# Laguerre transform

1. Jun 23, 2007

### Klaus_Hoffmann

If we have the ODE introduced by Laguerre with n any Real number so

$$x\,y'' + (1 - x)\,y' + n\,y = 0\,$$

then i introduce the Laguerre transform (changing n from 0 to oo) in the form

$$g(n)=\int_{0}^{\infty}dx\ f(x) L_{n} (x)$$

$$f(x)=\int_{0}^{\infty}dn\frac{ g(n)}{\Gamma (n+1)^{2}} L_{n} (x)$$

where in the last integral the variable is n and you integrate over all the possible positive index Laguerre function for x fixed, (although with a bit of inmodesty i call them Hoffmann-Laguerre transform )