Hypergeometric Solutions to Ultradiscrete Painlevé Equations

We propose new solutions to ultradiscrete Painlevé equations that cannot be derived using the ultradiscretization method. In particular, we show the third ultradiscrete Painelevé equation possesses hypergeometric solutions. We show this by considering 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 several methods, we derive a family of hypergeometric solutions. The area of integrable discrete mappings has blossomed within the last few years. Integrable discrete versions of the Painlevé equations is an area of active research[2]. Analogous to the continuous Painlevé equations, as a hallmark of integrability, the discrete Painlevé equations admit rational solutions [10] and also hypergeometric solutions [11]. The ultradiscrete Painelevé equations are obtained via a limiting process called ultradiscretization [16]. The ultradiscretization process sends a rational function of some variables f(a1, . . . , an) to a rational function over a semiring in a new set of ultradiscrete variables by the following limit F (A1, . . . , An) = lim ǫ→0 ǫ log(f(a1, . . . , an) where the new ultradiscrete variables are related to the old variables by the equations ai = e i. Such a process successfully related an integrable cellular automata known as the box-ball system [17] to integrable q-difference equations [16]. Roughly speaking it is a transformation bringing the following binary operations to their ultradiscrete equivalent given by a+ b → max(A,B) (1a) ab → A+B (1b) a/b → A−B (1c) where there is no analog of subtraction. The inequivalence of subtraction often restricts any calculations to a subtraction free framework, in that many of the methods for studying the resultant equations come from subtraction free methods associated with the mapping it is derived from. One derives an ultradiscrete Painlevé equation by applying the ultradiscretization method to a subtraction free version of the q-difference analogs of a Painlevé equation[5].There are ultradiscrete analogs of all six of the Painlevé equations [4]. We will focus on the ultradiscretization of the equation (2) w(qt)w(t/q) = a3a4(w(t) + a1t)(w(t) + a2t) (w(t) + a3)(w(t) + a4) which is given by W +W = A3 +A4 +max(W,A1 + T ) + max(W,A2 + T ) (3) −max(W,A3)−max(W,A4). These equations are known as q-PIII and u-PIII respectively. There is evidence that such equations are integrable, such as the existence of a Lax Pair [8]. There 1