Discrete Painlev\'e equations, Orthogonal Polynomials on the Unit Circle and N-recurrences for averages over U(N) -- \PIIIa and \PV $\tau$-functions

TL;DR

This paper links Toeplitz determinants related to Painlevé systems with discrete Painlevé equations via orthogonal polynomials on the unit circle, revealing new recurrence relations and algebraic structures.

Contribution

It establishes a novel connection between Toeplitz determinants, orthogonal polynomials, and discrete Painlevé equations using affine Weyl group symmetry.

Findings

Recurrences for reflection coefficients are equivalent to discrete Painlevé equations.

Derived explicit hypergeometric function expressions for reflection coefficients.

Unified framework for Toeplitz determinants and Painlevé systems through algebraic and lattice methods.

Abstract

In this work we show that the $ N\times N $ Toeplitz determinants with the symbols $ z^{\mu}\exp(-{1/2}\sqrt{t}(z+1/z)) $ and $ (1+z)^{\mu}(1+1/z)^{\nu}\exp(tz) $ -- known $\tau$-functions for the \PIIIa and \PV systems -- are characterised by nonlinear recurrences for the reflection coefficients of the corresponding orthogonal polynomial system on the unit circle. It is shown that these recurrences are entirely equivalent to the discrete Painlev\'e equations associated with the degenerations of the rational surfaces $ D^{(1)}_{6} \to E^{(1)}_{7} $ (discrete Painlev\'e {\rm II}) and $ D^{(1)}_{5} \to E^{(1)}_{6} $ (discrete Painlev\'e {\rm IV}) respectively through the algebraic methodology based upon of the affine Weyl group symmetry of the Painlev\'e system, originally due to Okamoto. In addition it is shown that the difference equations derived by methods based upon the Toeplitz lattice and Virasoro constraints, when reduced in order by exact summation, are equivalent to our recurrences. Expressions in terms of generalised hypergeometric functions $ {{}^{\vphantom{(1)}}_0}F^{(1)}_1, {{}^{\vphantom{(1)}}_1}F^{(1)}_1 $ are given for the reflection coefficients respectively.