Asymptotic Zero Behavior of Laguerre Polynomials with Negative Parameter

By Arno B. J. Kuijlaars, K. Mclaughlin. 2002

61 citations recorded by the source corpus.

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 $r= - \!\lim_n {1}/{n} \log [\dist\!(\alpha_n, \mbox{\footnotesize{\bf Z}})]$ exists, we show that the zeros accumulate on $\Gamma_r \cup [\beta_1,\beta_2]$ with $\beta_1$ and $\beta_2$ only depending on $A$. For $r \in [0,\infty)$, $\Gamma_r$ is a closed loop encircling the origin, which for $r = +\infty$, 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.

Subfields: Applied Mathematics, Geometry and Topology, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/d0e9821e3f462d27f4265123adb00d93b9af4650

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In Applied Mathematics

In Geometry and Topology

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Citations in the corpus

Quadratic Hermite–Padé Approximation to the Exponential Function: A Riemann–Hilbert Approach (2003, unknown): “See also [7], [8], [4], [23], [24], [25], [ 26 ], [27], [44] for recent developments.”
Orthogonality of Jacobi polynomials with general parameters (2003, background): “The case α, β > −1 is classical, and standard orthogonality on the interval [−1, 1] takes place, being the key for the study of many properties of Jacobi polynomials.”
Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials (2009, background): “[4], [24], [47], [49], [50], [64]; this is necessarily a very partial list).”
Riemann-Hilbert Analysis for Orthogonal Polynomials (2003, background): “Later developments related to orthogonal polynomials include [2, 3, 5, 18, 20, 21, 22, 23, 34].”
Strong asymptotics for Jacobi polynomials with varying nonstandard parameters (2003, methodology): “For Laguerre polynomials the same phenomenon was analyzed recently in [ 21 ].”