sub>1</sub> = 0 one can construct a closed formula for H<sub>n</sub>:
where (n â 1)!! = 1 × 3 × ... × (n â 1).
From the generating function of Hermitian polynomials it follows that
Reference gives a formal power series:
where formally the n-th power of H, H<sup>n</sup>, is the n-th Hermite number, H<sub>n</sub>. (See Umbral calculus.)