Computational Methods and Function Theory 1 (2001), No. 1, 205--233
Copyright Heldermann Verlag 2001
Arno B. J. Kuijlaars
arno@wis.kuleuven.ac.be, Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium.
Kenneth T.-R. McLaughlin
mcl@amath.unc.edu, mcl@math.arizona.edu, Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, U.S.A.
In this paper we study the asymptotic behavior of Laguerre polynomials $L_n^{(\alpha_n)}(nz)$ as $n \to \infty$, where $\alpha_n$ is a sequence of negative parameters such that $-\alpha_n/n$ tends to a limit $A > 1$ as $n \to \infty$. These polynomials \hbox{satisfy} a non-hermitian orthogonality on certain contours in the complex plane. This fact allows the formulation of a Riemann-Hilbert problem whose solution is given in terms of these Laguerre polynomials. The asymptotic analysis of the Riemann-Hilbert problem is carried out by the steepest descent method of Deift and Zhou, in the same spirit as done by Deift et al.\ for the case of orthogonal polynomials on the real line. A main feature of the present paper is the choice of the correct contour.
Keywords: Riemann-Hilbert problems, generalized Laguerre polynomials, strong asymptotics, steepest descent method.
MSC 2000: 30E15, 33C45.
[FullText-pdf (408 K)] [FullText-ps (632 K)]