In mathematics, Laguerre transform is an integral transform named after the mathematician Edmond Laguerre, which uses generalized Laguerre polynomials L n α ( x ) {\displaystyle L_{n}^{\alpha }(x)} as kernels of the transform. The Laguerre transform of a function f ( x ) {\displaystyle f(x)} is
L { f ( x ) } = f ~ α ( n ) = ∫ 0 ∞ e − x x α L n α ( x ) f ( x ) d x {\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
L − 1 { f ~ α ( n ) } = f ( x ) = ∑ n = 0 ∞ ( n + α n ) − 1 1 Γ ( α + 1 ) f ~ α ( n ) L n α ( x ) {\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)}
Some Laguerre transform pairs
References