In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. Typically they are defined for physicists' Hermite polynomials.
Formal definition The numbers Hn = Hn(0), where Hn(x) is a Hermite polynomial of order n, may be called Hermite numbers. The first Hermite numbers are:
H 0 = 1 {\displaystyle H_{0}=1\,}
H 1 = 0 {\displaystyle H_{1}=0\,}
H 2 = − 2 {\displaystyle H_{2}=-2\,}
H 3 = 0 {\displaystyle H_{3}=0\,}
H 4 = + 12 {\displaystyle H_{4}=+12\,}
H 5 = 0 {\displaystyle H_{5}=0\,}
H 6 = − 120 {\displaystyle H_{6}=-120\,}
H 7 = 0 {\displaystyle H_{7}=0\,}
H 8 = + 1680 {\displaystyle H_{8}=+1680\,}
H 9 = 0 {\displaystyle H_{9}=0\,}
H 10 = − 30240 {\displaystyle H_{10}=-30240\,}
Recursion relations Are obtained from recursion relations of Hermitian polynomials for x = 0:
H n = − 2 ( n − 1 ) H n − 2 . {\displaystyle H_{n}=-2(n-1)H_{n-2}.\,\!}
Since H0 = 1 and H1 = 0 one can construct a closed formula for Hn:
H n = { 0 , if n is odd ( − 1 ) n / 2 2 n / 2 ( n − 1 ) ! ! , if n is even {\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}}}
where (n − 1)!! = 1 × 3 × ... × (n − 1).
Usage From the generating function of Hermitian polynomials it follows that
exp ( − t 2 + 2 t x ) = ∑ n = 0 ∞ H n ( x ) t n n ! {\displaystyle \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 + 2 x ) n {\displaystyle H_{n}(x)=(H+2x)^{n}\,\!}
where formally the n-th power of H, Hn, is the n-th Hermite number, Hn. (See Umbral calculus.)
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