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Year 2021, Volume: 4 Issue: 1, 10 - 16, 01.03.2021

Kahraman Esen Özen Murat Tosun

https://doi.org/10.33401/fujma.811058

Abstract

References

  • [1] B. L. van der Waerden, Hamilton’s discovery of quaternions, Math. Mag., 49 (1976), 227-234.
  • [2] W. R. Hamilton, Lectures on quaternions, Hodges and Smith, Dublin, 1853.
  • [3] A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Am. Math. Mon., 70 (1963), 289-291.
  • [4] S. Halıcı, On complex Fibonacci quaternions, Adv. Appl. Clifford Algebras, 23 (2013), 105-112.
  • [5] M. R. Iyer, Some results on Fibonacci quaternions, Fibonacci Q., 7 (1969), 201-210.
  • [6] T. Erişir, M. A. Güngör, On Fibonacci spinors, Int. J. Geom. Methods Mod. (2020), DOI: 10.1142/S0219887820500656.
  • [7] M. Akyiğit, H. H. Kösal, M. Tosun, Fibonacci generalized quaternions, Adv. Appl. Clifford Algebras, 24 (2014), 631-641.
  • [8] C. Flaut, V. Shpakivskyi, Real matrix representations for the complex quaternions, Adv. Appl. Clifford Algebras, 23 (2013), 657-671.
  • [9] K. E. Özen, M. Tosun, Elliptic biquaternion algebra, AIP Conf. Proc., 1926 (2018), 020032.
  • [10] I. M. Yaglom, Complex Numbers in Geometry, Academic Press, Newyork, 1968.
  • [11] A. A. Harkin, J. B. Harkin, Geometry of generalized complex numbers, Math. Mag., 77(2) (2004), 118-129.
  • [12] K. Eren, S. Ersoy, Burmester theory in Cayley-Klein planes with affine base, J. Geom., 109(3) (2018), 45.
  • [13] K. Eren, S. Ersoy, Revisiting Burmester theory with complex forms of Bottema’s instantaneous invariants, Complex Var. Elliptic Equ., 62(4) (2017), 431-437.
  • [14] Z. Derin, M. A. Güngör, On Lorentz transformations with elliptic biquaternions. In: Hvedri, I. (ed.) Tblisi-Mathematics, pp 121-140. Sciendo, Berlin (2020).
  • [15] Y. Kulaç, M. Tosun, Some equations on p-complex Fibonacci numbers, AIP Conf. Proc., 1926 (2018), 020024.
  • [16] R. A. Dunlap, The golden ratio and Fibonacci numbers, World Scientific, 1997.
  • [17] T. Koshy, Fibonacci and Lucas numbers with applications, A Wiley-Interscience publication, U.S.A, 2001.
  • [18] K. E. Özen, M. Tosun, Further results for elliptic biquaternions, Conf. Proc. Sci. Technol., 1 (2018), 20-27.
  • [19] K. E. Özen, M. Tosun, Elliptic matrix representations of elliptic biquaternions and their applications, Int. Electron. J. Geom., 11 (2018), 96-103.
  • [20] K. E. Özen, M. Tosun, A general method for solving linear elliptic biquaternion equations, Complex Var. Elliptic Equ., (2020), 1-12, DOI: 10.1080/17476933.2020.1738409.
  • [21] K. E. Özen, A general method for solving linear matrix equations of elliptic biquaternions with applications, AIMS Math., 5 (2020), 2211–2225.

Fibonacci Elliptic Biquaternions

Year 2021, Volume: 4 Issue: 1, 10 - 16, 01.03.2021

Kahraman Esen Özen Murat Tosun

https://doi.org/10.33401/fujma.811058

Abstract

A. F. Horadam defined the complex Fibonacci numbers and Fibonacci quaternions in the middle of the 20th century. Half a century later, S. Hal{\i}c{\i} introduced the complex Fibonacci quaternions by inspiring from these definitions and discussed some properties of them. Recently, the elliptic biquaternions, which are  generalized form of the complex and real quaternions, have been presented. In this study, we introduce the set of Fibonacci elliptic biquaternions that includes the set of complex Fibonacci quaternions as a special case and investigate some properties of Fibonacci elliptic biquaternions. Furthermore, we give the Binet formula and Cassini's identity in terms of Fibonacci elliptic biquaternions. Finally, we give elliptic and real matrix representations of Fibonacci elliptic biquaternions.

