American Institute of Mathematical Sciences

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May  2007, 7(3): 661-682. doi: 10.3934/dcdsb.2007.7.661

Global asymptotics of Hermite polynomials via Riemann-Hilbert approach

R. Wong 1, and  L. Zhang 1,

1. 

Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China, China

Received  September 2006 Revised  December 2006 Published  February 2007

In this paper, we study the asymptotic behavior of the Hermite polynomials $H_{n}((2n+1)^{1/2}z)$ as $n\rightarrow \infty$. A globally uniform asymptotic expansion is obtained for $z$ in an unbounded region containing the right half-plane Re $z \geq 0$. A corresponding expansion can also be given for $z$ in the left half-plane by using the symmetry property of the Hermite polynomials. Our approach is based on the steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou.
Keywords: Riemann-Hilbert problems, airy functions., Global asymptotics, Hermite polynomials.
Mathematics Subject Classification: 33C45, 41A6.
Citation: R. Wong, L. Zhang. Global asymptotics of Hermite polynomials via Riemann-Hilbert approach. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 661-682. doi: 10.3934/dcdsb.2007.7.661
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