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

TL;DR

This paper establishes a semiclassical asymptotic formula for specific elements in a non-polynomial hyperbolic q-Askey scheme, linking it to Painlevé equations and their tau functions.

Contribution

It proves a new semiclassical asymptotic formula and connects it to canonical transformations in Painlevé equations, expanding understanding of the q-Askey scheme.

Findings

Asymptotic formula for elements $\

$ extrm{M}$ and $ extrm{Q}$ in the scheme.

Exponent as a generating function for canonical transformations.

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.