Some Combinatorial Properties of the k-Fibonacci and the k-Lucas Quaternions

Ramírez, José L.

doi:10.1515/auom-2015-0037

Cited by 41 publications

(36 citation statements)

References 16 publications

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“…From the algebraic expression and by using the mathematical definition formulated by Ramirez (2015), we can still obtain:…”

Section: The Fibonacci Quaternions and Fibonacci Octonions' Researchmentioning

confidence: 99%

“…We recall that, even with a symbolic notational system of Fibonacci quaternions and Fibonacci octonions, the verification form of certain properties (and identities) may incur any mathematical errors present in some papers. For example, we have the case of the conjecture about Catalan's formula that proved wrong (Polatli & Kesim, 2015;Ramirez, 2015). Certainly, technological support can help us to produce conjectures with greater chances of success, given that, Mathematics does not progress only by the correct and precise arguments.…”

Section: The Quaternions Conjugate Is Indicated By X_00:=dgconjugate(mentioning

confidence: 99%

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The Quaterniontonic and Octoniontonic Fibonacci Cassini’s Identity: An Historical Investigation with the Maple’s Help

Alves

2018

Int Elect J Math Ed

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This paper discusses a proposal for exploration and verification of numerical and algebraic behavior correspondingly to Generalized Fibonacci model. Thus, it develops a special attention to the class of Fibonacci quaternions and Fibonacci octonions and with this assumption, the work indicates an investigative and epistemological route, with assistance of software CAS Maple. The advantage of its use can be seen from the algebraic calculation of some Fibonacci's identities that showed unworkable without the technological resource. Moreover, through an appreciation of some mathematical definitions and recent theorems, we can understand the current evolutionary content of mathematical formulations discussed over this writing. On the other hand, the work does not ignore some historical elements which contributed to the discovery of quaternions by the mathematician William Rowan Hamilton (1805 -1865). Finally, with the exploration of some simple software's commands allows the verification and, above all, the comparison of the numerical datas with the theorems formally addressed in some academic articles.

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“…From the algebraic expression and by using the mathematical definition formulated by Ramirez (2015), we can still obtain:…”

Section: The Fibonacci Quaternions and Fibonacci Octonions' Researchmentioning

confidence: 99%

Section: The Quaternions Conjugate Is Indicated By X_00:=dgconjugate(mentioning

confidence: 99%

The Quaterniontonic and Octoniontonic Fibonacci Cassini’s Identity: An Historical Investigation with the Maple’s Help

Alves

2018

Int Elect J Math Ed

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“…In this paper, he examined some of the interesting properties of the k −Lucas numbers themselves as well as looking at its close relationship with the k ‐Fibonacci numbers. In 2015, Ramírez defined the k ‐Fibonacci and the k −Lucas quaternions. The author investigated the generating functions and Binet's formulas for these quaternions and he derived some sums formulas and identities such as Cassini's identity.…”

Section: Introductionmentioning

confidence: 99%

On metallic ratio in

Akbiyik

Yüce

2019

Math Methods in App Sciences

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Metallic ratio is a root of the simple quadratic equation x2 = kx + 1 for k is any positive integer which is the characteristic equation of the recurrence relation of k‐Fibonacci (k‐Lucas) numbers. This paper is about the metallic ratio in Zp. We define k‐Fibonacci and k‐Lucas numbers in Zp, and we show that metallic ratio can be calculated in Zp if and only if p≡ ± 1 mod (k2 + 4), which is the generalization of the Gauss reciprocity theorem for any integer k. Also, we obtain that the golden ratio, the silver ratio, and the bronze ratio, the three together, can be calculated in Z79 for the first time. Moreover, we introduce k‐Fibonacci and k‐Lucas quaternions with some algebraic properties and some identities for them.

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“…[22,23] introduced a new generalization of the Fibonacci and Lucas quaternions, named as, the bi-periodic Fibonacci and Lucas quaternions. They are emerged as a generalization of the best known quaternions in the literature, such as classical Fibonacci and Lucas quaternions in [10], Pell and Pell-Lucas quaternions in [5], k−Fibonacci and k−Lucas quaternions in [18].…”

Section: Introductionmentioning

confidence: 99%

The generalized bi-periodic Fibonacci quaternions and octonions

2018

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In this paper, we present a further generalization of the biperiodic Fibonacci quaternions and octonions. We give the generating function, the Binet formula, and some basic properties of these quaternions and octonions. The results of this paper not only give a generalization of the bi-periodic Fibonacci quaternions and octonions, but also include new results such as the matrix representation and the norm value of the generalized bi-periodic Fibonacci quaternions.2000 Mathematics Subject Classification. 11B39, 05A15, 11R52.

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Some Combinatorial Properties of the k-Fibonacci and the k-Lucas Quaternions

Abstract: In this paper, we define the k-Fibonacci and the k-Lucas quaternions. We investigate the generating functions and Binet formulas for these quaternions. In addition, we derive some sums formulas and identities such as Cassini's identity.

Cited by 41 publications

References 16 publications

The Quaterniontonic and Octoniontonic Fibonacci Cassini’s Identity: An Historical Investigation with the Maple’s Help

The Quaterniontonic and Octoniontonic Fibonacci Cassini’s Identity: An Historical Investigation with the Maple’s Help

On metallic ratio in

The generalized bi-periodic Fibonacci quaternions and octonions

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