Title: Asymptotic zero behavior of Laguerre polynomials with negative parameter
Authors: Kuijlaars, Arno ×
McLaughlin, KTR #
Issue Date: 2004
Publisher: SPRINGER
Series Title: Constructive approximation vol:20 issue:4 pages:497-523
Abstract: We consider Laguerre polynomials L_n^{(alpha_n)}(nz) with varying negative parameters alpha_n, such that the limit A = - lim_n alpha_n/n exists and belongs to (0,1). For A > 1, it is known that the zeros accumulate along an open contour in the complex plane. For every A in (0,1), we describe a one-parameter family of possible limit sets of the zeros. Under the condition that the limit r = - lim-n 1/n log[dist(alpha-n, Z)] exists, we show that the zeros accumulate on the union of Gamma_r and [beta_1, beta_2] with beta_1 and beta_2 only depending on A. For r in 0,infinity), Gamma_r is a closed loop encircling the origin, which for r = +infinity, reduces to the origin. This shows a great sensitivity of the zeros to alpha_n's proximity to the integers. We use a Riemann-Hilbert formulation for the Laguerre polynomials, together with the steepest descent method of Deift and Zhou to obtain asymptotics for the polynomials, from which the zero behavior follows.
ISSN: 0176-4276
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Analysis Section
× corresponding author
# (joint) last author

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