Constructive Approximation

, Volume 25, Issue 1, pp 109–123 | Cite as

Remainder Pade Approximants for the Exponential Function

Article

Abstract

Following our earlier research, we propose a new method for obtaining the complete Pade table of the exponential function. It is based on an explicit construction of certain Pade approximants, not for the usual power series for exp at 0 but for a formal power series related in a simple way to the remainder term of the power series for exp. This surprising and nontrivial coincidence is proved more generally for type II simultaneous Pade approximants for a family \((\exp(a_jz))_{j=1,\ldots, r}\) with distinct complex a's and we recover Hermite's classical formulas. The proof uses certain discrete multiple orthogonal polynomials recently introduced by Arvesu, Coussement, and van Assche, which generalize the classical Charlier orthogonal polynomials.

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Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Universite du Littoral, Cote d'Opale, Centre Universitaire de la Mi-Voix, Bat H. Poincare, 50 rue F. Buisson, BP 699, 62228 Calais CedexFrance
  2. 2.Institut Fourier, CNRS UMR 5582, Universite Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint-Martin d'Heres CedexFrance

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