On complex Fibonacci quaternions
type: Item
instance of: scholarly article
title: On complex Fibonacci quaternions
zbMATH Open document ID: 1270.11012
author: Serpil Halici
published in: Advances in Applied Clifford Algebras
publication date: 2013-04-08T00:00:00Z
review text: Let \(\{F_n\}\) be the Fibonacci sequence and \[ Q_n = F_n + F_{n+1}e_1 + F_{n+2}e_2 + F_{n+3}e_3 \] be the \(n\)-th Fibonacci quaternion, where \(e_1, e_2, e_3\) are the standard orthonormal basis in \(R^3\) and they satisfy equalities \[ e_1^2 = e_2^2 = e_3^2 = e_1e_2e_3 = -1. \] The \(n\)-th complex Fibonacci quaternion is defined by \[ R_n = Q_n + iQ_{n+1},\quad i^2 = -1. \] Some interesting properties of these numbers are studied, e.g., for \(n \geq 1\), the Binet formula has the form (Theorem 2.1) \[ R_n = R_1F_n + R_0F_{n-1}, \] where \[ R_0 = C_0 + C_1e_1 + C_2e_2 + C_2e_3, \] \[ R_1= C_1 + C_2e_1 + C_3e_2 + C_4e_3 \] and \[ C_n = F_n + iF_{n+1}. \]
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 6151444
zbMATH Keywords: recurrence relations
zbMATH Keywords: Fibonacci numbers
zbMATH Keywords: quaternions
reviewed by: Krassimir Atanassov
MaRDI profile type: Publication
full work available at URL: s00006-012-0337-5
OpenAlex ID: W2097583789
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DOI: 10.1007/S00006-012-0337-5
description: scientific article; zbMATH DE number 6151444