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5 - Discrete Painlevé Equations and Orthogonal Polynomials

Published online by Cambridge University Press:  05 July 2011

By
Alexander Its
Edited by
Decio Levi ,
Peter Olver ,
Zora Thomova  and
Pavel Winternitz
Alexander Its
Affiliation:
Indiana University–Purdue University Indianapolis
Decio Levi
Affiliation:
Università degli Studi Roma Tre
Peter Olver
Affiliation:
University of Minnesota
Zora Thomova
Affiliation:
SUNY Institute of Technology
Pavel Winternitz
Affiliation:
Université de Montréal
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Summary

Abstract

Random matrices and orthogonal polynomials have been, for more than a decade, one of the principal sources of the important analytical ideas and exciting problems in the theory of discrete Painlevé equations. In the orthogonal polynomial setting, the discrete Painlevé equations appear in the form of the nonlinear difference relations satisfied by the relevant recurrence coefficients. The principal analytical question is the analysis of certain double-scaling limits of the solutions of the discrete Painlevé equations. In these notes we will present a review on the subject using the Riemann–Hilbert formalism as a main analytic tool.

General setting

These notes are devoted to the orthogonal polynomials and Painlevé equations: both continuous and discrete. In the theory of orthogonal polynomials, the Painlevé equations, both continuous and discrete, appear as the equations satisfied by the recurrence coefficients of orthogonal polynomials. Our main goal is to discuss some of the results concerned with the global asymptotic analysis of the solutions of discrete Painlevé equations generated by the recurrence coefficients. We shall start with the setting of the Riemann–Hilbert formalism for orthogonal polynomials which has been used to achieve these results. Simultaneously, this formalism will allow us to introduce the discrete Painlevé equations in a very natural way. There will be no new facts in this part of the notes, except, perhaps, the way in which the accents between the different aspects of the subject are distributed.

Type
Chapter
Information
Symmetries and Integrability of Difference Equations , pp. 139 - 159
Publisher: Cambridge University Press
Print publication year: 2011

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