Type II Hermite-Padé approximation to the exponential function

Details Type II Hermite-Padé approximation to the exponential function Type II Hermite-Padé approximation to the exponential function

2005-10-13

arxiv.org/abs/math/0510278v1

Journal of Computational and Applied Mathematics 207 (2007), 227--244

Mathematics

Classical Analysis and ODEs

20 pages, 5 figures

Scientific paper

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.

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