Semiclassical limit of a non-polynomial $q$-Askey scheme

Jonatan Lenells; Julien Roussillon

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arXiv · 2407.03464

Semiclassical limit of a non-polynomial $q$-Askey scheme

Jonatan Lenells·Julien Roussillon

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Abstract

We prove a semiclassical asymptotic formula for the two elements $\mathcal M$ and $\mathcal Q$ lying at the bottom of the recently constructed non-polynomial hyperbolic $q$-Askey scheme. We also prove that the corresponding exponent is a generating function of the canonical transformation between pairs of Darboux coordinates on the monodromy manifold of the Painlev\'e I and $\textrm{III}_3$ equations, respectively. Such pairs of coordinates characterize the asymptotics of the tau function of the corresponding Painlev\'e equation. We conjecture that the other members of the non-polynomial hyperbolic $q$-Askey scheme yield generating functions associated to the other Painlev\'e equations in the semiclassical limit.

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BibTeX RIS

Jonatan Lenells, Julien Roussillon. 2024-07-03. Semiclassical limit of a non-polynomial $q$-Askey scheme. https://arxiv.org/abs/2407.03464

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