Hermite transform

In mathematics, Hermite transform is an integral transform named after the mathematician Charles Hermite, which uses Hermite polynomials $H_{n}(x)$ as kernels of the transform. This was first introduced by Lokenath Debnath in 1964.[1][2][3][4]

The Hermite transform of a function $F(x)$ is

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

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

$F(x)\,$	$f_{H}(n)\,$
$x^{m}$	${\begin{cases}{\frac {m!{\sqrt {\pi }}}{2^{m-n}\left({\frac {m-n}{2}}\right)!}},&(m-n){\text{ even and}}\geq 0\\0,&{\text{otherwise}}\end{cases}}$ [5]
$e^{ax}\,$	${\sqrt {\pi }}a^{n}e^{a^{2}/4}\,$
$e^{2xt-t^{2}},\ \|t\|<{\frac {1}{2}}\,$	${\sqrt {\pi }}(2t)^{n}$
$H_{m}(x)\,$	${\sqrt {\pi }}2^{n}n!\delta _{nm}\,$
$x^{2}H_{m}(x)\,$	$2^{n}n!{\sqrt {\pi }}{\begin{cases}1,&n=m+2\\\left(n+{\frac {1}{2}}\right),&n=m\\(n+1)(n+2),&n=m-2\\0,&{\text{otherwise}}\end{cases}}$
$e^{-x^{2}}H_{m}(x)\,$	$\left(-1\right)^{p-m}2^{p-1/2}\Gamma (p+1/2),\ m+n=2p,\ p\in \mathbb {Z}$
$H_{m}^{2}(x)\,$	${\begin{cases}2^{m+n/2}{\sqrt {\pi }}{\binom {m}{n/2}}{\frac {m!n!}{(n/2)!}},&n{\text{ even and}}\leq 2m\\0,&{\text{otherwise}}\end{cases}}$ [6]
$H_{m}(x)H_{p}(x)\,$	${\begin{cases}{\frac {2^{k}{\sqrt {\pi }}m!n!p!}{(k-m)!(k-n)!(k-p)!}},&m+n+p=2k,\ k\geq m,n,p,\ k\in \mathbb {Z} \\0,&{\text{otherwise}}\end{cases}}\,$ [7]
$H_{n+p+q}(x)H_{p}(x)H_{q}(x)\,$	${\sqrt {\pi }}2^{n+p+q}(n+p+q)!\,$
${\frac {d^{m}}{dx^{m}}}F(x)\,$	$f_{H}(n+m)\,$
$x{\frac {d^{m}}{dx^{m}}}F(x)\,$	$nf_{H}(n+m-1)+{\frac {1}{2}}f_{H}(n+m+1)\,$
$e^{x^{2}}{\frac {d}{dx}}\left[e^{-x^{2}}{\frac {d}{dx}}F(x)\right]\,$	$-2nf_{H}(n)\,$
$F(x)*G(x)\,$	${\sqrt {\pi }}(-1)^{n}\left[2^{2n+1}\Gamma \left(n+{\frac {3}{2}}\right)\right]^{-1}f_{H}(n)g_{H}(n)\,$ [8]
$e^{z^{2}}\sin({\sqrt {2}}xz),\ \|2z\|<1\,$	${\begin{cases}{\sqrt {\pi }}\sum _{m=0}^{\infty }(-1)^{m}(2z)^{2m+1},&n=2m+1\\0,&n\neq 2m+1\end{cases}}\,$
$(1-z^{2})^{-1/2}\exp \left[{\frac {2xyz-(x^{2}+y^{2})z^{2}}{(1-z^{2})}}\right]\,$	${\sqrt {\pi }}\sum _{m=0}^{\infty }z^{m}H_{m}(y)\delta _{nm}\,$

References

Debnath, L. (1964). "On Hermite transform". Matematički Vesnik. 1 (30): 285–292.
Debnath; Lokenath; Bhatta, Dambaru (2014). Integral transforms and their applications. CRC Press. ISBN 9781482223576.
Debnath, L. (1968). "Some operational properties of Hermite transform". Matematički Vesnik. 5 (43): 29–36.
Dimovski, I. H.; Kalla, S. L. (1988). "Convolution for Hermite transforms". Math. Japonica. 33: 345–351.
McCully, Joseph Courtney; Churchill, Ruel Vance (1953). "Hermite and Laguerre integral transforms : preliminary report". {{cite journal}}: Cite journal requires |journal= (help)
Feldheim, Ervin (1938). "Quelques nouvelles relations pour les polynomes d'Hermite". Journal of the London Mathematical Society (in French). s1-13: 22–29. doi:10.1112/jlms/s1-13.1.22.
Bailey, W. N. (1939). "On Hermite polynomials and associated Legendre functions". Journal of the London Mathematical Society. s1-14 (4): 281–286. doi:10.1112/jlms/s1-14.4.281.
Glaeske, Hans-Jürgen (1983). "On a convolution structure of a generalized Hermite transformation" (PDF). Serdica Bulgariacae Mathematicae Publicationes. 9 (2): 223–229.

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[1] Debnath, L. (1964). "On Hermite transform". Matematički Vesnik. 1 (30): 285–292.

[2] Debnath; Lokenath; Bhatta, Dambaru (2014). Integral transforms and their applications. CRC Press. ISBN 9781482223576.

[3] Debnath, L. (1968). "Some operational properties of Hermite transform". Matematički Vesnik. 5 (43): 29–36.

[4] Dimovski, I. H.; Kalla, S. L. (1988). "Convolution for Hermite transforms". Math. Japonica. 33: 345–351.

[5] McCully, Joseph Courtney; Churchill, Ruel Vance (1953). "Hermite and Laguerre integral transforms : preliminary report". {{cite journal}}: Cite journal requires |journal= (help)

[6] Feldheim, Ervin (1938). "Quelques nouvelles relations pour les polynomes d'Hermite". Journal of the London Mathematical Society (in French). s1-13: 22–29. doi:10.1112/jlms/s1-13.1.22.

[7] Bailey, W. N. (1939). "On Hermite polynomials and associated Legendre functions". Journal of the London Mathematical Society. s1-14 (4): 281–286. doi:10.1112/jlms/s1-14.4.281.

[8] Glaeske, Hans-Jürgen (1983). "On a convolution structure of a generalized Hermite transformation" (PDF). Serdica Bulgariacae Mathematicae Publicationes. 9 (2): 223–229.