Notes on algebraic functions - Guan Ke-Ying; Lei Jinzhi (International Journal of Mathematics and Mathematical Sciences)
Consideration of the monodromy group of the hypergeometric equation z(1−z)w″+[γ−(1+α+β)z]w′−αβw=0, in the case of α=1/6, β=5/6, γ=7/6, shows that the global hypergeometric function solution F(1/6;5/6;7/6;z) is nonalgebraic although it has only algebraic singularities. Therefore, the proposition given in [2,4] that a function is algebraic if it has only the algebraic singularities on the extended z-plane is not true. Through introduction of the concept of singular element criterion for deciding when a function is algebraic on the basis of properties of its singularities is given.