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Painlevé IV Asymptotics for Orthogonal Polynomials with Respect to a Modified Laguerre Weight
Author(s) -
Dai D.,
Kuijlaars A. B. J.
Publication year - 2009
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2008.00423.x
Subject(s) - mathematics , orthogonal polynomials , laguerre polynomials , parametrix , scaling limit , jacobi polynomials , classical orthogonal polynomials , limit (mathematics) , discrete orthogonal polynomials , weight function , wilson polynomials , pure mathematics , method of steepest descent , mathematical analysis , scaling , partial differential equation , geometry , hyperbolic partial differential equation
We study polynomials that are orthogonal with respect to the modified Laguerre weight z − n +ν e − Nz ( z − 1) 2 b , in the limit where n , N →∞ with N / n → 1 and ν is a fixed number in . With the effect of the factor ( z − 1) 2 b , the local parametrix near the critical point z = 1 can be constructed in terms of Ψ functions associated with the Painlevé IV equation. We show that the asymptotics of the recurrence coefficients of orthogonal polynomials can be described in terms of specified solution of the Painlevé IV equation in the double scaling limit. Our method is based on the Deift/Zhou steepest decent analysis of the Riemann–Hilbert problem associated with orthogonal polynomials.