Discrete Painlevé equations, orthogonal polynomials on the unit circle, and N-recurrences for averages over U(N) —PIII′ and PV τ-functions

Abstract

We show that the N × N Toeplitz determinants with the symbols zμexp(−(1/2)t(z+1/z)) and (1+z)μ(1+1/z)ν exp(tz)—known τ-functions for the PIII′ 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 D6(1)→E7(1) (discrete Painlevé II) and D5(1)→E6(1) (discrete Painlevé IV), respectively, through the algebraic methodology based upon 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 0F1(1),1F1(1) are given for the reflection coefficients, respectively.