Hermite transform

In mathematics, Hermite transform is an integral transform named after the mathematician Charles Hermite, which uses Hermite polynomials [math]\displaystyle{ H_n(x) }[/math] as kernels of the transform. This was first introduced by Lokenath Debnath in 1964. The Hermite transform of a function [math]\displaystyle{ F(x) }[/math] is [math]\displaystyle{ H\{F(x)\} = f_H(n) = \int_{-\infty}^\infty e^{-x^2} \ H_n(x)\ F(x) \ dx }[/math]

The inverse Hermite transform is given by [math]\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) }[/math]

Some Hermite transform pairs

[math]\displaystyle{ F(x)\, }[/math]	[math]\displaystyle{ f_H(n)\, }[/math]
[math]\displaystyle{ x^m }[/math]	[math]\displaystyle{ \begin{cases} \frac{m!\sqrt{\pi} }{2^{m-n} \left(\frac{m-n}{2}\right)!}, & (m-n)\text{ even and} \geq0 \\ 0, & \text{otherwise} \end{cases} }[/math]
[math]\displaystyle{ e^{ax}\, }[/math]	[math]\displaystyle{ \sqrt\pi a^n e^{a^2/4}\, }[/math]
[math]\displaystyle{ e^{2xt-t^2}, \ |t|\lt \frac{1}{2}\, }[/math]	[math]\displaystyle{ \sqrt\pi (2t)^n }[/math]
[math]\displaystyle{ H_m(x)\, }[/math]	[math]\displaystyle{ \sqrt\pi 2^n n!\delta_{nm}\, }[/math]
[math]\displaystyle{ x^2H_m(x)\, }[/math]	[math]\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} }[/math]
[math]\displaystyle{ e^{-x^2}H_m(x)\, }[/math]	[math]\displaystyle{ \left(-1\right)^{p-m} 2^{p-1/2} \Gamma(p+1/2),\ m+n=2p,\ p\in\mathbb{Z} }[/math]
[math]\displaystyle{ H_m^2(x)\, }[/math]	[math]\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} }[/math]
[math]\displaystyle{ H_m(x)H_p(x)\, }[/math]	[math]\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}\, }[/math]
[math]\displaystyle{ H_{n+p+q}(x)H_p(x)H_q(x)\, }[/math]	[math]\displaystyle{ \sqrt\pi 2^{n+p+q} (n+p+q)!\, }[/math]
[math]\displaystyle{ \frac{d^m}{dx^m}F(x)\, }[/math]	[math]\displaystyle{ f_H(n+m)\, }[/math]
[math]\displaystyle{ x\frac{d^m}{dx^m}F(x)\, }[/math]	[math]\displaystyle{ nf_H(n+m-1)+\frac{1}{2}f_H(n+m+1)\, }[/math]
[math]\displaystyle{ e^{x^2}\frac{d}{dx}\left[e^{-x^2}\frac{d}{dx}F(x)\right]\, }[/math]	[math]\displaystyle{ -2nf_H(n)\, }[/math]
[math]\displaystyle{ F(x - x_0) }[/math]	[math]\displaystyle{ \sqrt{\pi}\sum^\infty_{k=0}\frac{(-x_0)^k}{k!}f_H(n+k) }[/math]
[math]\displaystyle{ F(x)*G(x)\, }[/math]	[math]\displaystyle{ \sqrt\pi(-1)^n\left[2^{2n+1}\Gamma \left(n+\frac{3}{2}\right)\right]^{-1}f_H(n) g_H(n)\, }[/math]
[math]\displaystyle{ e^{z^2} \sin(x z), \ |z|\lt \frac 12\ \, }[/math]	[math]\displaystyle{ \begin{cases} \sqrt\pi (-1)^{\lfloor\frac{n}{2}\rfloor}(2z)^{n} , & n\,\mathrm{odd}\\ 0 , & n\,\mathrm{even} \end{cases}\, }[/math]
[math]\displaystyle{ (1-z^2)^{-1/2} \exp\left[\frac{2xyz-(x^2+y^2)z^2}{(1-z^2)}\right]\, }[/math]	[math]\displaystyle{ \sqrt\pi z^n H_n(y) }[/math]
[math]\displaystyle{ \frac{H_m(y)H_{m+1}(x)-H_m(x)H_{m+1}(y)}{2^{m+1}m!(x-y)} }[/math]	[math]\displaystyle{ \begin{cases}\sqrt{\pi}H_n(y) & n \leq m\\ 0 & n \gt m \end{cases} }[/math]

Sources

• Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1955), Higher transcendental functions, II, McGraw-Hill, ISBN 978-0-07-019546-2, http://apps.nrbook.com/bateman/Vol2.pdf