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

Abstract

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 wellposed. 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 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.

Dwork–carlitz Exponential and Overconvergence for Additive Functions in Positive Characteristic

“…These problems are much simpler than those in the characteristic zero case. The reason is that the above functions satisfy differential equations with the Carlitz derivatives (see [10,11,12,13,17]); the difference structure of the latter leads immediately to overconvergence properties of some linear combinations of solutions.…”

“…holds for any a ∈ K c (see [13]). holds for any values of the variable and parameters, such that all the terms of (24) make sense.…”