# A review of elliptic difference Painlev\'e equations

## TL;DR

This paper reviews the development of elliptic difference Painlevé equations, emphasizing recent advances in their construction on the $E_8$ lattice, highlighting their mathematical structure and classification.

## Contribution

It provides a comprehensive review of elliptic discrete Painlevé equations, including recent constructions and examples on the $E_8$ lattice, expanding understanding of their diversity.

## Key findings

- New examples of elliptic discrete Painlevé equations
- Enhanced understanding of their construction on the $E_8$ lattice
- Summarization of recent developments in the field

## 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.