Asymptotics for quadratic Hermite-Padé polynomials associated with the exponential function
type: Item
instance of: scholarly article
title: Asymptotics for quadratic Hermite-Padé polynomials associated with the exponential function
zbMATH Open document ID: 1030.41013
publication date: 2004-02-08T00:00:00Z
full work available at URL: https://eudml.org/doc/123200
full work available at URL: pp195-222.html
review text: The (diagonal quadratic) Hermite-Padé polynomials \(p_n, q_n, r_n \in\mathcal P_n\) of type I associated with the exponential function \(e^z\) are defined by the relation \[ e_n(z):=p_n(z)+q_n(z)e^z+r_n(z)e^{2z} = O(z^{3n+2})\quad\text{as}\quad z \to 0, \] and the (diagonal quadratic) Hermite-Padé polynomials type of II of degree \(2n\) \({\mathfrak p}_{2n}, {\mathfrak q}_{2n}, {\mathfrak r}_{2n}\in\mathcal P_{2n}\) are defined by the two relations \[ {\mathfrak e}_{1,2n}(z):={\mathfrak p}_{2n}(z)e^z-{\mathfrak q}_{2n}(z)=O(z^{3n+1}) \quad\text{as}\quad z \to 0, \] \[ {\mathfrak e}_{2,2n}(z):={\mathfrak p}_{2n}(z)e^{2z}-{\mathfrak r}_{2n}(z)=O(z^{3n+1}) \quad\text{as}\quad z \to 0. \] The interesting regularity in distributions of zeros of these polynomials, displayed in the paper for \(n=30\) (stars, boxes, diamonds), suggests that there should exist analytic expressions that allow to describe the asymptotic behavior of the zeros, and also the asymptotic behavior of the Hermite-Padé polynomials and the error term \(e_n\) themselves. Results in this direction are the main topic of the paper. Analytic descriptions are given for the arcs, on which the contracted zeros of both sets of these polynomials cluster as \(n\to\infty\). Analytic expressions are also given for the density functions of the asymptotic distributions of these zeros. The description is based on an algebraic function of third degree and on a harmonic function defined on the Riemann surface associated with the algebraic function. The existence and basic properties of the asymptotic distributions of the zeros and the arcs on which these distributions live are proved. The asymptotic relations themselves are only conjectured, but numerical calculations are presented, which demonstrate the plausibility of these conjectures.
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 1893819
zbMATH Keywords: diagonal quadratic Hermite-Padé polynomials of type I and type II
zbMATH Keywords: exponential function
zbMATH Keywords: German and Latin polynomials
zbMATH Keywords: Hermite-Padé simultaneous rational approximants
zbMATH Keywords: algebraic Hermite-Padé approximants
zbMATH Keywords: orthogonality
zbMATH Keywords: Mahler relations
zbMATH Keywords: asymptotic behavior of polynomials
zbMATH Keywords: asymptotics distributions of zeros
zbMATH Keywords: error terms
zbMATH Keywords: harmonic function on Riemann surface
zbMATH Keywords: algebraic function of third degree
zbMATH Keywords: probability measures
zbMATH Keywords: numerical methods
author: Herbert Stahl
reviewed by: Eleonora A. Storozhenko
MaRDI profile type: Publication
published in: ETNA - Electronic Transactions on Numerical Analysis
description: scientific article; zbMATH DE number 1893819