= Solution
The <Sturm-Liouville eigenfunction expansion> of a suitable real function $f$ is
$$
f(x)=\sum_ny_n(x)\int_a^bw(\xi)y_n(\xi)f(\xi)\,d\xi=\int_a^b\left[w(\xi)\sum_ny_n(\xi)y_n(x)\right]f(\xi)\,d\xi.
$$
Comparing this identity with the sampling property of the <Dirac delta distribution> gives the completeness kernel
$$
\boxed{w(\xi)\sum_ny_n(\xi)y_n(x)=\delta(x-\xi).}
$$
This is an identity of <distributions>, rather than a pointwise-convergent series of ordinary functions.
Assuming zero is not an <eigenvalue>, define the <spectral Green function of a regular Sturm-Liouville problem> by
$$
\boxed{G(x;\xi)=\sum_n\frac{y_n(x)y_n(\xi)}{\lambda_n}.}
$$
Applying $\mathcal L_x$ gives $w(x)\sum_ny_n(x)y_n(\xi)=\delta(x-\xi)$: multiplying the preceding completeness kernel by $w(x)/w(\xi)$ leaves the <Dirac delta distribution> unchanged. Each term also obeys the same homogeneous <boundary conditions>. There is no extra weight factor in $G$ for the ordinary source measure $d\xi$.
If zero is an <eigenvalue>, the inverse <Green's function> does not exist on all forcing functions: the forcing must obey the <solvability condition at a Sturm-Liouville eigenvalue>. Omitting the zero mode instead constructs a reduced inverse, with its projection removed from the right-hand side. This exception occurs in part (c) at $\alpha=1$.
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