Abstract
We show that an ultradiscrete analogue of the third Painlevé equation admits discrete Riccati type solutions. We derive these solutions by considering a framework in which the ultradiscretization process arises as a restriction of a non-archimedean valuation over a field. Using this framework we may relax the conditions one requires to apply the ultradiscretization process. We derive a family of transcendental solutions that appear as the non-archimedean field valuation of hypergeometric functions.
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