Construction of generalized bicomplex Leonardo numbers

Murat Turan and Sıddıka Özkaldı Karakuş
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 1, Pages 43–51
DOI: 10.7546/nntdm.2026.32.1.43-51
Full paper (PDF, 220 Kb)

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Authors and affiliations

Murat Turan
Department of Mathematics, Faculty of Engineering and Natural Sciences, Osmaniye Korkut Ata University
Osmaniye, Türkiye

Sıddıka Özkaldı Karakuş
Department of Mathematics, Faculty of Science, Bilecik Seyh Edebali University
Bilecik, Türkiye

Abstract

In this paper, we introduce a new class of bicomplex numbers whose components are expressed in terms of bicomplex Leonardo numbers. The motivation for this study arises from the growing interest in generalizations of well-known integer sequences within hypercomplex number systems, which reveal deeper algebraic and geometric properties. First, we define the bicomplex Leonardo numbers and establish their fundamental recurrence relation. Then, we derive a Binet-like formula, which serves as a powerful analytical tool for exploring further identities and relationships.

By employing this Binet-like representation, we obtain several new results, including summation formulas, d’Ocagne’s identity, Catalan’s identity, and Cassini’s identity for bicomplex Leonardo numbers. These identities not only extend classical number-theoretic properties into the bicomplex domain but also demonstrate structural consistencies across related algebraic systems. Furthermore, we establish an important connection between the Catalan and Cassini identities, revealing an intrinsic relationship that enhances the understanding of their interdependence within the bicomplex setting.

Keywords

• Leonardo numbers
• Bicomplex numbers
• Bicomplex Leonardo numbers

2020 Mathematics Subject Classification

• 11B37
• 11B39
• 05A15

References

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Manuscript history

• Received: 16 October 2025
• Revised: 9 February 2026
• Accepted: 16 February 2026
• Online First: 19 February 2026

Copyright information

Ⓒ 2026 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).