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 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:

[math]\displaystyle{ H_0 = 1\, }[/math]

[math]\displaystyle{ H_1 = 0\, }[/math]

[math]\displaystyle{ H_2 = -2\, }[/math]

[math]\displaystyle{ H_3 = 0\, }[/math]

[math]\displaystyle{ H_4 = +12\, }[/math]

[math]\displaystyle{ H_5 = 0\, }[/math]

[math]\displaystyle{ H_6 = -120\, }[/math]

[math]\displaystyle{ H_7 = 0\, }[/math]

[math]\displaystyle{ H_8 = +1680\, }[/math]

[math]\displaystyle{ H_9 =0\, }[/math]

[math]\displaystyle{ H_{10} = -30240\, }[/math]

Recursion relations

Are obtained from recursion relations of Hermitian polynomials for x = 0:

[math]\displaystyle{ H_{n} = -2(n-1)H_{n-2}.\,\! }[/math]

Since H₀ = 1 and H₁ = 0 one can construct a closed formula for H_n:

[math]\displaystyle{ H_n = \begin{cases} 0, & \mbox{if }n\mbox{ is odd} \\ (-1)^{n/2} 2^{n/2} (n-1)!! , & \mbox{if }n\mbox{ is even} \end{cases} }[/math]

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

Usage

From the generating function of Hermitian polynomials it follows that

[math]\displaystyle{ \exp (-t^2 + 2tx) = \sum_{n=0}^\infty H_n (x) \frac {t^n}{n!}\,\! }[/math]

Reference gives a formal power series:

[math]\displaystyle{ H_n (x) = (H+2x)^n\,\! }[/math]

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