Finding a complicated recursion relation by Frobenius' method

Question

I am trying to find quasinormal modes of Kerr black holes, following https://www.edleaver.com/Misc/EdLeaver/Publications/AnalyticRepresentationForQuasinormalModesOfKerrBlackHoles.pdf. To do this, I need to solve a radial differential equation (eqn 17) by ansatz (eqn 23) to obtain a recursion relation (eqn 24), but I am failing and cant figure it out. I am trying to follow the examples on the wikipedia page on Frobenius method, but I think I am failing to understand how to deal with the fact that our ansatz power series is of form $\sum_n d_n (\frac{r-r_+}{r-r_-})^n$ instead of $\sum_n d_n z^n$. At the end of the procedure I am left with various sums of different powers of $(\frac{r-r_+}{r-r_-})$, but the problem is that they are also accompanied by various different powers of $(r-r_+)$ and so I am not able to group the terms. Perhaps substituting $z=(\frac{r-r_+}{r-r_-})$ should simplify, but doing it this way I still did not obtain sums of only powers of z although it does look a lot more promising. Any help would be much appreciated!