2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73640The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference equations of certain coefficient functions appearing in the theory. A natural formulation of the Riemann-Hilbert problem is presented which has as its solution the above system of bi-orthogonal polynomials and associated functions. In particular for the case of regular semi-classical weights on the unit circle $ w(z) = \prod^m_{j=1}(z-z_j(t))^{ρ_j} $, consisting of $ m \in \mathbb{Z}_{> 0} $ finite singularities, difference equations with respect to the bi-orthogonal polynomial degree $ n $ (Laguerre-Freud equations or discrete analogs of the Schlesinger equations) and differential equations with respect to the deformation variables $ z_j(t) $ (Schlesinger equations) are derived completely characterising the system.This extends and supersedes math-ph/0308036Classical Analysis and ODEsMathematical Physics05E35; 39A05; 37F10; 33C45; 34M55text