Quadratic Hermite–Padé Approximation to the Exponential Function: A Riemann–Hilbert Approach

By Arno B. J. Kuijlaars, Walter Van Assche, Franck Wielonsky. 2003

47 citations recorded by the source corpus.

Abstract We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite–Padé approximation to the exponential function, defined by p(z)e-z + q(z) + r(z) ez = O(z3n+2) as z → 0. These polynomials are characterized by a Riemann–Hilbert problem for a 3 × 3 matrix valued function. We use the Deift–Zhou steepest descent method for Riemann–Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and functions derived from it. Our work complements the recent results of Herbert Stahl.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/816ef461d9eaea76f7a57f266ea450817d8cac02

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

Multiple orthogonal polynomial ensembles (2009, background): “MOPs were also studied from the point of view of new special functions [6, 19, 22, 49, 64, 67].”
Large n Limit of Gaussian Random Matrices with External Source, Part I (2004, methodology): “As in [ 20 ] a main tool in the analysis is an appropriate three sheeted Riemann surface.”
Asymptotics of Hermite–Padé Rational Approximants for Two Analytic Functions with Separated Pairs of Branch Points (Case of Genus 0) (2007, methodology): “Concluding this survey of results which have important and direct influence on this paper, we would like to mention the papers [6, 7, 10, 16, 17, 25, 50, 51, 52] which have been written during the work on the present paper and in which some of the ideas and methods elaborated here have already been ”
Strong asymptotics for Jacobi polynomials with varying nonstandard parameters (2003, methodology): “The optimal curves are trajectories of a quadratic differential and were used for steepest descent analysis of Riemann-Hilbert problems first in [3] and later in [2, 19, 21, 22, 24 ].”