Hermite transform explained

H _{n(x)} as kernels of the transform.

The Hermite transform

H\{F(x)\} \equiv f _{H} (n) of a function F(x) is$$H\backslash \; \backslash equiv\; f\_H(n)\; =\; \backslash int\_^\backslash infty\; e^\; \backslash \; H\_n(x)\backslash \; F(x)\; \backslash \; dx$$

The inverse Hermite transform

H ^{-1} \{f _{H(n)\}} is given by$$H^\backslash \; \backslash equiv\; F(x)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; f\_H(n)\; H\_n(x)$$

Some Hermite transform pairs

F(x)	f _{H(n)}
x ^{m}	\begin{cases} (m!\sqrt{\pi)/(}{ 2) ^{m-n} \left( (m - n)/(2) \right)!}, & (m - n) evenand \geq 0 \\ 0 , & otherwise \end{cases}
e ^{ax}	\sqrt\pi a ^{n} e ^{a ^{2/4}}
e ^{2 xt - t ^{2}} ,	t	<\frac\,	\sqrt\pi ( 2 t) ^{n}
H _{m(x)}	\sqrt\pi 2 ^{n} n!\delta _{nm}
x ^{2 H} _{m(x)}	2 ^{n} n! \sqrt{\pi}\begin{cases} 1 , & n = m +2 \\ \left(n + (1)/(2) \right), & n = m \\ (n +1 )(n +2 ),& n = m -2 \\ 0 , & otherwise \end{cases}
e ^{- x ^{2}} H _{m(x)}	\left( -1 \right) ^{p-m} 2 ^{p-1/2} \Gamma(p +1 / 2 ), m + n =2 p, p\in Z
H ^{2 (x)} _{m}	\begin{cases} 2 ^{m+n/2} \sqrt\pi \binom m{n/ 2 } (m!n!)/((n/ 2 )!) , & n evenand \leq 2 m \\ 0 , & otherwise \end{cases} ^[1]
H _{m(x)H} _{p(x)}	\begin{cases} (2 ^{k\sqrt\pi} m!n!p!)/((k - m)!(k - n)!(k - p)!) , & n + m + p =2 k, k\in Z ;	m-p	\leq n\leq m+p\\ 0, & \text \end\,^[2]
H _{n+p+q} (x)H _{p(x)H} _{q(x)}	\sqrt\pi 2 ^{n+p+q} (n + p + q)!
(d ^{m})/(dx ^{m}) F(x)	f _{H(n+m)}
x (d ^{m})/(dx ^{m}) F(x)	nf _{H(n + m -1 ) + (1)/(2)} f _{H(n+m+1)}
e ^{x ^{2}} (d)/(dx) \left[e ^{- x ^{2}} (d)/(dx) F(x)\right]	-2 nf _{H(n)}
F(x - x _{0)}	\sqrt{\pi}\sum ^{inf ty} _{k =0} (( - x ^{k} _{0 )})/(k!) f _{H(n+k)}
F(x)*G(x)	\sqrt\pi( -1 ) ^{n\left[2} ^{2n+1} \Gamma \left(n + (3)/(2) \right)\right] ^{-1} f _{H(n)} g _{H(n)} ^[3]
e ^{z ^{2}} \sin(x z),	z	<\frac 12\ \,	\begin{cases} \sqrt\pi ( -1 ) ^{\lfloor (n)/(\rfloor 2)} ( 2 z) ^{n} , & n odd \\ 0 , & n even \end{cases}
( 1- z ^{2)} ^{-1/2} \exp\left[ (2 xyz - (x ^{2+y} ^{2)z} ^{2})/(( 1- z ^{2)}) \right]	\sqrt\pi z ^{n} H _{n(y)} ^[4] ^[5]
(H _{m(y)H} _{m+1} (x) - H _{m(x)H} _{m+1} (y))/(2 ^{m+1} m!(x - y))	\begin{cases}\sqrt{\pi}H _{n(y)} & n \leq m\\ 0 & n > m \end{cases}

Notes and References

1. Feldheim . Ervin . 1938 . Quelques nouvelles relations pour les polynomes d'Hermite . Journal of the London Mathematical Society . fr . s1-13 . 22–29 . 10.1112/jlms/s1-13.1.22.
2. Bailey . W. N. . On Hermite polynomials and associated Legendre functions . Journal of the London Mathematical Society . 4 . 1939 . 281–286 . s1-14 . 10.1112/jlms/s1-14.4.281.
3. Glaeske . Hans-Jürgen . On a convolution structure of a generalized Hermite transformation . Serdica Bulgariacae Mathematicae Publicationes . 9 . 2 . 1983 . 223–229 .
4. , 10.13 (22).
5. . See p. 174, eq. (18) and p. 173, eq. (13).