**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

**Full paper (PDF, 230 Kb)**

## Details

### 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.