# Does a general procedure exist for reducing ${_2F_1}(a,b;c;z)$ when $a,b,c\in\Bbb Q$?

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.

A general procedure for arbitrary rational parameters does not exist. The values of parameters for which hypergeometric function $$_2F_1$$ becomes algebraic are known and classified by the Schwarz table. The idea behind this list is that the monodromy group of the relevant hypergeometric equation should be finite.