The hunting for the discrete Painlev\'e VI is over

Abstract

We present the discrete, q-, form of the Painlev\'e VI equation written as a three-point mapping and analyse the structure of its singularities. This discrete equation goes over to P_{VI} at the continuous limit and degenerates towards the discrete q-P_{V} through coalescence. It possesses special solutions in terms of the q-hypergeometric function. It can bilinearised and, under the appropriate assumptions, ultradiscretised. A new discrete form for P_{V} is also obtained which is of difference type, in contrast with the `standard' form of the discrete P_{V}. Finally, we present the `asymmetric' form of q-P_{VI}$ as a system of two first-order mappings involving seven arbitrary parameters.