Hypergeometric differential equation for $c=1$ and $a+b+1=0$ I would like to find the base of solutions for the following differential equation,
$$z(1-z)f''(z)+f'(z)+ \alpha \cdot f(z)=0$$
where $\alpha$ is a parameter and the prime indicates derivative w.r.t z.  The above equation is the hypergeometric one say, $$z(1-z)f''(z)+(c-z(a+b+1))f'(z) -ab\cdot f(z)=0$$ in the case with $c=1$ and $a+b+1=0$. BUT it is not clear at all what are the two independent solutions. One has to be the hypergeometric but  the other one I do not know beacuse of the degeneracy of the parameters [ $c=1$ and $a+b+1=0$].
Many thanks!!!
 A: WolframAlpha gives two solutions (link). One of them is the hypergeometric function $f(z)={_2}F_1(a,b;c;z)$ that you expected, with
$$a,b=-\frac12 \pm \frac12\sqrt{4\alpha+1},\quad c=1$$
(The order of $a,b$ is irrelevant.) The second is expressed in terms of the MeijerG function. If I use Mathematica and force it to expand the MeijerG function (via the FunctionExpand command) I obtain
$$f(z)=\frac12 (z-1)^2 {_2}F_1(1+a,1+b;3;1-z)$$
where $a,b$ are chosen as above. However, Mathematica indicates that this representation is only valid for $|z|<1$. For $|z|>1$, it instead concludes that the relevant MeijerG function is identically zero. I take this to mean that a different second solution is valid outside the unit circle. As such, the utility of this basis will depend on where you're trying to solve $f(z)$.
A: If you want just some expression in terms of special functions for the general solution, then you can use as two independent solutions
$$f_\pm(z)= z^{\frac12\pm\beta}{}_2F_1\left(-\tfrac12\mp\beta,-\tfrac12\mp\beta,1\mp2\beta,z^{-1}\right), $$
where $\alpha=\beta^2-\tfrac14$.
Explanation: For hypergeometric equation, the singular points $z=0,1,\infty$ play similar roles. Your parameters are "pathological" for the bases "adapted" to $z=0$ and $z=1$ but for the basis "adapted" to $z=\infty$ they are okay.
On the other hand, if you do want the bases adapted to the description of local behavior of solutions at $z=0$ or $z=1$, you may find them in the book of Abramowitz-Stegun, formulas 15.5.16-15.5.17; or equation 15.10.8 in DLMF.
