Chebyshev derivative Sturm-Liouville pair
= Chebyshev derivative Sturm-Liouville pair
{c}
The Chebyshev equation has self-adjoint form
$$
\left(\sqrt{1-x^2}\,T_n'\right)'
+\frac{n^2}{\sqrt{1-x^2}}T_n=0.
$$
For $U_n=T_n'$, differentiation gives
$$
\left((1-x^2)^{3/2}U_n'\right)'
+(n^2-1)\sqrt{1-x^2}\,U_n=0.
$$
The respective orthogonality weights are $(1-x^2)^{-1/2}$ and $(1-x^2)^{1/2}$.