A scalar Riemann boundary value problem approach to orthogonal polynomials on the circle

8 pages, no figures.-- MSC2000 codes: 33C47; 42C05.MR#: MR2252097 (2007k:33010)Zbl#: Zbl 1130.42025A scalar Riemann boundary value problem defining orthogonal polynomials on the unit circle and the corresponding functions of the second kind is obtained. The Riemann problem is used for the asymptotic analysis of the polynomials orthogonal with respect to an analytical real-valued weight on the circle.The research was supported by INTAS Research Network NeCCA 03-51-6637. The first author was also supported by the Grants RFBR 05-01-00522, NSh-1551.2003.1 and by the program N1 DMS, RAS. The second authorwas supported by Ministerio de Ciencia y Tecnología under Grant number MTM2005-01320. The third author was supported by Ministerio de Ciencia y Tecnología under Grant number BFM2003-06335-C03-02.Publicad

Rational Solutions of the Painlev\'e-II Equation Revisited

The rational solutions of the Painlev\'e-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for establishing these properties. In particular, Riemann-Hilbert representations have proven to be useful for extracting the asymptotic behavior of the rational solutions in the limit of large degree (equivalently the large-parameter limit). We review the elementary properties of the rational Painlev\'e-II functions, and then we describe three different Riemann-Hilbert representations of them that have appeared in the literature: a representation by means of the isomonodromy theory of the Flaschka-Newell Lax pair, a second representation by means of the isomonodromy theory of the Jimbo-Miwa Lax pair, and a third representation found by Bertola and Bothner related to pseudo-orthogonal polynomials. We prove that the Flaschka-Newell and Bertola-Bothner Riemann-Hilbert representations of the rational Painlev\'e-II functions are explicitly connected to each other. Finally, we review recent results describing the asymptotic behavior of the rational Painlev\'e-II functions obtained from these Riemann-Hilbert representations by means of the steepest descent method

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