For $\alpha,\beta$, nonpositive integers at least one of which is non-zero, define $\omega\colon (-1,1)\to \mathbb{R}$ as $\omega(x) = (1-x)^\alpha(1+x)^\beta$. Then $\omega$ blows at at $x=-1$ or $x=1$ or both. I am interested in knowing more about the following Sturm–Liouville problem.
Find a all pairs of functions $u\colon [-1,1]\to \mathbb{R}$ and numbers $\lambda$ such that
$$ 0= \frac{\mathrm{d}}{\mathrm{d}x} \left((1-x^2)\omega(x) u'(x) \right) + \lambda \omega(x) u(x) $$ and boundary conditions $$0= \left(\frac{\mathrm{d}}{\mathrm{d}x}\right)^nu(1),$$ for $n=0,\ldots, |\alpha|-1$ and
$$0= \left(\frac{\mathrm{d}}{\mathrm{d}x}\right)^nu(-1),$$ for $n=0,\ldots, |\beta|-1$.
This isn't a ``regular'' Sturm-Liouville problem according to Wikipedia because the weight function isn't defined on the closed interval, and we don't have correct number of boundary conditions.
However, this eigenvalue problem actually is solved by the Jacobi polynomials $u=P_n^{(\alpha,\beta)}, \lambda= n(n+\alpha+\beta+1),$ with integer $n\ge |\alpha| + |\beta|$. I confirmed this with Mathematica, and this is actually where the problem comes from.
So my question is where can I learn more about Sturm-Liouville-like problems with singular weight functions and different boundary conditions. I am especially interested in knowing what sort of inner product "falls out" of this problem.
I appreciate any pointers to references where this Jacobi differential equation is handled.