The hunting for the discrete Painlevé VI is over

Details The hunting for the discrete Painlevé VI is over The hunting for the discrete Painlevé VI is over

1999-01-19

arxiv.org/abs/solv-int/9901006v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

6 pages, Plain-TeX

Scientific paper

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.

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