The differential equation is $$\frac{d^2g}{dz^2}-\frac{l(l+1)}{z(z-1)}g=0.$$
After using the Frobenius method I have gotten to indicial roots $r=1,0$. The recursion formula for $r=0$ is $$n(n+1)c_{n+1} = [n(n-1)-l(l+1)]c_n.$$
And for $r = 1$ the recursion is $$(n+1)(n+2)c_{n+1}=[k(k+1)-l(l+1)]c_n.$$
Now my problem is that I cannot obtain the first independent solution whose first few terms are $$u_0=z, u_1=z(1-z), u_2=z(z-1)(2z-1), \dots,$$
where $$g_1(z)=\sum_{n=0}^{\infty}c_n^l u_l(z).$$
I will be very grateful to anyone that can explain how to obtain the above solution.
An alternative method to solve the equation is shown below. This involves the $_2\text{F}_1$ hypergeometric function : http://mathworld.wolfram.com/HypergeometricFunction.html