Hermitepolynom

Differensekvation

${\displaystyle H_{n+1}(x)=2xH_{n}(x)-H_{n}'(x).\,\!}$

Explicit formel

${\displaystyle H_{n}(x)=n!\sum _{m=0}^{\lfloor n/2\rfloor }{\frac {(-1)^{m}}{m!(n-2m)!}}(2x)^{n-2m}}$

Speciella värden

${\displaystyle H_{n}(0)={\begin{cases}0,&{\mbox{om }}n{\mbox{ är udda}}\\(-1)^{n/2}2^{n/2}(n-1)!!,&{\mbox{om }}n{\mbox{ är jämnt}}\end{cases}}}$

Genererande funktion

${\displaystyle \exp(2xt-t^{2})=\sum _{n=0}^{\infty }H_{n}(x){\frac {t^{n}}{n!}}\,\!}$

Övrigt

Multiplikationsteoremet:

${\displaystyle {\mathit {H}}_{n}(\gamma x)=\sum _{i=0}^{\lfloor n/2\rfloor }\gamma ^{n-2i}(\gamma ^{2}-1)^{i}{n \choose 2i}{\frac {(2i)!}{i!}}{\mathit {H}}_{n-2i}(x)}$

${\displaystyle H_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}H_{k}(x)(2y)^{(n-k)}=2^{-{\frac {n}{2}}}\cdot \sum _{k=0}^{n}{n \choose k}H_{n-k}\left(x{\sqrt {2}}\right)H_{k}\left(y{\sqrt {2}}\right)}$

${\displaystyle {\mathit {H}}_{n}^{(m)}(x)=2^{m}\cdot {\frac {n!}{(n-m)!}}\cdot {\mathit {H}}_{n-m}(x)=2^{m}\cdot m!\cdot {n \choose m}\cdot {\mathit {H}}_{n-m}(x)\,\!}$

${\displaystyle H_{n+1}(x)=2xH_{n}(x)-2nH_{n-1}(x)\,\!}$

Laguerrepolynom

Hermitepolynomen är relaterade till Laguerrepolynomen enligt

${\displaystyle H_{2n}(x)=(-4)^{n}\,n!\,L_{n}^{(-1/2)}(x^{2})=4^{n}\,n!\sum _{i=0}^{n}(-1)^{n-i}{n-{\frac {1}{2}} \choose n-i}{\frac {x^{2i}}{i!}}\,\!}$

${\displaystyle H_{2n+1}(x)=2(-4)^{n}\,n!\,x\,L_{n}^{(1/2)}(x^{2})=2\cdot 4^{n}\,n!\sum _{i=0}^{n}(-1)^{n-i}{n+{\frac {1}{2}} \choose n-i}{\frac {x^{2i+1}}{i!}}\,\!}$.

Relation till generaliserade hypergeometriska serier

${\displaystyle H_{2n}(x)=(-1)^{n}\,{\frac {(2n)!}{n!}}\,_{1}F_{1}\left(-n,{\frac {1}{2}};x^{2}\right)}$

${\displaystyle H_{2n+1}(x)=(-1)^{n}\,{\frac {(2n+1)!}{n!}}\,2x\,_{1}F_{1}\left(-n,{\frac {3}{2}};x^{2}\right)}$

där ${\displaystyle \,_{1}F_{1}(a,b;z)}$ är en generaliserad hypergeometrisk funktion.