Padé approximants and balanced hypergeometric series.
type: Item
instance of: scholarly article
title: Padé approximants and balanced hypergeometric series.
zbMATH Open document ID: 1064.11053
DOI: 10.1016/S0021-7824(03)00027-8
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2003-12-03T00:00:00Z
review text: Les auteurs construisent des approximants de Padé simultanés pour des fonctions polylogarithmes aux points 0, 1 et \(\infty\). Ils donnent des formules explicites faisant intervenir des fonctions hypergéométriques. Par spécialisation ils retrouvent les outils utilisés dans presque tous les travaux concernant l'irrationalité des \(\zeta(2n + 1)\) (voir l'article de \textit{T. Rivoal}, Acta Arith. 103, No.~2, 157--167 (2002; Zbl 1015.11033)) pour une bibliographie). Un des corollaires est que, parmi les \(\lambda Li_s(\alpha) + \mu\;\frac{(\log \alpha)^s}{(s-1)!}\) pour \(s\) parcourant \(\mathbb{N}\setminus \{0\}\), il y a une infinité de nombres \(\mathbb{Q}\)-linéairement indépendants, si \(\alpha\in[0,1]\cap\mathbb{Q}\) et \((\lambda,\mu)\in \mathbb{R}^2\setminus \{(0,0)\}\) (ce qui contient la trans\-cendance de \(\log \alpha\)).
Mathematics Subject Classification ID: 11J72
Mathematics Subject Classification ID: 11J91
zbMATH DE Number: 2012741
zbMATH Keywords: Padé approximants
zbMATH Keywords: hypergeometric series
zbMATH Keywords: Diophantine approximation
zbMATH Keywords: Riemann zeta function
author: Stéphane Fischler
author: T. Rivoal
reviewed by: François Gramain
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2012741