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Strong Asymptotics for Multiple Laguerre Polynomials

Authors:	V. Lysov F. Wielonsky

Affiliation:	(1) Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya Sq., 4, Moscow, Russia;(2) Laboratoire de Mathematiques P. Painleve, UMR CNRS 8524 - Bat. M2, Universite des Sciences et Technologies Lille, F-59655, Villeneuve d'Ascq Cedex, France

Abstract:	We consider multiple Laguerre polynomials l_n of degree 2n orthogonal on (0,∞) with respect to the weights $x^{alpha}e^{-beta_{1}x}$ and $x^{alpha}e^{-beta_{2}x}$ , where -1 < α, 0 < β₁ < β₂, and we study their behavior in the large n limit. The analysis differs among three different cases which correspond to the ratio β₂/β₁ being larger, smaller, or equal to some specific critical value κ. In this paper, the first two cases are investigated and strong uniform asymptotics for the scaled polynomials l_n(nz) are obtained in the entire complex plane by using the Deift-Zhou steepest descent method for a (3 × 3)-matrix Riemann-Hilbert problem.

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