Irrationality measures for the series of reciprocals from recurrence sequences.
type: Item
instance of: scholarly article
title: Irrationality measures for the series of reciprocals from recurrence sequences.
zbMATH Open document ID: 1069.11030
author: Tapani Matala-Aho
author: M. Prevost
published in: Journal of Number Theory
publication date: 2003-03-23T00:00:00Z
review text: The paper under review deals with rational approximation (or more precisely, approximations by quotients of integers in an imaginary quadratic field) for complex numbers of the shape \(\sum_{n=0}^\infty t^n/W_n\), where \(W_n\) satisfies a second-order recurrence \(W_{n+2}= rW_{n+1}+sW_n\), (\(r,s,W_1,W_2\in\mathbb{Q}^*\) and \(t\) lying in a subset of an imaginary quadratic field. Using Padd approximations of Heine's \(q\)-hypergeometric series, the authors obtain upper bounds (and this, uniformly with respect to \(t\)) for the irrationality measure of the values of such series. In the particular cases where \((W_n)_{n>0}\) is the Fibonacci or the Lucas sequence, the bounds are \(\frac{2\pi^2}{\pi^2-3} =2,87341\dots\) and \(\frac{3\pi^2}{\pi^2-6}=7,65163\dots\), respectively. This substantially improves earlier result's obtained for these two sequences.
Mathematics Subject Classification ID: 11J82
zbMATH DE Number: 1886719
zbMATH Keywords: Padé approximation
zbMATH Keywords: Heine \(q\)-hypergeometric series
zbMATH Keywords: irrationality measure
zbMATH Keywords: recurrence
zbMATH Keywords: Fibonomials
reviewed by: Boris Adamczewski
MaRDI profile type: Publication
OpenAlex ID: W1980030248
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DOI: 10.1006/JNTH.2002.2795
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description: scientific article; zbMATH DE number 1886719