I have been trying to find a solution on any of these Fuchsian differential equations...
$v''(z)+\frac{A_1+B_1z }{z (z-1) }v'(z)+\frac{C_1+ D_1 z}{z^2 (z-1)^2 }v(z)=0$
$u''(z)+\frac{A_0+B_0 z^2 }{z(z-1) (z+1)}u'(z)+\frac{C_0+D_0 z^2 }{z^2(z-1)^2 (z+1)^2}u(z)=0$
$A_0,A_1,B_0,B_1,C_0,C_1,D_0,D_1$ $\in \mathbb{C}$ and aren't integers.
I would say that the first one looks exactly as the Fuchsian form of the Hypergeometric differential equation, since the singular points it has, are in $0$,$1$ and $\infty$
The second one seems more complicated to me. Based on what I have read, it should be possible to express it as the Heun differential equation, but I do not really understand how to do that. I understand that this form is one of the most "canonical" since the regular points are in $0$,$1$, $-1$ and $\infty$
How could I try to solve this 2 differential equations? And which one could be easier to solve?
Also, I wonder, I tried to solve these differential equations using Mathematica. And it gave results for both in terms of the Hypergeometric functions. So I guess the solutions of the Heun differential equation (The heun functions )can be expressed as Hypergeometric functions as well, Is there any advantage of doing so?
Could you please recommend me an introductory book on this topic? I recently started in this topic, and some books are kind of much advanced for a newbie.
Something that I have been using as reference: https://www.ams.org/journals/mcom/2007-76-258/S0025-5718-06-01939-9/S0025-5718-06-01939-9.pdf
Thanks in advance, for any comments/suggestions/ideas!