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 $\u2102\setminus \{0,1\}$ which satisfy the following properties:
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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 ${\varphi}_{0},{\psi}_{0}$ 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 to our sheaf.
Then the sheaf consists of solutions to a hypergeometric equation, hence the function elements^{} are hypergeometric functions.
Title  global characterization of hypergeometric function 

Canonical name  GlobalCharacterizationOfHypergeometricFunction 
Date of creation  20141231 15:15:16 
Last modified on  20141231 15:15:16 
Owner  rspuzio (6075) 
Last modified by  rspuzio (6075) 
Numerical id  6 
Author  rspuzio (6075) 
Entry type  Definition 
Classification  msc 33C05 