On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions

Orhan Dışkaya; Hamza Menken; Paula Maria Machado Cruz Catarino

doi:10.53570/jnt.1199465

EN

On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions

Abstract

In this paper, we present a new definition, referred to as the Francois sequence, related to the Lucas-like form of the Leonardo sequence. We also introduce the hyperbolic Leonardo and hyperbolic Francois quaternions. Afterward, we derive the Binet-like formulas and their generating functions. Moreover, we provide some binomial sums, Honsberger-like, d’Ocagne-like, Catalan-like, and Cassini-like identities of the hyperbolic Leonardo quaternions and hyperbolic Francois quaternions that allow an understanding of the quaternions' properties and their relation to the Francois sequence and Leonardo sequence. Finally, considering the results presented in this study, we discuss the need for further research in this field.

Keywords

Fibonacci numbers, Leonardo numbers, Lucas numbers, Francois numbers, hyperbolic quaternions

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Details

Primary Language

English

Subjects

Applied Mathematics

Journal Section

Research Article

Authors

Orhan Dışkaya
0000-0001-5698-7834
Türkiye

Hamza Menken ^*
0000-0003-1194-3162
Türkiye

Paula Maria Machado Cruz Catarino
0000-0001-6917-5093
Portugal

Publication Date

March 31, 2023

Submission Date

November 4, 2022

Acceptance Date

January 12, 2023

Published in Issue

Year 2023 Number: 42

DOI

https://doi.org/10.53570/jnt.1199465

IZ

https://izlik.org/JA35NR33UF

APA

Dışkaya, O., Menken, H., & Cruz Catarino, P. M. M. (2023). On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions. Journal of New Theory, 42, 74-85. https://doi.org/10.53570/jnt.1199465

AMA

1.Dışkaya O, Menken H, Cruz Catarino PMM. On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions. JNT. 2023;(42):74-85. doi:10.53570/jnt.1199465

Chicago

Dışkaya, Orhan, Hamza Menken, and Paula Maria Machado Cruz Catarino. 2023. “On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions”. Journal of New Theory, nos. 42: 74-85. https://doi.org/10.53570/jnt.1199465.

EndNote

Dışkaya O, Menken H, Cruz Catarino PMM (March 31, 2023) On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions. Journal of New Theory 42 74–85.

IEEE

[1]O. Dışkaya, H. Menken, and P. M. M. Cruz Catarino, “On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions”, JNT, no. 42, pp. 74–85, Mar. 2023, doi: 10.53570/jnt.1199465.

ISNAD

Dışkaya, Orhan - Menken, Hamza - Cruz Catarino, Paula Maria Machado. “On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions”. Journal of New Theory. 42 (March 31, 2023): 74-85. https://doi.org/10.53570/jnt.1199465.

JAMA

1.Dışkaya O, Menken H, Cruz Catarino PMM. On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions. JNT. 2023;:74–85.

MLA

Dışkaya, Orhan, et al. “On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions”. Journal of New Theory, no. 42, Mar. 2023, pp. 74-85, doi:10.53570/jnt.1199465.

Vancouver

1.Orhan Dışkaya, Hamza Menken, Paula Maria Machado Cruz Catarino. On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions. JNT. 2023 Mar. 31;(42):74-85. doi:10.53570/jnt.1199465