Determinantal and permanental representations of companion sequences associated to the r-Fibonacci sequence

Hacène Belbachir and Ihab-Eddine Djellas
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 1, Pages 64–74
DOI: 10.7546/nntdm.2022.28.1.64-74
Full paper (PDF, 181 Kb)

Details

Authors and affiliations

Hacène Belbachir
USTHB, Faculty of Mathematics, RECITS Laboratory
Po. Box 32, El Alia, 16111, Bab Ezzouar, Algiers, Algeria

Ihab-Eddine Djellas
USTHB, Faculty of Mathematics, RECITS Laboratory
Po. Box 32, El Alia, 16111, Bab Ezzouar, Algiers, Algeria

Abstract

Recently companion sequences of -Fibonacci sequence were defined. The aim of this paper is to give some determinantal and permanental representations of these sequences via Hessenberg matrices. Several representations of classical sequences and polynomials are established. We conclude by using our representations to give consecutive terms of companion sequences simultaneously.

Keywords

• Generalized bivariate r-Fibonacci polynomials
• Generalized bivariate r-Lucas polynomials
• Companion sequences
• Determinant
• Permanent
• Hessenberg matrix

2020 Mathematics Subject Classification

• 15A15
• 11B39
• 11B37

References

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Manuscript history

• Received: 22 January 2021
• Revised: 24 November 2021
• Accepted: 11 February 2022
• Online First: 14 February 2022