<p>In this paper, we present two main objectives. Firstly, we introduce a novel generalization of the generalized Fibonacci quaternion sequences, providing an explicit analytic Binet formula that significantly extends previous formulations in the literature. This advancement offers a powerful mathematical tool for analyzing and computing quaternion sequences efficiently. Secondly, we explore quaternion sequences associated with real sequences that serve as solutions of a linear difference equation of order <i>r</i>, whose coefficients exhibit periodicity with period <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11227_2025_7283_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$p\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We derive comprehensive results that not only extend existing work but also unify previously disparate approaches under a common mathematical framework. Our formulation elegantly handles all cases of root multiplicities and provides a systematic approach to analyzing these complex quaternion sequences. Furthermore, we highlight the special case <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11227_2025_7283_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$r=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11227_2025_7283_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$p=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, offering illustrative numerical examples that demonstrate the practical utility of our theoretical framework. Our results provide mathematicians and applied scientists with new computational tools for working with quaternion sequences in settings where periodic coefficients arise naturally.</p>

On the generalized Fibonacci quaternions and the linear difference equations with periodic coefficients

期刊论文
发表日期：

2025年5月3日

查看全文

R. Ben Taher,
M. Lassri

摘要

In this paper, we present two main objectives. Firstly, we introduce a novel generalization of the generalized Fibonacci quaternion sequences, providing an explicit analytic Binet formula that significantly extends previous formulations in the literature. This advancement offers a powerful mathematical tool for analyzing and computing quaternion sequences efficiently. Secondly, we explore quaternion sequences associated with real sequences that serve as solutions of a linear difference equation of order r, whose coefficients exhibit periodicity with period $p\ge 2$ $p \geq 2$ . We derive comprehensive results that not only extend existing work but also unify previously disparate approaches under a common mathematical framework. Our formulation elegantly handles all cases of root multiplicities and provides a systematic approach to analyzing these complex quaternion sequences. Furthermore, we highlight the special case $r=2$ $r = 2$ and $p=2$ $p = 2$ , offering illustrative numerical examples that demonstrate the practical utility of our theoretical framework. Our results provide mathematicians and applied scientists with new computational tools for working with quaternion sequences in settings where periodic coefficients arise naturally.