I have an ODE of the form $$f^2 p^{\prime \prime}(r) + ff^{\prime}p^{\prime}(r) + V(r)p(r) = 0$$ where $f(r) = (1 - \frac{2M}{r} + \frac{Q^2}{r^2}) = \frac{1}{r^2}(r^2 - 2Mr + Q^2) = \frac{(r-r_+)(r-r_-)}{r^2}$ where $r_+,r_-$ are roots of $f$ and $V(r) = (\omega - \frac{qQ}{r})^2 - f(\frac{l(l+1)}{r^2} + \frac{f^{\prime}}{r})$. It arises from a problem in GR.
If I bring it to standard form, ($p^{\prime \prime}(r) + \frac{f^{\prime}}{f}p^{\prime}(r) + \frac{V}{f^2}p(r) = 0$), I am able to show that $$\lim_{r \to r_+} (r - r_+)\frac{f^{\prime}}{f}$$ and $$\lim_{r \to r_+} (r - r_+)^2\frac{V(r)}{f^2}$$ are both finite and well defined. Therefore, the point is $r_+$ is a regular singular point (I hope this term is correct), and by Fuch's theorem I can have a solution of the form: $$p(r) = \sum_n a_n(r - r_+)^{n+s}$$ However, when I use this series and compute the derivatives of $p$ and substitute them into the ODE above, I seem to get the following:
$$\sum_{n = 0}a_n[ (n+s)(n+s-1)(r - r_+)^{n+s-2} + \frac{2(Mr-Q^2)}{r(r-r_-)}(n+s)(r - r_+)^{n+s-2} + \frac{r^4V}{(r-r_-)^2}(r - r_+)^{n+s-2}] = 0$$
However, I am unsure how to proceed from here. It has been a while since I have used this method, so I am a little rusty but I do not think I have made an error regarding the validity/application of the method. Further, the reason I think this is wrong is because if I attempt to write down an indicial equation from this, I will have $r$ dependent terms, unless I group all the $r$ terms before I decide the degree of each term. If I do that I am unsure how to rename indices with terms of $(r-r_+)$ and $(r-r_-)$ and $r$ all giving me some final degree to the term.