Type II Hermite-Pade approximation to the exponential function

[摘要] 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-Pade 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 -> 0. Our analysis relies on a characterization of these polynomials in terms of a 3 x 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-Pade 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. (C) 2006 Elsevier B.V. All rights reserved.

[发布日期] 2007-10-15

[效力级别]  Proceedings Paper

[关键词] Hermite-pade approximation;Riemann-hilbert problems;steepest descent analysis