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 Page view： 721        Net amount： 735 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 Download PDF Full-Text