Hermite
Hypergeometric representation
Orthogonality relation(s)
Recurrence relation
Monic recurrence relation

Differential equation
Forward shift operator
Backward shift operator

Rodrigues-type formula
Generating functions







Limit relations
Meixner-Pollaczek polynomial to Hermite polynomial

Jacobi polynomial to Hermite polynomial

Gegenbauer / Ultraspherical polynomial to Hermite polynomial

Krawtchouk polynomial to Hermite polynomial

Laguerre polynomial to Hermite polynomial
![{\displaystyle {\displaystyle
\lim_{\alpha\rightarrow\infty}
\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n}
\Laguerre[\alpha]{n}@{(2\alpha}^{\frac{1}{2}}x+\alpha)=\frac{(-1)^n}{n!}\Hermite{n}@{x}
}}](/index.php?title=Special:MathShowImage&hash=bfd0bcc40637e88abeb16806d94999f4&mode=latexml)
Charlier polynomial to Hermite polynomial

![{\displaystyle {\displaystyle
\Hermite{2n}@{x}=(-1)^nn!\,2^{2n}\Laguerre[-\frac{1}{2}]{n}@{x^2}
}}](/index.php?title=Special:MathShowImage&hash=779facce98fdd8dc13ce4fcb10c70ae4&mode=latexml)
![{\displaystyle {\displaystyle
\Hermite{2n+1}@{x}=(-1)^nn!\,2^{2n+1}x\Laguerre[\frac{1}{2}]{n}@{x^2}
}}](/index.php?title=Special:MathShowImage&hash=de77e403b08ef0c095d996022a9a83ea&mode=latexml)

