From Continuous to Discrete Painlevé Equations

Discrete analogues of the Painlevé equations have recently been derived using either a method based on orthogonal polynomials or the singularity confinement method. Here we analyze additional methods for deriving integrable non-autonomous discrete equations. These methods are based on the discrete AKNS approach, on the use of Bäcklund and Schlesinger transforms of the (continuous) Painlevé equations, and on discrete analogues of Miura transformations.