Global asymptotics of Hermite polynomials via Riemann-Hilbert approach

R. Wong; L. Zhang

doi:10.3934/dcdsb.2007.7.661

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2007, Volume 7, Issue 3: 661-682. Doi: 10.3934/dcdsb.2007.7.661

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Global asymptotics of Hermite polynomials via Riemann-Hilbert approach

R. Wong^1, and
L. Zhang¹

1.
Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Received: August 31, 2006

Revised: November 30, 2006

Published: April 2007

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Abstract

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.

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Mathematics Subject Classification: 33C45, 41A60.

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