Elliptic asymptotic behaviour of $q$-Painlev\'e transcendents
Classical Analysis and ODEs
2025-06-09 v1 Mathematical Physics
math.MP
Abstract
The discrete Painlev\'e equations have mathematical properties closely related to those of the differential Painlev\'e equations. We investigate the appearance of elliptic functions as limiting behaviours of -Painlev\'e transcendents, analogous to the asymptotic theory of classical Painlev\'e transcendents. We focus on the -difference second Painlev\'e equation in the asymptotic regime , showing that generic leading-order behaviour is given in terms of elliptic functions and that the slow modulation in this behaviour is approximated in terms of complete elliptic integrals.
Keywords
Cite
@article{arxiv.2506.05724,
title = {Elliptic asymptotic behaviour of $q$-Painlev\'e transcendents},
author = {Joshua Holroyd},
journal= {arXiv preprint arXiv:2506.05724},
year = {2025}
}