Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight  

作　　者：Zhi-Tao WEN Roderick WONG Shuai-Xia XU Zhi-Tao WEN;Roderick WONG;Shuai-Xia XU(Department of Mathematics,Taiyuan University of Technology,Taiyuan 030024,China;Liu Bie Ju Centre for Mathematical Sciences,City University of Hong Kong,Hong Kong,China;Institut Franco-Chinois de l’Energie Nucleaire,Sun Yat-sen University,Guangzhou 510275,China)

机构地区：[1]Department of Mathematics,Taiyuan University of Technology,Taiyuan 030024,China [2]Liu Bie Ju Centre for Mathematical Sciences,City University of Hong Kong,Hong Kong,China [3]Institut Franco-Chinois de l’Energie Nucleaire,Sun Yat-sen University,Guangzhou 510275,China

出　　处：《Chinese Annals of Mathematics,Series B》2018年第3期553-596,共44页数学年刊（B辑英文版）

基　　金：supported by the National Natural Science Foundation of China(Nos.11771090,11571376)

摘　　要：In this paper, the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w（x） = /x/2αe-（x4＋tx2）, x ∈R, where α is a constant larger than - 1/2 and t is any real number. They consider this problem in three separate cases： （i） c 〉 -2, （ii） c = -2, and （iii） c 〈 -2, where c ：= tN-1/2 is a constant, N = n ＋ a and n is the degree of the polynomial. In the first two cases, the support of the associated equilibrium measure μ is a single interval, whereas in the third case the support of μt consists of two intervals. In each case, globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou （1993）.

关 键 词：Orthogonal polynomials Globally uniform asymptotics Riemann-Hilbertproblems The second Painlev6 transcendent Theta function 

分 类 号：O175[理学—数学]