Laguerre transform

In mathematics, Laguerre transform is an integral transform named after the mathematician Edmond Laguerre, which uses generalized Laguerre polynomials $L_n^{\alpha }(x)$ as kernels of the transform.

The Laguerre transform of a function $f(x)$ is

$L\{f(x)\}={\tilde{f}}_{\alpha }(n)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-x}{x}^{\alpha }{L}_n^{\alpha }(x)f(x)dx$

The inverse Laguerre transform is given by

$L^{-1}\{{\tilde{f}}_{\alpha }(n)\}=f(x)=\sum _{n=0}^{\mathrm{\infty }}{(\genfrac{}{}{0ex}{}{n+\alpha }{n})}^{-1}\frac{1}{\mathrm{\Gamma }(\alpha +1)}{\tilde{f}}_{\alpha }(n)L_n^{\alpha }(x)$

Some Laguerre transform pairs

$f(x)$	${\tilde{f}}_{\alpha }(n)$
$x^{a-1},a>0$	$\frac{\mathrm{\Gamma }(a+\alpha )\mathrm{\Gamma }(n-a+1)}{n!\mathrm{\Gamma }(1-a)}$
$e^{-ax},a>-1$	$\frac{\mathrm{\Gamma }(n+\alpha +1)a^n}{n!(a+1{)}^{n+\alpha +1}}$
$\sin ax,a>0,\alpha =0$	$\frac{a^n}{(1+a^2{)}^{\frac{n+1}{2}}}\sin [n{\tan }^{-1}\frac{1}{a}+{\tan }^{-1}(-a)]$
$\cos ax,a>0,\alpha =0$	$\frac{a^n}{(1+a^2{)}^{\frac{n+1}{2}}}\cos [n{\tan }^{-1}\frac{1}{a}+{\tan }^{-1}(-a)]$
$L_m^{\alpha }(x)$	$(\genfrac{}{}{0ex}{}{n+\alpha }{n})\mathrm{\Gamma }(\alpha +1){\delta }_{mn}$
$e^{-ax}{L}_m^{\alpha }(x)$	$\frac{\mathrm{\Gamma }(n+\alpha +1)\mathrm{\Gamma }(m+\alpha +1)}{n!m!\mathrm{\Gamma }(\alpha +1)}\frac{(a-1{)}^{n-m+\alpha +1}}{a^{n+m+2\alpha +2}}{}_2{F}_1(n+\alpha +1;\frac{m+\alpha +1}{\alpha +1};\frac{1}{a^2})$
$f(x)x^{\beta -\alpha }$	$\sum _{m=0}^n(m!{)}^{-1}(\alpha -\beta {)}_m{L}_{n-m}^{\beta }(x)$
$e^x{x}^{-\alpha }\mathrm{\Gamma }(\alpha ,x)$	$\sum _{n=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{n+\alpha }{n})\frac{\mathrm{\Gamma }(\alpha +1)}{n+1}$
$x^{\beta },\beta >0$	$\mathrm{\Gamma }(\alpha +\beta +1)\sum _{n=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{n+\alpha }{n})(-\beta {)}_n\frac{\mathrm{\Gamma }(\alpha +1)}{\mathrm{\Gamma }(n+\alpha +1)}$
$(1-z{)}^{-(\alpha +1)}\exp \left(\frac{xz}{z-1}\right),|z|<1,\alpha \ge 0$	$\sum _{n=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{n+\alpha }{n})\mathrm{\Gamma }(\alpha +1)z^n$
$(xz{)}^{-\alpha /2}{e}^z{J}_{\alpha }[2(xz{)}^{1/2}],|z|<1,\alpha \ge 0$	$\sum _{n=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{n+\alpha }{n})\frac{\mathrm{\Gamma }(\alpha +1)}{\mathrm{\Gamma }(n+\alpha +1)}{z}^n$
$\frac{d}{dx}f(x)$	${\tilde{f}}_{\alpha }(n)-\alpha \sum _{k=0}^n{\tilde{f}}_{\alpha -1}(k)+\sum _{k=0}^{n-1}{\tilde{f}}_{\alpha }(k)$
$x\frac{d}{dx}f(x),\alpha =0$	$-(n+1){\tilde{f}}_0(n+1)+n{\tilde{f}}_0(n)$
$\underset{0}{\overset{x}{\int }}f(t)dt,\alpha =0$	${\tilde{f}}_0(n)-{\tilde{f}}_0(n-1)$
$e^x{x}^{-\alpha }\frac{d}{dx}\left[e^{-x}{x}^{\alpha +1}\frac{d}{dx}\right]f(x)$	$-n{\tilde{f}}_{\alpha }(n)$
${\left\{e^x{x}^{-\alpha }\frac{d}{dx}\left[e^{-x}{x}^{\alpha +1}\frac{d}{dx}\right]\right\}}^{k}f(x)$	$(-1{)}^k{n}^k{\tilde{f}}_{\alpha }(n)$
$L_n^{\alpha }(x),\alpha >-1$	$\frac{\mathrm{\Gamma }(n+\alpha +1)}{n!}$
$x{L}_n^{\alpha }(x),\alpha >-1$	$\frac{\mathrm{\Gamma }(n+\alpha +1)}{n!}(2n+1+\alpha )$
$\frac{1}{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t}f(t)dt\underset{0}{\overset{\pi }{\int }}{e}^{\sqrt{xt}\cos \theta }\cos (\sqrt{xt}\sin \theta )g(x+t-2\sqrt{xt}\cos \theta )d\theta ,\alpha =0$	${\tilde{f}}_0(n){\tilde{g}}_0(n)$
$\frac{\mathrm{\Gamma }(n+\alpha +1)}{\sqrt{\pi }\mathrm{\Gamma }(n+1)}\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t}{t}^{\alpha }f(t)dt\underset{0}{\overset{\pi }{\int }}{e}^{-\sqrt{xt}\cos \theta }{\sin }^{2\alpha }\theta g(x+t+2\sqrt{xt}\cos \theta )\frac{J_{\alpha -1/2}(\sqrt{xt}\sin \theta )}{[(\sqrt{xt}\sin \theta )/2{]}^{\alpha -1/2}}d\theta$	${\tilde{f}}_{\alpha }(n){\tilde{g}}_{\alpha }(n)$