= Regular solution of a forced Chebyshev equation
{title2=$[(1-x^2)D^2-xD+\gamma^2]^{-1}F=\sum_{n\geq0}\frac{F_nT_n(x)}{\gamma^2-n^2}$}
For noninteger real $\gamma$, expand a smooth forcing as $F=\sum_{n\geq0}F_nT_n$ in <Chebyshev polynomials>. Because $(1-x^2)T_n''-xT_n'=-n^2T_n$, division of each coefficient by $\gamma^2-n^2$ gives the regular solution. The coefficients use the <weighted inner product> with weight $(1-x^2)^{-1/2}$. Choosing the same endpoint domain as the regular <eigenfunctions> of the <Sturm-Liouville problem> is essential: finite values alone also allow homogeneous solutions $\cos(\gamma\arccos x)$ and $\sin(\gamma\arccos x)$, which generally have divergent endpoint derivatives. Setting $x=\cos\theta$ converts the regular endpoint condition into the <Neumann boundary conditions> $Z_\theta(0)=Z_\theta(\pi)=0$. The homogeneous equation then has no nonzero solution when $\gamma$ is noninteger, proving uniqueness in this domain.
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