Hermite number

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:

:$H\_0\; =\; 1\backslash ,$

:$H\_1\; =\; 0\backslash ,$

:$H\_2\; =\; -2\backslash ,$

:$H\_3\; =\; 0\backslash ,$

:$H\_4\; =\; +12\backslash ,$

:$H\_5\; =\; 0\backslash ,$

:$H\_6\; =\; -120\backslash ,$

:$H\_7\; =\; 0\backslash ,$

:$H\_8\; =\; +1680\backslash ,$

:$H\_9\; =0\backslash ,$

:$H\_\{10\}\; =\; -30240\backslash ,$

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

:$H\_\{n\}\; =\; -2(n-1)H\_\{n-2\}.\backslash ,\backslash !$

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

:$H\_n\; =$

\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}

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

From the generating function of Hermitian polynomials it follows that

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

Reference gives a formal power series:

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

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

Category:Integer sequences