# Elliptic Painlev\'e equations from next-nearest-neighbor translations on the $E_8^{(1)}$ lattice

## TL;DR

This paper introduces a new elliptic discrete Painlevé equation derived from next-nearest-neighbor translations on the $E_8^{(1)}$ lattice, expanding the class of known elliptic difference equations.

## Contribution

It presents a novel elliptic Painlevé equation from next-nearest-neighbor lattice translations, generalizing previous models and connecting to Adler's Q4 equation.

## Key findings

- New elliptic Painlevé equation with 8 parameters
- Reduction of Sakai's equation to a projective form
- Connection to Adler's Q4 equation

## Abstract

The well known elliptic discrete Painlev\'e equation of Sakai is constructed by a standard translation on the $E_8^{(1)}$ lattice, given by nearest neighbor vectors. In this paper, we give a new elliptic discrete Painlev\'e equation obtained by translations along next-nearest-neighbor vectors. This equation is a generic (8-parameter) version of a 2-parameter elliptic difference equation found by reduction from Adler's partial difference equation, the so-called Q4 equation. We also provide a projective reduction of the well known equation of Sakai.