Manipulating Hypergeometric functions I have a differential equation:
$$(f\Phi')'-\frac{l(l+1)}{r^2}\Phi=0,$$
which I've solved with a program that yields 
$$\Phi(r)=C_1 (\frac{r}{R})^{2}{_2 }F_1 (1-l,2+l,3,\frac{r}{R}).$$ 
While this technically works, I want this to be in the following form:
$$\Phi_<(r)=C_1 (\frac{r}{r_0})^{l+1}_{} {_2 }F_1 (-l-1,-l+1,-2l,\frac{R}{r})$$ when $r<r_0$ 
and 
$$\Phi_>(r)=C_2 (\frac{r_0}{r})^{l}{_2 }F_1 (l,l+2,2l+2,\frac{R}{r})$$ when $r>r_0$ 
for some arbitrary distance $r_0 >R$. Also, $f=1-\frac{R}{r}.$
Both of the above functions satisfy my differential equation, so there must be a way to write each of these functions in terms of the one that my program found. Can someone give me a few pointers?
For a little extra context, my solution corresponds to the potential that some particle feels outside a black hole in Schwarzschild spacetime where $R$ is the location of the event horizon and $r_0$ is the instantaneous position of the particle.
 A: I think you want the connection formulae that link solutions about the three regular singular points to each other. There is a brief discussion starting on page 408 of my lecture notes:
http://courses.physics.illinois.edu/phys509/sp2017/bmaster.pdf
A: With a substitution $x\equiv r/R$ one can transform the ODE into a hypergeometric equation:
$$x(1-x)\phi''(x)-\phi'(x)+l(l+1)\phi(x)=0.$$
This corresponds to $(a,b,c)=(-1-l,l,-1)$ or equivalently $(l,-1-l,-1)$. The Hypergeometric ODE has two linear independent solutions $\phi_1$ and $\phi_2$.
For $|x|<1$ and $c\neq 0,1,2,\dots$:
$$\phi_1^<(x)= {}_2 F_1(a,b,c,x)$$
$$\phi_2^<(x)= x^{1-c}{}_2 F_1(a-c+1,b-c+1,2-c,x)$$
For $|x|>1$ and $a-b\neq 0,1,2,\dots$:
$$\phi_1^>(x)= x^{-a}{}_2 F_1(a,a-c+1,a-b+1,x^{-1})$$
$$\phi_2^>(x)= x^{-b}{}_2 F_1(b,b-c+1,b-a+1,x^{-1})$$
Plugging in the definition for $x$ and the values of $(a,b,c)$ one can get the solutions OP is looking for. I can recommend A. Jeffrey and H. Dai - 2008 - Handbook of mathematical formulars and integrals - 4th ed. The formulas I gave come from this book sec. 22.17.
