# Hermite numbers

The Hermite numbers $H_{n}$  may be defined by the generating function

 $\displaystyle e^{-t^{2}}\;:=\;\sum_{n=0}^{\infty}H_{n}\frac{t^{n}}{n!}$ (1)

which is same as the generating function of Hermite polynomials at the value 0 of the argument $z$.  After expanding the left hand side of (1) to Taylor series $1-\frac{t^{2}}{1!}+\frac{t^{4}}{2!}-\frac{t^{6}}{3!}+-\ldots$, one can write

 $\displaystyle 1-2\!\cdot\!\frac{t^{2}}{2!}+12\!\cdot\!\frac{t^{4}}{4!}-120\!% \cdot\!\frac{t^{6}}{6!}+-\ldots\;\;=\;\sum_{n=0}^{\infty}H_{n}\frac{t^{n}}{n!}.$ (2)

Thus one sees that

 $H_{0}=1,\quad H_{1}=0,\quad H_{2}=-2,\quad H_{3}=0,\quad H_{4}=12,\quad H_{5}=% 0,\quad H_{6}=-120,\quad\ldots$

Evidently,

 $\displaystyle H_{n}\;=\;\begin{cases}\frac{(-1)^{\frac{n}{2}}n!}{(\frac{n}{2})% !}\textrm{ when }2\mid n\\ 0\;\qquad\textrm{ when }2\nmid n\end{cases}$ (3)

The Hermite numbers form the sequence (http://www.research.att.com/ njas/sequences/index.html?q=A067994&language=english&go=SearchSloane A067994)

 $1,\,0,\,-2,\,0,\,12,\,0,\,-120,\,0,\,1680,\,0,\,-30240,\,0,\,665280,\,0,\,-172% 97280,\,0,\,\ldots$

which obeys the recurrence relation

 $\displaystyle H_{n}\;=\;2(1\!-\!n)H_{n-2}.$ (4)

According to (1), the Hermite numbers satisfy  $H_{n}\,=\,H_{n}(0)$  where $H_{n}(x)$ is the Hermite polynomial of degree $n$.  The of Hermite numbers and Hermite polynomials may be expressed also by using symbolic powers

 $H^{\nu}\;=:\;H_{\nu}$

as follows:

 $\displaystyle(2x+H)^{n}\;=\;H_{n}(x).$ (5)

This means e.g. that

 $(2x+H)^{2}\;=\;(2x)^{2}+2\cdot 2xH^{1}+H^{2}\;=\;4x^{2}+4xH_{1}+H_{2}\;=\;4x^{% 2}-2\;=\;H_{2}(x).$
Title Hermite numbers HermiteNumbers 2013-03-22 19:08:32 2013-03-22 19:08:32 pahio (2872) pahio (2872) 8 pahio (2872) Definition msc 11B68 EulerNumbers2 AppellSequence