On the generalized p-periodic linear recursive sequences via the Fibonacci–Horner decomposition of the matrix powers

Mustapha Rachidi, László Szalay and Fatih Yilmaz
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 2, Pages 311–325
DOI: 10.7546/nntdm.2025.31.2.311-325
Full paper (PDF, 197 Kb)

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Authors and affiliations

Mustapha Rachidi
Instituto de Matemática INMA, Federal University of Mato Grosso do Sul
Campo Grande – MS, Brazil

László Szalay
Department of Mathematics, Institute of Basic Sciences, University of Sopron
Sopron, Hungary

Department of Mathematics, Jan Selye University
Komárno, Slovakia

Fatih Yilmaz
Department of Mathematics, Ankara Hacı Bayram Veli University
Ankara, 06900, Turkey

Abstract

In this study, we investigate the matrix formulation of the generalized -periodic linear recursive sequences. To reach our goal, we consider the properties of the Fibonacci–Horner decomposition of the matrix powers and those of the weighted linear recursive sequence of Fibonacci type. We provide the linear, the combinatorial, and the analytic representations of the generalized -periodic linear recursive sequences. For illustrating our general results, properties of some special cases are studied and numerical example are furnished.

Keywords

• Generalized -periodic linear recursive sequences
• Analytic representation
• Linear combinatorial representation
• Fibonacci–Horner decomposition

2020 Mathematics Subject Classification

• 11B39
• 11B75
• 11C20
• 65Q10
• 65Q30

References

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Manuscript history

• Received: 6 April 2025
• Revised: 14 May 2025
• Accepted: 20 May 2025
• Online First: 2 June 2025

Copyright information

Ⓒ 2025 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).