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Bi-orthogonal Polynomials on the Unit Circle, regular semi-classical Weights and Integrable Systems | | P.J. Forrester
; N.S. Witte
; | | Date: |
20 Dec 2004 | | Subject: | Classical Analysis and ODEs; Mathematical Physics MSC-class: 05E35; 39A05; 37F10; 33C45; 34M55 | math.CA math-ph math.MP | | Abstract: | The 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))^{
ho_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. | | Source: | arXiv, math.CA/0412394 | | Services: | Forum | Review | PDF | Favorites | |
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