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, $f=1-\frac{R}{r}$, and a prime indicates differentiation with respect to 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?
The hypergeometric function $_2F_1(a,b,c,z)$ is solution of an hypergeometric differential equation which solutions are known on several forms :
(from http://mathworld.wolfram.com/HypergeometricFunction.html )
In the case of your result, with $\begin{cases}a=1-l \\b=2+l \\c=3 \\z=\frac{r}{R}\end{cases}$
$\Phi(r)=C_1 (\frac{r}{R})^{2} {_2 }F_1 (1-l,2+l,3,\frac{r}{R}) = C_1 (\frac{r}{R})^{2}\:_2F_1(a,b,c,z)$
Equation (38) gives another form of solution :
$ C'_1 (\frac{r}{R})^{2}z^{-a}\:_2F_1(a,a+1-c,a+1-b,z^{-1})= C'_1 (\frac{r}{R})^{2} (\frac{r}{R})^{-(1-l)} {_2 }F_1 (1-l,-1-l,-2l,\frac{R}{r}) = C'_1 (\frac{r}{R})^{l+1} {_2 }F_1 (1-l,-1-l,-2l,\frac{R}{r}) $
This corresponds to your first case, with different constant factors $\frac{C_1}{r_0^{l+1}}$ and $\frac{C'_1}{R^{l+1}}$. The relationship between the factors has to be determined according to some boundary conditions which are not clearly stated in the wording of the question (as far as I can understand).
As well, Equation (39) gives another form of solution :
$ C'_2 (\frac{r}{R})^{2}z^{-b}\:_2F_1(b+1-c,b,b+1-a,z^{-1})= C'_2 (\frac{r}{R})^{2} (\frac{r}{R})^{-(2+l)} {_2 }F_1 (l,2+l,2+2l,\frac{R}{r}) = C'_2 (\frac{R}{r})^{l} {_2 }F_1 (l,2+l,2+2l,\frac{R}{r}) $
This corresponds to your second case, with different constant factors $ C_2r_0^l$ and $C'_2R^l$ which relationship has to be determined according to some boundary conditions.