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# global characterization of hypergeometric function

Riemann noted that the hypergeometric function can be characterized by its global properties, without reference to power series, differential equations, or any other sort of explicit expression. His characterization is conveniently restated in terms of sheaves:

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

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It is closed under analytic continuation.

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It is closed under taking linear combinations.

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

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There exists a neighborhood $D_{0}$ such that $0\in D)$, 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, hence the function elements are hypergeometric functions.

## Mathematics Subject Classification

33C05*no label found*

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