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

Abstract

In this work we show that the $ N\times N $ Toeplitz determinants with the symbols $ z^μ\exp(-{1/2}\sqrt{t}(z+1/z)) $ and $ (1+z)^μ(1+1/z)^ν\exp(tz) $ -- known $τ$-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 Painlevé equations associated with the degenerations of the rational surfaces $ D^{(1)}_{6} \to E^{(1)}_{7} $ (discrete Painlevé {\rm II}) and $ D^{(1)}_{5} \to E^{(1)}_{6} $ (discrete Painlevé {\rm IV}) respectively through the algebraic methodology based upon of the affine Weyl group symmetry of the Painlevé 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.