Show that the differential equation for \(F\) given in the text can also be written as

\[\begin{equation*}\xi \frac{d^{2} F}{d \xi^{2}}+\frac{d F}{d \xi}+\lambda^{2} \xi\left(1-\xi^{2}\right) F=0 \tag{12.44}\end{equation*}\]

Show that this belongs to a class of Sturm-Liouville problem. Verify that the \(F\) functions are orthogonal with a weighting factor of \(\xi\left(1-\xi^{2}\right)\). This permits the calculation of the series coefficients by the integrals shown in the text.

The solution can be expressed in terms of the hypergeometric functions shown in the text, but some transformations are needed in order to reduce it to a standard Kummer equation. Mathematically motivated students may wish to pursue all the details leading to the hypergeometric functions.