Elliptic and Hyperelliptic Solutions of Discrete Painlev\'e I and Its Extensions to Higher Order Difference Equations

TL;DR

This paper constructs elliptic and hyperelliptic solutions for discrete Painlevé I and explores higher order difference equations that extend its framework, revealing new algebraic structures and solution types.

Contribution

It introduces elliptic and hyperelliptic solutions for discrete Painlevé I and extends the analysis to higher order difference equations with genus two curves.

Findings

Solutions expressed in terms of elliptic and hyperelliptic functions

Higher order difference equations include discrete Painlevé I as a special case

New algebraic structures for discrete integrable systems

Abstract

The solutions of the discrete Painlev\'e equation I were constructed in terms of elliptic and hyperelliptic $\psi$ functions for algebraic curves of genera one and two. For the case of genus two, there appear higher order difference equations which naturally contain the discrete Painlev\'e equation I as a special case.