A review of elliptic difference Painlevé equations

Abstract: Discrete Painlev\'e equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable $n$ and there are three different types of equations according to whether the coefficient functions are linear, exponential or elliptic functions of $n$. In this paper, we focus on the elliptic type and give a review of the construction of such equations on the $E_8$ lattice. The first such construction was given by Sakai \cite{SakaiH2001:MR1882403}. We focus on recent developments giving rise to more examples of elliptic discrete Painlev\'e equations.