Quicksilver Asymptotics and Pole Arrays for q-Painlevé II
Abstract
We construct local families of solutions of the q-difference second Painlevé equation that vanish at infinity with a common algebraic expansion, parameterised by the amplitude of an exponentially small correction. Considering an appropriate boundary region, a formal expansion that retains two independently excited exponential perturbations is developed, and subsequent transseries summation predicts the location of simple poles. Adopting a q-theta function realisation of the exponential factors, we prove an asymptotic pole-location formula for a one-parameter family of true solutions. These poles approach the aforementioned boundaries where exponential behaviour fundamentally changes, providing an analogue of pole arrays near anti-Stokes directions in classical Painlevé asymptotics.
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BibTeX
@misc{holroyd2026quicksilver,
title = {{Quicksilver Asymptotics and Pole Arrays for $q$-Painlevé II}},
author = {Joshua Holroyd},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27525},
eprint = {2610.27525},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27525},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quicksilver Asymptotics and Pole Arrays for q-Painlevé II
AU - Joshua Holroyd
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27525
UR - https://arxiv.org/abs/2610.27525
AB - We construct local families of solutions of the q-difference second Painlevé equation that vanish at infinity with a common algebraic expansion, parameterised by the amplitude of an exponentially small correction. Considering an appropriate boundary region, a formal expansion that retains two independently excited exponential perturbations is developed, and subsequent transseries summation predicts the location of simple poles. Adopting a q-theta function realisation of the exponential factors, we prove an asymptotic pole-location formula for a one-parameter family of true solutions. These poles approach the aforementioned boundaries where exponential behaviour fundamentally changes, providing an analogue of pole arrays near anti-Stokes directions in classical Painlevé asymptotics.
ER -