title: Hermite number
text: # 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:
  
      $H_{0}=1\,$
      $H_{1}=0\,$
      $H_{2}=-2\,$
      $H_{3}=0\,$
      $H_{4}=+12\,$
      $H_{5}=0\,$
      $H_{6}=-120\,$
      $H_{7}=0\,$
      $H_{8}=+1680\,$
      $H_{9}=0\,$
      $H_{10}=-30240\,$
  
  ## 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 $H_1$ = 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).
  
  ## Usage
  
  From the generating function of Hermitian polynomials it follows that
  
      $\exp(-t^{2}+2tx)=\sum _{n=0}^{\infty }H_{n}(x){\frac {t^{n}}{n!}}\,\!$
  
  Reference gives a formal power series:
  
      $H_{n}(x)=(H+2x)^{n}\,\!$
  
  where formally the n-th power of H, $H^n,$ is the n-th Hermite number, $H_n.$ (See Umbral calculus.)
  
  