Hermite number explained

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.^[1]

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{cases} 0 , & if n isodd \\ ( -1 ) ^{n/2} 2 ^{n/2} (n -1 )!! , & if n iseven \end{cases}

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

Usage

From the generating function of Hermitian polynomials it follows that

\exp ( - t ^{2} + 2 tx) = \sum ^{inf ty} _{n =0} H _{n} (x) (t ^{n})/(n!)

Reference ^[1] gives a formal power series:

H _{n} (x) = (H +2 x) ^{n}

where formally the n-th power of H, Hⁿ, is the n-th Hermite number, H_n. (See Umbral calculus.)

Notes

1. Weisstein, Eric W. "Hermite Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HermiteNumber.html