= Laguerre derivative square identity
{c}
{title2=$\int_0^\infty t(L_n'-L_n)(L_m'-L_m)e^{-t}=(n+1)\int_0^\infty L_nL_me^{-t}$}
For <polynomial> solutions of the <Laguerre differential equation>, put $D_n=L_n'-L_n$. The equation gives $D_n+tD_n'=-(n+1)L_n$. Integrating $\int tD_n(L_m'-L_m)e^{-t}$ 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.
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