Lauricella's theorem

In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. Typically they are defined for physicists' Hermite polynomials.

Formal definition

The numbers H_n = H_n(0), where H_n(x) is a Hermite polynomial of order n, may be called Hermite numbers.

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 H_n:

$H_n=\{\begin{array}{ll}0,& \text{if }n\text{ is odd}\\ (-1{)}^{n/2}{2}^{n/2}(n-1)!!,& \text{if }n\text{ is even}\end{array}$

where (n - 1)!! = 1 × 3 × ... × (n - 1).

Usage

From the generating function of Hermitian polynomials it follows that

$\exp (-t^2)=\sum _{n=0}^{\mathrm{\infty }}{H}_n\frac{t^n}{n!}$

Reference 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, H_n. (See Umbral calculus.)