
 
Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight 
 
Citation： 
ZhiTao WEN,Roderick WONG,ShuaiXia XU.Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight[J].Chinese Annals of Mathematics B,2018,39(3):553~596 
Page view： 991
Net amount： 932 
Authors： 
ZhiTao WEN; Roderick WONG;ShuaiXia XU 
Foundation： 
This work was supported by the National Natural Science
Foundation of China (Nos.11771090, 11571376). 


Abstract： 
In this paper, the authors consider the asymptotic behavior of the
monic polynomials orthogonal with respect to the weight function
$w(x)=x^{2\alpha}\rme^{(x^4+tx^2)}$, $x\in \mathbb{R}$, where
$\alpha$ is a constant larger than $\frac{1}{2}$ 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 $c:=tN^{\frac 12}$ is
a constant, $N=n+\alpha$ and $n$ is the degree of the polynomial. In
the first two cases, the support of the associated equilibrium
measure $\mu_t$ is a single interval, whereas in the third case the
support of $\mu_t$ 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
RiemannHilbert problems introduced by Deift and Zhou (1993). 
Keywords： 
Orthogonal polynomials, Globally uniform asymptotics,RiemannHilbert problems, The second Painlev'e transcendent, Thetafunction 
Classification： 
41A60, 30E15 

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