Evolution Equations and Functions of Hypergeometric Type over Fields of Positive Characteristic

We consider a class of partial differential equations with Carlitz derivatives over a local field of positive characteristic, for which an analog of the Cauchy problem is well-posed. Equations of such type correspond to quasi-holonomic modules over the ring of differential operators with Carlitz der...

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Bibliographic Details
Published in:Bulletin of the Belgian Mathematical Society, Simon Stevin Simon Stevin, 2007-01, Vol.14 (5), p.947-959
Main Author: Kochubei, Anatoly N.
Format: Article
Language:English
Subjects:
12H99
16S32
33E50
Carlitz derivative
F_q-linear function
hypergeometric function
quasi-holonomic module
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Summary:We consider a class of partial differential equations with Carlitz derivatives over a local field of positive characteristic, for which an analog of the Cauchy problem is well-posed. Equations of such type correspond to quasi-holonomic modules over the ring of differential operators with Carlitz derivatives. The above class of equations includes some equations of hypergeometric type. Building on the work of Thakur, we develop his notion of the hypergeometric function of the first kind (whose parameters belonged initially to \mathbb Z) in such a way that it becomes fully an object of the function field arithmetic, with the variable, parameters and values from the field of positive characteristic.
ISSN:1370-1444
2034-1970
DOI:10.36045/bbms/1197908905