One of the definition of the hypergeometric function $_2 F_1$ rely only on its global properties around the singularities (and not on a differential equation or a serie expansion)
In modern language (and here I'm just copy pasting from PlanetMath):
Suppose that we have a sheaf of holomorphic functions over $\mathbb{C} \setminus \{0,1\}$ 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 $0 \in D_0$, holomorphic functions $\phi_0, \psi_0$ 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 $z \mapsto z^{\alpha_0} \phi(z)$ and $z \mapsto z^{\beta_0} \psi(z)$ belong to our sheaf.
Then the sheaf consists of solutions to a hypergeometric equation
This characterization seems to have been made by Riemann (at least Deligne and Mostow attribute it to him (p. 2)). But I cannot find any modern reference/proof for this fact, except for the original paper by Riemann (and I don't speak German).
It looks like common knowledge and I'm probably missing a keyword, or a well known reference. Any help is appreciated.
I'm looking for a proof of this result, and I'm also interested by any generalization of this theorem when the space of function elements is more than two dimensional.
