Dual Fibonacci Quaternions

Nurkan, Semra Kaya; Güven, İkay Arslan

doi:10.1007/s00006-014-0488-7

Cited by 35 publications

(28 citation statements)

References 11 publications

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“…In addition, other mathematicians are still interested in the dual form of quaternions (Nurkan & Güven, 2015) and octonions (Savin, 2015) or the split of quaternions or octonions that we will not discuss here (Halici, 2015). Similarly to what happened way in the sixties, with properties related to extension of the subscripts to the integers numbers, Halice (2012), explains the determination of the set of numbers { − }`∈ .…”

Section: The Fibonacci Quaternions and Fibonacci Octonions' Researchmentioning

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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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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Section: The Fibonacci Quaternions and Fibonacci Octonions' Researchmentioning

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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“…Many interesting properties of Fibonacci and Lucas quaternions can be found in [3,7]. Nurkan and Gven in [8] defined dual Fibonacci quaternions and dual Lucas quaternions. In [2] Halici investigated complex Fibonacci quaternions.…”

mentioning

confidence: 99%

A Note on Jacobsthal Quaternions

Szynal‐Liana

Włoch

2015

Adv. Appl. Clifford Algebras

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“…Halici obtained the Binet formula, generating function, matrix representation, second‐order recursion, and extension to negative indices. In 2015, Kaya Nurkan and Güven Arslan defined the dual (coefficient) Fibonacci quaternion and dual Lucas quaternion with i 2 = j 2 = k 2 = ijk = −1, a standard orthonormal basis in

R^{3},

and with dual Fibonacci numbers ( i 2 = 0). They gave the Binet's formulas, Cassini identities, and some relations between the dual Fibonacci and Lucas quaternions.…”

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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Dual Fibonacci Quaternions

Cited by 35 publications

References 11 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

A Note on Jacobsthal Quaternions

On metallic ratio in

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