Elliptic asymptotic behaviour of $q$-Painlev\'e transcendents
Elliptic asymptotic behaviour of $q$-Painlev\'e transcendents
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 $q$-Painlev\'e transcendents, analogous to the asymptotic theory of classical Painlev\'e transcendents. We focus on the $q$-difference second Painlev\'e equation in the asymptotic regime $|q-1|\ll1$, 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.
Joshua Holroyd
数学
Joshua Holroyd.Elliptic asymptotic behaviour of $q$-Painlev\'e transcendents[EB/OL].(2025-06-06)[2025-07-09].https://arxiv.org/abs/2506.05724.点此复制
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