Definition:GenHermite

<< Definition:f

Main Page

Definition:GottliebLaguerre >>

The LaTeX DLMF and DRMF macro \GenHermite represents the generalized Hermite polynomial.

This macro is in the category of polynomials.

In math mode, this macro can be called in the following ways:

\GenHermite[\alpha]{n} produces ${\displaystyle{\displaystyle{\displaystyle H^{\alpha}_{n}}}}$

\GenHermite{n} produces ${\displaystyle{\displaystyle{\displaystyle H_{n}}}}$

\GenHermite[\alpha]{n}@{x} produces ${\displaystyle{\displaystyle{\displaystyle H^{\alpha}_{n}\left(x\right)}}}$

\GenHermite{n}@{x} produces ${\displaystyle{\displaystyle{\displaystyle H_{n}\left(x\right)}}}$

These are defined by

${\displaystyle{\displaystyle H^{\mu}_{2m}\left(x\right):={\rm const}\times L^{\mu-\frac{1}{2}}_{m}\left(x^{2}\right),\qquad}}$
${\displaystyle{\displaystyle H^{\mu}_{2m+1}\left(x\right):={\rm\const}\times xL^{\mu+\frac{1}{2}}_{m}\left(x^{2}\right).}}$
Then for ${\displaystyle{\displaystyle\mu>-\tfrac{1}{2}}}$ we have orthogonality relation ${\displaystyle{\displaystyle\int_{-\infty}^{\infty}H^{\mu}_{m}\left(x\right)H^{\mu}_{n}\left(x\right)|x|^{2\mu}e^{-x^{2}}dx,=0}}$
for ${\displaystyle{\displaystyle m\neq n}}$.

Bibliography

Page 156 of CHI.

Symbols List

${\displaystyle{\displaystyle{\displaystyle H_{n}^{\mu}}}}$ : generalized Hermite polynomial : http://drmf.wmflabs.org/wiki/Definition:GenHermite
${\displaystyle{\displaystyle{\displaystyle L_{n}^{(\alpha)}}}}$ : Laguerre (or generalized Laguerre) polynomial : http://dlmf.nist.gov/18.3#T1.t1.r27
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv

<< Definition:f

Main Page

Definition:GottliebLaguerre >>