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

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 Painlevé I and ${\text{III}}_3$ equations, respectively. Such pairs of coordinates characterize the asymptotics of the tau function of the corresponding Painlevé equation. We conjecture that the other members of the non-polynomial hyperbolic $q$-Askey scheme yield generating functions associated to the other Painlevé equations in the semiclassical limit.