Bi-orthogonal systems on the unit circle, regular semi-classical weights and integrable systems - II

We derive the Christoffel-Geronimus-Uvarov transformations of a system of bi-orthogonal polynomials and associated functions on the unit circle, that is to say the modification of the system corresponding to a rational modification of the weight function. In the specialisation of the weight function to the regular semi-classical case with an arbitrary number of regular singularities {z 1 ,. .. , z M } the bi-orthogonal system is known to be isomonodromy preserving with respect to deformations of the singular points. If the zeros and poles of the Christoffel-Geronimus-Uvarov factors coincide with the singularities then we have the Schlesinger transformations of this isomonodromic system. Compatibility of the Schlesinger transformations with the other structures of the system-the recurrence relations, the spectral derivatives and deformation derivatives is explicitly deduced. Various forms of Hirota-Miwa equations are derived for the τ-functions or equivalently Toeplitz determinants of the system. 1. Motivations The unitary group U(N) with Haar (uniform) measure has eigenvalue probability density function (see e.g. [12, Chapter 2]) (1.1) 1 (2π) N N! 1≤ j<k≤N |z k − z j | 2 , z l ≔ e iθ l ∈ T, θ l ∈ (−π, π], where T = {z ∈ C : |z| = 1}. One of the motivations of our study is to charac-terise averages over U ∈ U(N) of class functions w(U) (i.e. symmetric functions of the eigenvalues of U only) which have the factorization property N l=1 w(z l) for {z 1 ,. .. , z N } ∈ Spec(U). Such functions w(z) can be interpreted as weights. Introducing the Fourier components {w l } l∈Z of the weight w(z) = ∞ l=−∞ w l z l , due to the well known Heine identity [48] (1.2) N l=1 w(z l) we are equivalently studying Toeplitz determinants. Such averages over the unitary group are ubiquitous in many applications to mathematical physics, in particular the gap probabilities and characteristic polynomial averages in the circular ensembles of random matrix theory [12],[1],[16], the spin-spin correlations of the planar Ising model [39],[31], the density matrix of a system of impenetrable bosons on the ring [14] and probability distributions