Laguerre derivative square identity (${\int }_0^{\infty }t(L_n^{\prime }-L_n)(L_m^{\prime }-L_m)e^{-t}=(n+1){\int }_0^{\infty }{L}_n{L}_m{e}^{-t}$)

For polynomial solutions of the Laguerre differential equation, put $D_n=L_n^{\prime }-L_n$. The equation gives $D_n+t{D}_n^{\prime }=-(n+1)L_n$. Integrating $\int t{D}_n(L_m^{\prime }-L_m)e^{-t}$ by parts gives the displayed identity; the factor $t$ kills the endpoint at zero and exponential decay kills the endpoint at infinity. Its left side is symmetric in $n,m$, so subtracting the swapped identities proves orthogonality for distinct degrees. This provides a direct energy proof of orthogonality for the Laguerre polynomials without assuming their norm formula.