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 which satisfy the following properties:
- •
-
•
It is closed under taking linear combinations
.
-
•
The space of function elements over any open set is two dimensional.
-
•
There exists a neighborhood such that , holomorphic functions defined on , and complex numbers
such that, for an open set of not containing , it happens that and 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 | 2014-12-31 15:15:16 |
Last modified on | 2014-12-31 15:15:16 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 6 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 33C05 |