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


 * $$L\{f(x)\} = \tilde f_\alpha(n) = \int_{0}^\infty e^{-x} x^\alpha \ L_n^\alpha(x)\ f(x) \ dx$$

The inverse Laguerre transform is given by


 * $$L^{-1}\{\tilde f_\alpha(n)\} = f(x) = \sum_{n=0}^\infty \binom{n+\alpha}{n}^{-1} \frac{1}{\Gamma(\alpha+1)} \tilde f_\alpha(n) L_n^\alpha(x)$$