On some identities for the DGC Leonardo sequence

Çiğdem Zeynep Yılmaz and Gülsüm Yeliz Saçlı
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 2, Pages 253–270
DOI: 10.7546/nntdm.2024.30.2.253-270
Full paper (PDF, 261 Kb)

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Authors and affiliations

Çiğdem Zeynep Yılmaz
Department of Mathematics, Faculty of Engineering and Natural Sciences,
Istanbul Bilgi University, 34440, Istanbul, Türkiye

Gülsüm Yeliz Saçlı
Department of Computer Engineering, Faculty of Engineering and Architecture,
Istanbul Gelisim University, 34310, Istanbul, Türkiye

Abstract

In this study, we examine the Leonardo sequence with dual-generalized complex () coefficients for . Firstly, we express some summation formulas related to the Fibonacci, Lucas, and Leonardo sequences. Secondly, we present some order- characteristic relations, involving d’Ocagne’s, Catalan’s, Cassini’s, and Tagiuri’s identities. The essential point of the paper is that one can reduce the calculations of the Leonardo sequence by considering . This generalization gives the dual-complex Leonardo sequence for , hyper-dual Leonardo sequence for , and dual-hyperbolic Leonardo sequence for .

Keywords

• Binet’s formula
• Leonardo numbers
• Dual-generalized complex numbers

2020 Mathematics Subject Classification

• 11B37
• 11B39
• 11B83

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Manuscript history

• Received: 9 February 2023
• Revised: 30 March 2024
• Accepted: 7 May 2024
• Online First: 8 May 2024

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Ⓒ 2024 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).