Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers
type: Item
instance of: scholarly article
title: Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers
zbMATH Open document ID: 1104.11040
author: M. Prevost
author: Tapani Matala-Aho
published in: The Ramanujan Journal
publication date: 2006-11-03T00:00:00Z
review text: Let \(W_n=A \alpha ^n+B\beta ^n\) be either the Fibonacci or the Lucas sequence. Then under special conditions the authors prove that \[ \frac{\log (|\alpha|/|\beta|)}{\log (|\alpha|^{c-1}/|\beta|)} \] is an upper bound of the irrationality measure for the sum of the series \(\sum_{n=0}^\infty \frac{t^n}{W_{an+b}}\). Here \[ c=\frac 12 +\Bigl(\frac{\phi(a)}{2a^2}-\frac 3{\pi^2}\sum_{l=1, (l,a)=1}^{a-1}\frac 1{l^2}\Bigr)\prod_{p\mid a} \frac{p^2}{p^2-1} \] for the Fibonacci sequence and \[ c=\frac 12 +\Bigl(\frac{\phi(a)\gcd(a,2)^2}{6a^2}-\frac 4{\pi^2} \sum_{l=1, (l,2a)=1}^{a-1}\frac 1{l^2}\Bigr)\prod_{p\mid a, p\geq 3} \frac{p^2}{p^2-1} \] for the Lucas sequence where \(\varphi (x)\) is the Euler phi-function.
reviewed by: Jaroslav Hančl
Mathematics Subject Classification ID: 11J82
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 5070737
zbMATH Keywords: irrationality measure
zbMATH Keywords: Fibonacci and Lukas sequences
MaRDI profile type: Publication
full work available at URL: s11139-006-6511-4
OpenAlex ID: W2085475616
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DOI: 10.1007/S11139-006-6511-4
description: scientific article; zbMATH DE number 5070737