By hand, find the recurrence relation for a power series solution
\(H(\rho)\) of the equation:
\begin{equation}
\rho \frac{d^2 H}{d\rho^2} +(2\ell+2-\rho)\frac{dH}{d\rho}
+(\lambda-\ell-1) H=0
\end{equation}
where \(\ell\) is a known positive integer, and \(\lambda\) is an
unknown constant.
Suppose that you want a solution to (a) which is a polynomial of
degree 4. Assume that \(\ell=2\). What does that tell you about the
unknown constant \(\lambda\)?
Find the polynomial of degree 4 solution to the differential
equation in part (a) assuming \(\ell=2\). Assume anything you need to
about \(\lambda\).