1999/01/19 by B. Grammaticos, A. Ramani, Grammaticos, B. +1
Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #nlin.SI #solv-int
paper · pdf · doi:10.48550/arxiv.solv-int/9901006
6 pages, Plain-TeX
arxiv created 1999/01/19 · arxiv updated 2009/11/30
We present the discrete, q-, form of the Painlevé VI equation written as a three-point mapping and analyse the structure of its singularities. This discrete equation goes over to PVI at the continuous limit and degenerates towards the discrete q-PV 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 PV is also obtained which is of difference type, in contrast with the `standard' form of the discrete PV. Finally, we present the `asymmetric' form of q-PVI as a system of two first-order mappings involving seven arbitrary parameters.