global characterization of hypergeometric function

Riemann noted that the hypergeometric function can be characterizedby its global properties, without reference to power series, differentialequations, or any other sort of explicit expression. His characterizationis conveniently restated in terms of sheaves:

Suppose that we have a sheaf of holomorphic functions over $ℂ\setminus \{0,1\}$ which satisfy the following properties:

• It is closed under analytic continuation.

• It is closed under taking linear combinations.

• The space of function elements over any open set is two dimensional.

• There exists a neighborhood $D_0$ such that $0\in D)$, holomorphicfunctions ${\varphi }_0,{\psi }_0$ defined on $D_0$, and complex numbers ${\alpha }_0,{\beta }_0$ such that, for an open set of $d_0$ not containing $0$, it happens that$z\mapsto z^{{\alpha }_0}\varphi (z)$ and $z\mapsto z^{{\beta }_0}\psi (z)$ belong toour sheaf.

Then the sheaf consists of solutions to a hypergeometric equation, hencethe function elements are hypergeometric functions.