Irrationality results for reciprocal sums of certain Lucas numbers
type: Item
instance of: scholarly article
title: Irrationality results for reciprocal sums of certain Lucas numbers
zbMATH Open document ID: 0791.11031
DOI: 10.1007/BF01201780
author: Thomas Töpfer
author: P.-G. Becker-Landeck
published in: Archiv der Mathematik
publication date: 1994-07-21T00:00:00Z
review text: The authors prove the irrationality of the series \(\sum^ \infty_{n = 0} \varepsilon^ n / V_{2^ n}\), where \(\varepsilon \in \{\pm 1\}\) and \(V_{2^ n}\) are elements of a non-degenerate sequence of Lucas numbers \(\{V_ n\}_{n \in \mathbb{N}_ 0}\) satisfying the binary recurrence relation \(V_{n+2} = A_ 1 V_{n+1} + A_ 2 V_ n\), \(V_ 0 = 2\), \(V_ 1 = A_ 1\) with \(A_ 1, A_ 2 \in \mathbb{Z}\) and \(A^ 2_ 1 + 4A_ 2 \neq 0\). This extends results of \textit{C. Badea} [Glasg. Math. J. 29, 221-228 (1987; Zbl 0629.10027)] and \textit{R. André-Jeannin} [Fibonacci Q. 29, 132-136 (1991; Zbl 0725.11034)]. The proof is based on the study of certain dynamic properties of the map \(z \to z^ 2\) on the unit circle.
Mathematics Subject Classification ID: 11J72
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 563364
zbMATH Keywords: reciprocal sums of Lucas numbers
zbMATH Keywords: irrationality
zbMATH Keywords: binary recurrence relation
MaRDI profile type: Publication
cites work: Q5203107
cites work: The irrationality of certain infinite series
cites work: Transcendency Results for Sums of Reciprocals of Linear Recurrences
cites work: Zur Transzendenz gewisser Reihen. (On the transcendence of certain series)
cites work: Q3872528
cites work: Q4043118
cites work: Q4123397
cites work: On the Diophantine equation $ax^{2t}+bx^ty+cy^2=d$ and pure powers in recurrence sequences.
cites work: Q3744223
full work available at URL: https://doi.org/10.1007/bf01201780
OpenAlex ID: W2033820316
Recommended article: Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers
Recommended article: Q4308532
Recommended article: Q4455299
Recommended article: On the infinite sum of reciprocal Lucas number sequences
Recommended article: Q4281105
Recommended article: Q5203107
Recommended article: Q4283105
Recommended article: SOME IDENTITIES RELATED TO RECIPROCAL SUMS OF LUCAS NUMBERS
Recommended article: Linear independence results for sums of reciprocals of Fibonacci and Lucas numbers
description: scientific article; zbMATH DE number 563364