Hermite transform

In mathematics, the Hermite transform is an integral transform named after the mathematician Charles Hermite that uses Hermite polynomials ${\displaystyle H_{n}(x)}$ as kernels of the transform.

The Hermite transform ${\displaystyle H\{F(x)\}\equiv f_{H}(n)}$ of a function ${\displaystyle F(x)}$ is $${\displaystyle H\{F(x)\}\equiv f_{H}(n)=\int _{-\infty }^{\infty }e^{-x^{2}}\ H_{n}(x)\ F(x)\ dx}$$

The inverse Hermite transform ${\displaystyle H^{-1}\{f_{H}(n)\}}$ is given by $${\displaystyle H^{-1}\{f_{H}(n)\}\equiv F(x)=\sum _{n=0}^{\infty }{\frac {1}{{\sqrt {\pi }}2^{n}n!}}f_{H}(n)H_{n}(x)}$$

Some Hermite transform pairs

${\displaystyle F(x)\,}$	${\displaystyle f_{H}(n)\,}$
${\displaystyle x^{m}}$	${\displaystyle {\begin{cases}{\frac {m!{\sqrt {\pi }}}{2^{m-n}\left({\frac {m-n}{2}}\right)!}},&(m-n){\text{ even and}}\geq 0\\0,&{\text{otherwise}}\end{cases}}}$[1]
${\displaystyle e^{ax}\,}$	${\displaystyle {\sqrt {\pi }}a^{n}e^{a^{2}/4}\,}$
${\displaystyle e^{2xt-t^{2}},\ |t|<{\frac {1}{2}}\,}$	${\displaystyle {\sqrt {\pi }}(2t)^{n}}$
${\displaystyle H_{m}(x)\,}$	${\displaystyle {\sqrt {\pi }}2^{n}n!\delta _{nm}\,}$
${\displaystyle x^{2}H_{m}(x)\,}$	${\displaystyle 2^{n}n!{\sqrt {\pi }}{\begin{cases}1,&n=m+2\\\left(n+{\frac {1}{2}}\right),&n=m\\(n+1)(n+2),&n=m-2\\0,&{\text{otherwise}}\end{cases}}}$
${\displaystyle e^{-x^{2}}H_{m}(x)\,}$	${\displaystyle \left(-1\right)^{p-m}2^{p-1/2}\Gamma (p+1/2),\ m+n=2p,\ p\in \mathbb {Z} }$
${\displaystyle H_{m}^{2}(x)\,}$	${\displaystyle {\begin{cases}2^{m+n/2}{\sqrt {\pi }}{\binom {m}{n/2}}{\frac {m!n!}{(n/2)!}},&n{\text{ even and}}\leq 2m\\0,&{\text{otherwise}}\end{cases}}}$[2]
${\displaystyle H_{m}(x)H_{p}(x)\,}$	${\displaystyle {\begin{cases}{\frac {2^{k}{\sqrt {\pi }}m!n!p!}{(k-m)!(k-n)!(k-p)!}},&n+m+p=2k,\ k\in \mathbb {Z}$ ;\ |m-p|\leq n\leq m+p\\0,&{\text{otherwise}}\end{cases}}\,} [3]
${\displaystyle H_{n+p+q}(x)H_{p}(x)H_{q}(x)\,}$	${\displaystyle {\sqrt {\pi }}2^{n+p+q}(n+p+q)!\,}$
${\displaystyle {\frac {d^{m}}{dx^{m}}}F(x)\,}$	${\displaystyle f_{H}(n+m)\,}$
${\displaystyle x{\frac {d^{m}}{dx^{m}}}F(x)\,}$	${\displaystyle nf_{H}(n+m-1)+{\frac {1}{2}}f_{H}(n+m+1)\,}$
${\displaystyle e^{x^{2}}{\frac {d}{dx}}\left[e^{-x^{2}}{\frac {d}{dx}}F(x)\right]\,}$	${\displaystyle -2nf_{H}(n)\,}$
${\displaystyle F(x-x_{0})}$	${\displaystyle {\sqrt {\pi }}\sum _{k=0}^{\infty }{\frac {(-x_{0})^{k}}{k!}}f_{H}(n+k)}$
${\displaystyle F(x)*G(x)\,}$	${\displaystyle {\sqrt {\pi }}(-1)^{n}\left[2^{2n+1}\Gamma \left(n+{\frac {3}{2}}\right)\right]^{-1}f_{H}(n)g_{H}(n)\,}$[4]
${\displaystyle e^{z^{2}}\sin(xz),\ |z|<{\frac {1}{2}}\ \,}$	${\displaystyle {\begin{cases}{\sqrt {\pi }}(-1)^{\lfloor {\frac {n}{2}}\rfloor }(2z)^{n},&n\,\mathrm {odd} \\0,&n\,\mathrm {even} \end{cases}}\,}$
${\displaystyle (1-z^{2})^{-1/2}\exp \left[{\frac {2xyz-(x^{2}+y^{2})z^{2}}{(1-z^{2})}}\right]\,}$	${\displaystyle {\sqrt {\pi }}z^{n}H_{n}(y)}$[5][6]
${\displaystyle {\frac {H_{m}(y)H_{m+1}(x)-H_{m}(x)H_{m+1}(y)}{2^{m+1}m!(x-y)}}}$	${\displaystyle {\begin{cases}{\sqrt {\pi }}H_{n}(y)&n\leq m\\0&n>m\end{cases}}}$