Global asymptotics of Hermite polynomials via Riemann-Hilbert approach

Pages: 661 - 682, Volume 7, Issue 3, May 2007

doi:10.3934/dcdsb.2007.7.661       Abstract        Full Text (286.5K)       Related Articles

R. Wong - Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China (email)
L. Zhang - Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China (email)

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.

Keywords:  Global asymptotics, Hermite polynomials, Riemann-Hilbert problems, airy functions.
Mathematics Subject Classification:  33C45, 41A60.

Received: September 2006;      Revised: December 2006;      Published: February 2007.