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

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 "H"₀ = 1 and "H"₁ = 1 one can construct a closed formula for "H"_n:

:

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

Usage

From generating function of Hermitian polynomials it follows that

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

Reference gives a formal power series:

:$H\_n\; (x)\; =\; (H+2x)^n,!$

where formally the "n"-th power of "H", "H"^k, is the "n"-th Hermite number, "H"_n.

Notes