Ultradiscrete hypergeometric solutions to Painlevé cellular automata

Chris Ormerod

Abstract

We propose new solutions to the Painleve cellular automata that do not and cannot be derived using the ultradiscretization method. We consider a lift of these equations to a non-archimedean valuation field in which we may relax the subtraction free framework of previous explorations of the area. Using such a method, we derive a set of hypergeometric solutions to the Painleve equations which has long been thought impossible to derive.

Keywords: Ultradiscrete, Painleve, Hypergeometric, Tropical.