Split Fibonacci Quaternions

Akyiğit, Mahmut; Kösal, Hidayet Hüda; Tosun, Murat

doi:10.1007/s00006-013-0401-9

Cited by 51 publications

(34 citation statements)

References 7 publications

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“…In 2012, Halici investigated the generating functions, Binet's formulas, and some sums formulas for the Fibonacci and Lucas quaternions. In 2013, Akyigit et al defined the split Fibonacci quaternion,

Q_{n} = F_{n} + i F_{n + 1} + j F_{n + 2} + k F_{n + 3}, i^{2} = - 1, j^{2} = k^{2} = 1, i j = - j i = k,

the split Lucas quaternion, and the split generalized Fibonacci quaternion. They obtained Binet's formulas, Cassini identities, and some relations between the split Fibonacci, the split Lucas, and the split generalized Fibonacci quaternions.…”

Section: Introductionmentioning

confidence: 99%

“…In 2012, Halici 3 investigated the generating functions, Binet's formulas, and some sums formulas for the Fibonacci and Lucas quaternions. In 2013, Akyigit et al 4 defined the split Fibonacci quaternion, with i 2 = j 2 = k 2 = ijk = −1, a standard orthonormal basis in R 3 , and with Horadam's complex Fibonacci numbers (i 2 = −1, usual). The author also defined the complex Lucas quaternions.…”

Section: Introductionmentioning

confidence: 99%

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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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Q_{n} = F_{n} + i F_{n + 1} + j F_{n + 2} + k F_{n + 3}, i^{2} = - 1, j^{2} = k^{2} = 1, i j = - j i = k,

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On metallic ratio in

Akbiyik

Yüce

2019

Math Methods in App Sciences

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“…Several authors worked on different quaternions and their generalizations [12][13][14][15][16][17][18][19][20][21][22] . Also, some authors worked on dual quaternions and their generalizations [2][3][4][5][6] as follows:…”

Section: Introductionmentioning

confidence: 99%

Dual Pell Quaternions

Aydın¹,

Yüce²

2016

jusps-A

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“…Halici [14] derived the generating functions and many other identities for the Fibonacci and Lucas quaternions. Akyigit et al [1,2] introduced the Fibonacci generalized quaternions and split Fibonacci quaternions. Flaut and Shpakivskyi [13] gave some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions in a generalized quaternion algebra.…”

Section: Introductionmentioning

confidence: 99%