An introduction to harmonic complex numbers and harmonic hybrid Fibonacci numbers: A unified approach

Emel Karaca and Fatih Yılmaz
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 3, Pages 542–557
DOI: 10.7546/nntdm.2022.28.3.542-557
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Authors and affiliations

Emel Karaca
Department of Mathematics, Ankara Hacı Bayram Veli University
Ankara, Turkey

Fatih Yılmaz
Department of Mathematics, Ankara Hacı Bayram Veli University
Ankara, Turkey

Abstract

The purpose of this paper is to define and construct new number systems, called the harmonic complex Fibonacci sequences (HCF) and the harmonic hybrid Fibonacci (HHF) sequences. These sequences are defined by inspiring the well-known harmonic and hybrid numbers in literature. We give some fundamental definitions and theorems about these sequences in detail. Moreover, we examine some algebraic properties such as Binet-like-formula, partial sums related to these sequences. Finally, we provide a Maple 13 source code to verify the sequences easily.

Keywords

  • Fibonacci sequence
  • Hybrid numbers
  • Harmonic numbers

2020 Mathematics Subject Classification

  • 11B83
  • 11B37
  • 05A15

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Manuscript history

  • Received: 27 February 2022
  • Revised: 4 August 2022
  • Accepted: 18 August 2022
  • Online First: 20 August 2022

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Cite this paper

Karaca, E., & Yılmaz, F. (2022). An introduction to harmonic complex numbers and harmonic hybrid Fibonacci numbers: A unified approach. Notes on Number Theory and Discrete Mathematics, 28(3), 542-557, DOI: 10.7546/nntdm.2022.28.3.542-557.

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