Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight

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
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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).
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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