Hermite transform

In mathematics, Hermite transform is an integral transform named after the mathematician Charles Hermite, which uses Hermite polynomials ${\displaystyle H_{n}(x)}$ 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 pairs[]

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

References[]