Laguerre transform

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

The Laguerre transform of a function ${\displaystyle f(x)}$ is

${\displaystyle L\{f(x)\}={\tilde {f}}_{\alpha }(n)=\int _{0}^{\infty }e^{-x}x^{\alpha }\ L_{n}^{\alpha }(x)\ f(x)\ dx}$

The inverse Laguerre transform is given by

${\displaystyle L^{-1}\{{\tilde {f}}_{\alpha }(n)\}=f(x)=\sum _{n=0}^{\infty }{\binom {n+\alpha }{n}}^{-1}{\frac {1}{\Gamma (\alpha +1)}}{\tilde {f}}_{\alpha }(n)L_{n}^{\alpha }(x)}$