This question is related to a previous question of mine.
A quick visit to the Wolfram Functions site reveals a rather extensive list of reduction formulae for the hypergeometric function ${_2F_1}(a,b;c;z)$ when $a,b,c$ are rational numbers. I am curious about how these reduction formulae are derived and if there is a general procedure for finding them?
It was rather interesting that the link above includes reduction formulae for rational parameters when the parameters have denominators $1,2,3,4,5,6$, and $8$ but not $7$. Is there an interesting reason for this besides the list simply being incomplete?
I know of one trick for the case where $c=b+1$, which takes advantage of the differential formula $$ (z\partial_z+\beta_k-1){}_pF_q\left( \begin{array}{c}\alpha_1,\ldots,\alpha_p \\ \beta_1,\ldots,\beta_k,\ldots,\beta_q\end{array};z\right) =\left(\beta_k-1\right) {}_pF_q\left( \begin{array}{c}\alpha_1,\ldots,\alpha_p \\ \beta_1,\ldots,\beta_k-1,\ldots,\beta_q\end{array};z\right). \tag{1} $$
Take as an example $y(z)={_2F_1}(1,5/4;2;z)$. Then using $(1)$ we can derive the ODE $$ (z\partial_z+1)y=(1-z)^{-5/4}. $$ Coupling this equation with the initial condition $y(0)=1$ and the product rule for derivatives gives the simple result $$ \partial_z(zy)=(1-z)^{-5/4},\quad y(0)=1, $$ which is easily solved by integrating and using the initial condition to determine the constant of integration. Doing so yields $$ {_2F_1}(1,5/4;2;z)=\frac{4}{z}((1-z)^{-1/4}-1). $$
Of course, this is a very specialized case of the general approach I am interested in. If a general procedure for arbitrary rational parameters does not exist, I would also be interested in procedures for families of parameters, e.g. a procedure for the case where all parameters have denominator of $2$. Any references are also greatly appreciated.
