For continuous indices 'n' and 'm' is it possible to have
$$ \int_{0}^{\infty}dx e^{-x}L_{n}(x)L_{m}(x) = \delta (n-m)? $$
Another question: let $g(m)$ be the function defined via the transform
$$ \int_{0}^{\infty}dx e^{-x}f(x)L_{m}(x)=g(m). $$
Is its inverse given by $$ \int_{0}^{\infty}dm e^{-m}g(m)L_{m}(x)=f(x), $$
so that $$ \int_{0}^{\infty}e^{-m}dm\int_{0}^{\infty}dx e^{-x}L_{n}(x)L_{m}(x) =1? $$
to be more explicit , let be the integral transform
$$ g(m)= \int_{0}^{\infty}dxe^{-x}f(x)L_{m}(x) $$ , then how could we obtain $ f(x) $ from $ g(m) $ ??