The \(k\)-Fibonacci matrix and the Pascal matrix
type: Item
instance of: scholarly article
title: The \(k\)-Fibonacci matrix and the Pascal matrix
zbMATH Open document ID: 1285.11055
author: Sergio Falcon
published in: Central European Journal of Mathematics
publication date: 2011-12-12T00:00:00Z
review text: The classical Fibonacci matrix is extended to the concept of the \(k\)-Fibonacci matrix, defined as a unipotent lower triangular \(n\times n\) Toeplitz matrix \({\mathcal F}_n(k)=[f_{i,j}]_{i,j=1,\dots,n}\) with entries \(f_{i,j}=F_{k,i-j+1}\) if \(i\geq j\), 0 otherwise, where the \(k\)-Fibonacci numbers \(F_{k,n}, n\in{\mathbb N}\) verify the reccurence relations \[ F_{k,n+1}=kF_{k,n}+F_{k,n-1},\;n\geq 1, \] with the initial conditions \(F_{k,0}=0,\;F_{k,1}=1\). Two factorizations of the Pascal matrices are obtained, which involve the \(k\)-Fibonacci matrices and two types of matrices. Some combinatorial formulas containing the \(k\)-Fibonacci numbers are derived and the product of a \(k\)-Fibonacci matrix and its transpose is studied.
reviewed by: Valeriu Prepeliţă
Mathematics Subject Classification ID: 11C20
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 15A23
zbMATH DE Number: 5987910
zbMATH Keywords: Pascal matrix
zbMATH Keywords: \(k\)-Fibonacci matrices
zbMATH Keywords: \(k\)-Fibonacci numbers
zbMATH Keywords: factorization
MaRDI profile type: Publication
full work available at URL: s11533-011-0089-9
OpenAlex ID: W2006469504
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DOI: 10.2478/S11533-011-0089-9
description: scientific article; zbMATH DE number 5987910