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| Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight |
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Citation： |
Zhi-Tao WEN,Roderick WONG,Shuai-Xia 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： 2002
Net amount： 1605 |
Authors： |
Zhi-Tao WEN; Roderick WONG;Shuai-Xia XU |
Foundation： |
This work was supported by the National Natural Science
Foundation of China (Nos.11771090, 11571376). |
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| 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
Riemann-Hilbert problems introduced by Deift and Zhou (1993). |
Keywords： |
Orthogonal polynomials, Globally uniform asymptotics,Riemann-Hilbert problems, The second Painlev'e transcendent, Thetafunction |
Classification： |
41A60, 30E15 |
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