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Evolution equations and functions of hypergeometric type over fields of positive characteristics. (English) Zbl 1197.12004

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.

MSC:

12H99 Differential and difference algebra
33E50 Special functions in characteristic \(p\) (gamma functions, etc.)
16S32 Rings of differential operators (associative algebraic aspects)
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