Type II Hermite-Padé approximation to the exponential function

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-Padé approximants to the exponential function of respective degrees $2n+2$, $2n$ and $2n$, defined by $a(z)e^{-z}-b(z)=Ø(z^{3n+2})$ and $a(z)e^z-c(z)=Ø(z^{3n+2})$ 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-Padé 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.

20 pages, 5 figures

Cited by in corpus (1)

• Riemann-Hilbert Methods in the Theory of Orthogonal Polynomials