Riemann-Hilbert Analysis for Laguerre Polynomials with Large Negative Parameter

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

56 citations recorded by the source corpus.

In this paper we study the asymptotic behavior of Laguerre polynomials Ln(αn)(nz) as n → ∞, where αn is a sequence of negative parameters such that −αn/n tends to a limit A > 1 as n → ∞. These polynomials 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.

Subfields: Algebra and Number Theory, Applied Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/beb15a3fd6c081a182b1cfce1f3924943cfc7e5c

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In Algebra and Number Theory

In Applied Mathematics

In Discrete Mathematics and Combinatorics

Citations in the corpus

Asymptotic Zero Behavior of Laguerre Polynomials with Negative Parameter (2002, methodology): “In [21] we used the Riemann-Hilbert method to study Laguerre polynomials with varying negative parameters αn such that limn αn/n < −1, which is the case A > 1.”
Quadratic Hermite–Padé Approximation to the Exponential Function: A Riemann–Hilbert Approach (2003, result): “These results on the distribution of the zeros of Hermite‐Pad´ e approximants to exponentials may be seen as a continuation of the study initiated by Szeg˝ o in [42] concerning the distribution of the zeros of Taylor sections of the series for e z , subsequently generalized by Saff and Varga in [35]”
Riemann-Hilbert Analysis for Orthogonal Polynomials (2003, background): “Later developments related to orthogonal polynomials include [2, 3, 5, 18, 20, 21, 22, 23, 34].”
Padé approximants for functions with branch points — strong asymptotics of Nuttall–Stahl polynomials (2011, methodology): “It has also been noticed (see [7], [33], [3], [28], [34], [31], [13], [11]) that the method works for the non-Hermitian orthogonality in the complex plane with respect to complex-valued weights.”