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Strong asymptotics of orthogonal polynomials with respect to exponential weights
Author(s) -
Deift P.,
Kriecherbauer T.,
McLaughlin K. TR,
Venakides S.,
Zhou X.
Publication year - 1999
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199912)52:12<1491::aid-cpa2>3.0.co;2-#
Subject(s) - mathematics , orthogonal polynomials , mehler–heine formula , polynomial , order (exchange) , method of steepest descent , classical orthogonal polynomials , exponential function , jacobi polynomials , type (biology) , discrete orthogonal polynomials , combinatorics , complex plane , gegenbauer polynomials , real line , mathematical analysis , pure mathematics , ecology , finance , economics , biology
We consider asymptotics of orthogonal polynomials with respect to weights w ( x ) dx = e − Q ( x ) dx on the real line, where Q ( x ) = Σ 2 m k =0q k x k , q 2 m > 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann‐Hilbert problem following [22, 23]. We employ the steepest‐descent‐type method introduced in [18] and further developed in [17, 19] in order to obtain uniform Plancherel‐Rotach‐type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials. © 1999 John Wiley & Sons, Inc.