Parameterless discrete Painlevé equations and their Miura relations. Issue 1 (2nd January 2016) - Grammaticos, B.; Ramani, A.
Abstract : We present a study of discrete Painlevé equations which do not have any parameter, apart from those that can be removed by the appropriate scaling. We find four basic equations of this type as well as several more related to the basic ones by Miura transformations, which we derive explicitly. We obtain also the continuous limits of the basic parameterless equations and show that two of them are the discrete analogues of both the continuous Painlevé I and the zero-parameter Painlevé III.