On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions

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Journal of New Theory
Issue:42
On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions

On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions

Authors : Orhan DIŞKAYA, Hamza MENKEN, Paula Maria Machado CRUZ CATARİNO

Pages : 74-85

Doi:10.53570/jnt.1199465

View : 35 | Download : 8

Publication Date : 2023-03-31

Article Type : Research Paper

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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