\[\int_{0}^{x}L_{m}\left(y\right)L_{n}\left(x-y\right)\,\mathrm{d}y=\int_{0}^{x}%
L_{m+n}\left(y\right)\,\mathrm{d}y=L_{m+n}\left(x\right)-L_{m+n+1}\left(x%
\right).\]