I am trying to reduce this sum of these hypergeometric functions $${{}_2{F_1}\left( {\frac{1}{2}, - \frac{1}{3};\frac{1}{6};x} \right) + {}_2{F_1}\left( { - \frac{1}{2},\frac{1}{3};\frac{1}{6};x} \right) + {}_2{F_1}\left( { - \frac{1}{2}, - \frac{1}{3};\frac{1}{6};x} \right)}$$
I tried : $$\eqalign{ & 1.{}_2{F_1}\left( { - \frac{1}{2}, - \frac{1}{3};\frac{1}{6};x} \right)\overbrace = ^{{\text{Euler transformation}}}\left( {1 - x} \right){}_2{F_1}\left( {\frac{2}{3},\frac{1}{2};\frac{1}{6};x} \right) \cr & 2.{}_2{F_1}\left( {\frac{1}{2}, - \frac{1}{3};\frac{1}{6};x} \right) - \frac{2}{3}{}_2{F_1}\left( { - \frac{1}{2},\frac{2}{3};\frac{1}{6};x} \right) = - \frac{1}{3}x{\mkern 1mu} {}_2{F_1}\left( {\frac{1}{2},\frac{2}{3};\frac{1}{6};x} \right) + \frac{1}{3}{\mkern 1mu} {}_2{F_1}\left( {\frac{1}{2},\frac{2}{3};\frac{1}{6};x} \right) \cr & 3.{}_2{F_1}\left( {\frac{1}{2},\frac{1}{3};\frac{1}{6};x} \right) - {}_2{F_1}\left( { - \frac{1}{2},\frac{4}{3};\frac{1}{6};x} \right) = 5x{}_2{F_1}\left( {\frac{1}{2},\frac{4}{3};\frac{7}{6};x} \right) \cr} $$
But it seems to become harder. I notice that $\frac{1}{2},\frac{1}{3},\frac{1}{6}$ has a relation : $\frac{1}{2} = \frac{1}{6} + \frac{1}{3}$ and from this we have many derivations. Can I ask there is a method to reduce this sum? Thank you for your help and your time.