Semiclassical limit of a non-polynomial q-Askey scheme

We prove a semiclassical asymptotic formula for the two elements M and 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 III₃ 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.