Hermite transform

 In mathematics, Hermite transform is an integral transform named after the mathematician Charles Hermite, which uses Hermite polynomials  as kernels of the transform. This was first introduced by Lokenath Debnath in 1964.^[1][2][3][4]

The Hermite transform of a function ${\displaystyle F(x)}$ is

${\displaystyle H\{F(x)\}=f_{H}(n)=\int _{-\infty }^{\infty }e^{-x^{2}}\ H_{n}(x)\ F(x)\ dx}$

The inverse Hermite transform is given by

${\displaystyle H^{-1}\{f_{H}(n)\}=F(x)=\sum _{n=0}^{\infty }{\frac {1}{{\sqrt {\pi }}2^{n}n!}}f_{H}(n)H_{n}(x)}$

Some Hermite transform pairsEdit

${\displaystyle F(x)\,}$	${\displaystyle f_{H}(n)\,}$
${\displaystyle x^{m},\ n>m\,}$	${\displaystyle 0}$
${\displaystyle x^{n}\,}$	${\displaystyle {\sqrt {\pi }}n!P_{n}(1)}$
${\displaystyle e^{ax}\,}$	${\displaystyle {\sqrt {\pi }}a^{n}e^{a^{2}/4}\,}$
${\displaystyle e^{2xt-t^{2}},\ |t|<{\frac {1}{2}}\,}$	${\displaystyle {\sqrt {\pi }}\sum _{n=0}^{\infty }(2t)^{n}}$
${\displaystyle e^{x^{2}}{\frac {d}{dx}}\left[e^{-x^{2}}{\frac {d}{dx}}F(x)\right]\,}$	${\displaystyle -2nf_{H}(n)\,}$
${\displaystyle e^{-x^{2}}H_{n}(x)\,}$	${\displaystyle 2^{n-1/2}\Gamma (n+1/2)\,}$
${\displaystyle x^{2}H_{n}(x)\,}$	${\displaystyle (n+1/2)\delta _{n}\,}$
${\displaystyle {\frac {d^{m}}{dx^{m}}}F(x)\,}$	${\displaystyle f_{H}(n+m)\,}$
${\displaystyle x{\frac {d^{m}}{dx^{m}}}F(x)\,}$	${\displaystyle nf_{H}(n+m-1)+{\frac {1}{2}}f_{H}(n+m+1)\,}$
${\displaystyle F(x)*G(x)\,}$	${\displaystyle {\sqrt {\pi }}(-1)^{n}\left[2^{2n+1}\Gamma \left(n+{\frac {3}{2}}\right)\right]^{-1}f_{H}(n)g_{H}(n)\,}$[5]
${\displaystyle H_{m}(x)\,}$	${\displaystyle {\sqrt {\pi }}2^{n}n!\delta _{nm}\,}$
${\displaystyle H_{n}^{2}(x)\,}$	${\displaystyle {\sqrt {\pi }}\sum _{r=0}^{n}{\binom {n}{r}}2^{r+n}(2r)!n!\,}$
${\displaystyle H_{m}(x)H_{p}(x)\,}$	${\displaystyle {\begin{cases}{\frac {{\sqrt {\pi }}2^{k}m!n!p!}{(k-m)!(k-n)!(k-p)!}},&m+n+p=2k,\ k\geq m,n,p\\0,&{\text{otherwise}}\end{cases}}\,}$[6]
${\displaystyle H_{m}^{2}(x)H_{n}(x),\ m>n\,}$	${\displaystyle {\sqrt {\pi }}2^{n}2^{m}n!\sum _{k=0}^{n}{\binom {m}{k}}{\binom {n}{k}}{\binom {2k}{k}}\,}$[7]
${\displaystyle H_{n+p+q}(x)H_{p}(x)H_{q}(x)\,}$	${\displaystyle {\sqrt {\pi }}2^{n+p+q}(n+p+q)!\,}$
${\displaystyle e^{z^{2}}\sin({\sqrt {2}}xz),\ |2z|<1\,}$	${\displaystyle {\begin{cases}{\sqrt {\pi }}\sum _{m=0}^{\infty }(-1)^{m}(2z)^{2m+1},&n=2m+1\\0,&n\neq 2m+1\end{cases}}\,}$
${\displaystyle (1-z^{2})^{-1/2}\exp \left[{\frac {2xyz-(x^{2}+y^{2})z^{2}}{(1-z^{2})}}\right]\,}$	${\displaystyle {\sqrt {\pi }}\sum _{m=0}^{\infty }z^{m}H_{m}(y)\delta _{nm}\,}$

This article uses material from the Wikipedia article  Metasyntactic variable, which is released under the  Creative Commons Attribution-ShareAlike 3.0 Unported License.