A Bilinear Approach to the Discrete Painlevé I Equations

Abstract

We present the list of all known to date difference and q-discrete forms of the Painlevé I equations. A new, derived here for the first time, zero-parameter, q-discrete Painlevé equation is also given. We present interrelations between some of these equations, whenever they exist, in the form of Miura transformations. Finally we give the bilinearisation of these systems.