Expressing ${}_2F_1(a, b; c; z)^2$ as a single series Is there a way to express
$${}_2F_1\bigg(\frac{1}{12}, \frac{5}{12}; \frac{1}{2}; z\bigg)^2$$
as a single series a la Clausen? Note that Clausen's identity is not applicable here.

Using Maple, I get
$$
\sum_{n=0}^\infty \frac{\Gamma  \left( {\frac {7}{12}} \right) \Gamma  \left( {\frac {11}{12}}
 \right) 
{\mbox{$_4$F$_3$}(\frac{1}{12},{\frac {5}{12}},-n,\frac{1}{2}-n;\,1/2,-n+{\frac {7}{12}},-n+{\frac {11}{12}};\,1)\;(4z)^n }}
{16\,\Gamma  \left( -n+{\frac {11}{12}} \right) \Gamma  \left( -n+{
\frac {7}{12}} \right) \Gamma  \left( 2\,n+1 \right) \sin^2 \left( {\frac {5}{12}}\,\pi  \right) \sin^2 \left( \frac{1}{12}\,\pi  \right)}
\\
= 1+{\frac {5}{36}}z+{\frac {295}{3888}}{z}^{2}+{\frac {5525}{104976}}{
z}^{3}+{\frac {4281875}{105815808}}{z}^{4}+{\frac {564921305}{
17142160896}}{z}^{5}+O \left( {z}^{6} \right) 
$$
I don't know how much use that is.
