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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.

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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.

There is a little subtlety here: Schwarz table describes cases where both solutions of the relevant hypergeometric equation are algebraic. This situation is, in a sense, "generic". The remaining "pathological" cases where only one solution is algebraic correspond to infinite but reducible monodromy (these solutions are not only algebraic but in fact rational, and are essentially given by the Jacobi polynomials).

It is thus a relatively easy task to check whether a given hypergeometric function can be reduced to an algebraic expression. Constructing this expression explicitly is also straightforward but more involved. You may check this answer for details and useful references.

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