A review of elliptic difference Painlev\'e equations

Nalini Joshi; Nobutaka Nakazono

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arXiv · 1902.07842

A review of elliptic difference Painlev\'e equations

Nalini Joshi·Nobutaka Nakazono

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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.

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Nalini Joshi, Nobutaka Nakazono. 2019-02-21. A review of elliptic difference Painlev\'e equations. https://arxiv.org/abs/1902.07842

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