Symmetries of coefficients of three-term relations for the hypergeometric series (Q6137339)

In the footsteps of Vidunas, the purpose of this paper is to study symmetries of hypergeometric series by considering three-term relations of the form \[{_2}F_{1}\left(\begin{array}{c} a +k; b +l \\ c+m \end{array}\middle\vert x \right)= Q\cdot\ {_2}F_{1}\left(\begin{array}{c} a +1; b +1 \\ c+1 \end{array}\middle\vert x \right)+R\ {_2}F_{1}\left(\begin{array}{c} a ; b \\ c \end{array}\middle\vert x \right) ,\tag{He27} \] where \(Q,R\) are rational functions of \(a , b ,c\), and \(x\) is uniquely determined. The proofs use group action on a set, group operations on matrices, semidirect proct of groups, hypergeometric differential equation and relations between \(R\) and \(Q\) are proved.