Type II Hermite-Pad\'e approximation to the exponential function

TL;DR

This paper derives strong, uniform asymptotic formulas for scaled type II Hermite-Padé approximants to the exponential function across the complex plane, using Riemann-Hilbert problem techniques.

Contribution

It extends previous Riemann-Hilbert analysis to obtain asymptotics for type II Hermite-Padé approximants to the exponential function.

Findings

Uniform asymptotics in the complex plane for the approximants

Connection between type I and type II approximants via Mahler relations

Reusability of previous Riemann-Hilbert analysis methods

Abstract

We obtain strong and uniform asymptotics in every domain of the complex plane for the scaled polynomials $a (3nz)$, $b (3nz)$, and $c (3nz)$ where $a$, $b$, and $c$ are the type II Hermite-Pad\'e approximants to the exponential function of respective degrees $2n+2$, $2n$ and $2n$, defined by $a (z)e^{-z}-b (z)=\O (z^{3n+2})$ and $a (z)e^{z}-c (z)={\O}(z^{3n+2})$ as $z\to 0$. Our analysis relies on a characterization of these polynomials in terms of a $3\times 3$ matrix Riemann-Hilbert problem which, as a consequence of the famous Mahler relations, corresponds by a simple transformation to a similar Riemann-Hilbert problem for type I Hermite-Pad\'e approximants. Due to this relation, the study that was performed in previous work, based on the Deift-Zhou steepest descent method for Riemann-Hilbert problems, can be reused to establish our present results.