# Elliptic asymptotics in $q$-discrete Painlev\'{e} equations

## TL;DR

This paper investigates the asymptotic behavior of certain $q$-discrete Painlevé equations, revealing that their solutions tend to elliptic functions at infinity, and extends averaging methods to analyze their energy variations.

## Contribution

It introduces an extension of the averaging method to $q$-discrete Painlevé equations and derives the Picard-Fuchs equation for a specific case, advancing understanding of their asymptotics.

## Key findings

- Elliptic functions describe the generic asymptotics
- Energies vary slowly in these equations
- Derived the Picard-Fuchs equation for $q$-P$_{
m III}$

## Abstract

We study the asymptotic behaviour of two multiplicative- ($q$-) discrete Painlev\'e equations as their respective independent variable goes to infinity. It is shown that the generic asymptotic behaviours are given by elliptic functions. We extend the method of averaging to these equations to show that the energies are slowly varying. The Picard-Fuchs equation is derived for a special case of $q$-P$_{\rm III}$