# Proving Clausen hypergeometric identity

How do I show the Clausen identity $${}_2F_1\left(a, b; a+b+\frac{1}{2}; z\right)^2 = {}_3F_2\left(2a, 2b, a+b; a+b+\frac{1}{2}, 2a+2b, z\right)?$$ I saw this on MathWorld but am unsure how to progress. I tried the Cauchy product but that was no success. I am thinking maybe Apell functions due to the order on the left hand side, but am not entirely sure. A hint would be very much appreciated. Thank you!

Let $$y_1(z):={}_2F_1\left(a, b; a+b+\frac{1}{2}; z\right)^2$$ and $$y_2(z):= {}_3F_2\left(2a, 2b, a+b; a+b+\frac{1}{2}, 2a+2b, z\right)$$ You can verify that $$y_1$$ and $$y_2$$ both satisfy the following differential equation: $$z^2(z-1)y''' -3z \left (a+b+\frac{1}{2} - (a+b+1)z \right) y'' + ((2(a^2+b^2+4ab)+3(a+b)+1)z-(a+b)(2a+2b+1))y'+4ab(a+b)y=0$$ Then, you just need to find some point $$z_0$$ for which: $$y_1(z_0)=y_2(z_0); \, y_1'(z_0)=y_2'(z_0)$$ It is well known that there exists a unique solution $$y$$ to a liner ODE which satisfies $$y(z_0)=x_1$$, $$y'(z_0)=x_2$$ for some $$x_1$$, $$x_2$$. Consequently, $$y_1$$ and $$y_2$$ must be the same, and Clausen's identity follows.