First degree birational transformations of the Painlevé equations and their contiguity relations

Abstract

We present a consistent truncation, allowing us to obtain the first degree birational transformation found by Okamoto for the sixth Painlevé equation. The discrete equation arising from its contiguity relation is then just the sum of six simple poles. An algebraic solution is presented, which is equivalent to but simpler than the Umemura solution. Finally, the well known confluence provides a unified picture of all first degree birational transformations for the lower Painlevé equations, ranging them in two distinct sequences.