= Weighted orthogonality of Legendre polynomial derivatives
{title2=$\int_{-1}^1(1-x^2)P_n'(x)P_m'(x)\,dx=0\quad(n\ne m)$}
Differentiating the <Legendre differential equation> gives $((1-x^2)^2R_n')'+(n-1)(n+2)(1-x^2)R_n=0$ for $R_n=P_n'$. Multiply the equations for two distinct indices by the opposite polynomial and subtract. The endpoint term vanishes because of $(1-x^2)^2$, giving the displayed weighted <orthogonality>. The zero polynomial $R_0$ is not a nonzero eigenfunction.
Back to article page
