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Type II Hermite-Padé approximation to the exponential function | | A.B.J. Kuijlaars
; H. Stahl
; W. Van Assche
; F. Wielonsky
; | | Date: |
13 Oct 2005 | | Affiliation: | Leuven), H. Stahl (Berlin), W. Van Assche (Leuven), and F. Wielonsky (Lille | | 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 $a (z)e^{-z}-b (z)= (z^{3n+2})$ and $a (z)e^{z}-c (z)={O}(z^{3n+2})$ as $z o 0$. Our analysis relies on a characterization of these polynomials in terms of a $3 imes 3$ 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. | | Source: | arXiv, math.CA/0510278 | | Services: | Forum | Review | PDF | Favorites | |
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