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Bi-orthogonal Polynomials on the Unit Circle, Regular Semi-Classical Weights and Integrable Systems

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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))^{\rho_j}, $

consisting of

$ m \in {\bf 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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia

    P.J. Forrester & N.S. Witte

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  1. P.J. Forrester
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  2. N.S. Witte
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Correspondence to P.J. Forrester or N.S. Witte.

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Forrester, P., Witte, N. Bi-orthogonal Polynomials on the Unit Circle, Regular Semi-Classical Weights and Integrable Systems. Constr Approx 24, 201–237 (2006). https://doi.org/10.1007/s00365-005-0616-7

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  • DOI: https://doi.org/10.1007/s00365-005-0616-7

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