(\(\pm 1\))-invariant sequences and truncated Fibonacci sequences
type: Item
instance of: scholarly article
title: (\(\pm 1\))-invariant sequences and truncated Fibonacci sequences
zbMATH Open document ID: 1080.11015
author: Gyoung-Sik Choi
author: Suk-Geun Hwang
author: Ik-Pyo Kim
author: Bryan L. Shader
published in: Linear Algebra and its Applications
publication date: 2005-02-23T00:00:00Z
review text: Let \(P\) and \(D\) denote the Pascal matrix \(\bigl[\binom{i}{j}\bigr]\), (\(i,j=0,1,2,\dots\)) and the diagonal matrix \(\text{diag}((-1)^0,(-1)^1,(-1)^2,\dots)\), respectively. An infinite-dimensional real vector \(\mathbf x\) is called a \(\lambda\)-invariant sequence if \(PD\mathbf x=\lambda\mathbf x\). It is known that \(1\) and \(-1\) are the only eigenvalues of \(PD\). Various kinds of \((\pm 1)\)-invariant sequences are introduced by \textit{Z. -H. Sun} [Fibonacci Q. 39, 324--333 (2001; Zbl 0987.05013)]. In this paper the authors find some characterization of \((\pm 1)\)-invariant sequences in connection with the matrices \(P\) and \(Q\). They also introduce truncated Fibonacci and Lucas sequences and proves that an infinite dimensional real vector \(\mathbf x\) is \((-1)\)-invariant (\(1\)-invariant respectively) if and only if \(\mathbf x\) is expressible as a linear combination of truncated Fibonacci (truncated Lucas respectively) sequences.
reviewed by: Takao Komatsu
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11B50
zbMATH DE Number: 2137803
zbMATH Keywords: invariant sequence
zbMATH Keywords: truncated Fibonacci sequence
zbMATH Keywords: truncated Lucas sequence
describes a project that uses: OEIS
MaRDI profile type: Publication
full work available at URL: j.laa.2004.08.018
OpenAlex ID: W2034154222
cites work: \(k\)-Lucas numbers and associated bipartite graphs
cites work: On the \(k\)-generalized Fibonacci matrix \(Q_k\)
cites work: Q4834167
cites work: Invariant sequences under binomial transformation
DOI: 10.1016/J.LAA.2004.08.018
description: scientific article; zbMATH DE number 2137803