Cannot be submitted as a comment. Your question should be adjusted in view of the output of the Maple 2017.3 command

    dsolve(z*(z-1)*((D@@2)(w))(z)+(2*z-1)*(D(w))(z)+2*w(z) = 0);

$$w \left( z \right) ={\it \_C1}\,
{\mbox{$_2$F$_1$}(1/2-i/2\sqrt {7},1/2-i/2\sqrt {7};\,-i\sqrt {7}+1;\,{z}^{-1})}
{z}^{-1/2+i/2\sqrt {7}}+{\it \_C2}\,
{\mbox{$_2$F$_1$}(1/2+i/2\sqrt {7},1/2+i/2\sqrt {7};\,i\sqrt {7}+1;\,{z}^{-1})}
{z}^{-1/2-i/2\sqrt {7}}
 $$
as well as the output of the Mathematica 11.2.0.0 command

    DSolve[z*(z - 1)*w''[z] + (2 z - 1)*w'[z] + 2*w[z] == 0, w[z], z]
$$\left\{\left\{w(z)\to c_1 P_{\frac{1}{2} i \left(i+\sqrt{7}\right)}(2 z-1)+c_2 Q_{\frac{1}{2} i \left(i+\sqrt{7}\right)}(2 z-1)\right\}\right\} $$