Keywords

Fibonacci numbers Quaternions Elliptic biquaternions Matrix representations

References

  • [1] B. L. van der Waerden, Hamilton’s discovery of quaternions, Math. Mag., 49 (1976), 227-234.
  • [2] W. R. Hamilton, Lectures on quaternions, Hodges and Smith, Dublin, 1853.
  • [3] A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Am. Math. Mon., 70 (1963), 289-291.
  • [4] S. Halıcı, On complex Fibonacci quaternions, Adv. Appl. Clifford Algebras, 23 (2013), 105-112.
  • [5] M. R. Iyer, Some results on Fibonacci quaternions, Fibonacci Q., 7 (1969), 201-210.
  • [6] T. Erişir, M. A. Güngör, On Fibonacci spinors, Int. J. Geom. Methods Mod. (2020), DOI: 10.1142/S0219887820500656.
  • [7] M. Akyiğit, H. H. Kösal, M. Tosun, Fibonacci generalized quaternions, Adv. Appl. Clifford Algebras, 24 (2014), 631-641.
  • [8] C. Flaut, V. Shpakivskyi, Real matrix representations for the complex quaternions, Adv. Appl. Clifford Algebras, 23 (2013), 657-671.
  • [9] K. E. Özen, M. Tosun, Elliptic biquaternion algebra, AIP Conf. Proc., 1926 (2018), 020032.
  • [10] I. M. Yaglom, Complex Numbers in Geometry, Academic Press, Newyork, 1968.
  • [11] A. A. Harkin, J. B. Harkin, Geometry of generalized complex numbers, Math. Mag., 77(2) (2004), 118-129.
  • [12] K. Eren, S. Ersoy, Burmester theory in Cayley-Klein planes with affine base, J. Geom., 109(3) (2018), 45.
  • [13] K. Eren, S. Ersoy, Revisiting Burmester theory with complex forms of Bottema’s instantaneous invariants, Complex Var. Elliptic Equ., 62(4) (2017), 431-437.
  • [14] Z. Derin, M. A. Güngör, On Lorentz transformations with elliptic biquaternions. In: Hvedri, I. (ed.) Tblisi-Mathematics, pp 121-140. Sciendo, Berlin (2020).
  • [15] Y. Kulaç, M. Tosun, Some equations on p-complex Fibonacci numbers, AIP Conf. Proc., 1926 (2018), 020024.
  • [16] R. A. Dunlap, The golden ratio and Fibonacci numbers, World Scientific, 1997.
  • [17] T. Koshy, Fibonacci and Lucas numbers with applications, A Wiley-Interscience publication, U.S.A, 2001.
  • [18] K. E. Özen, M. Tosun, Further results for elliptic biquaternions, Conf. Proc. Sci. Technol., 1 (2018), 20-27.
  • [19] K. E. Özen, M. Tosun, Elliptic matrix representations of elliptic biquaternions and their applications, Int. Electron. J. Geom., 11 (2018), 96-103.
  • [20] K. E. Özen, M. Tosun, A general method for solving linear elliptic biquaternion equations, Complex Var. Elliptic Equ., (2020), 1-12, DOI: 10.1080/17476933.2020.1738409.
  • [21] K. E. Özen, A general method for solving linear matrix equations of elliptic biquaternions with applications, AIMS Math., 5 (2020), 2211–2225.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Kahraman Esen Özen 0000-0002-3299-6709

Murat Tosun 0000-0002-4888-1412

Publication Date March 1, 2021
Submission Date October 15, 2020
Acceptance Date January 6, 2021
Published in Issue Year 2021 Volume: 4 Issue: 1

Cite

APA Özen, K. E., & Tosun, M. (2021). Fibonacci Elliptic Biquaternions. Fundamental Journal of Mathematics and Applications, 4(1), 10-16. https://doi.org/10.33401/fujma.811058
AMA Özen KE, Tosun M. Fibonacci Elliptic Biquaternions. FUJMA. March 2021;4(1):10-16. doi:10.33401/fujma.811058
Chicago Özen, Kahraman Esen, and Murat Tosun. “Fibonacci Elliptic Biquaternions”. Fundamental Journal of Mathematics and Applications 4, no. 1 (March 2021): 10-16. https://doi.org/10.33401/fujma.811058.
EndNote Özen KE, Tosun M (March 1, 2021) Fibonacci Elliptic Biquaternions. Fundamental Journal of Mathematics and Applications 4 1 10–16.
IEEE K. E. Özen and M. Tosun, “Fibonacci Elliptic Biquaternions”, FUJMA, vol. 4, no. 1, pp. 10–16, 2021, doi: 10.33401/fujma.811058.
ISNAD Özen, Kahraman Esen - Tosun, Murat. “Fibonacci Elliptic Biquaternions”. Fundamental Journal of Mathematics and Applications 4/1 (March 2021), 10-16. https://doi.org/10.33401/fujma.811058.
JAMA Özen KE, Tosun M. Fibonacci Elliptic Biquaternions. FUJMA. 2021;4:10–16.
MLA Özen, Kahraman Esen and Murat Tosun. “Fibonacci Elliptic Biquaternions”. Fundamental Journal of Mathematics and Applications, vol. 4, no. 1, 2021, pp. 10-16, doi:10.33401/fujma.811058.
Vancouver Özen KE, Tosun M. Fibonacci Elliptic Biquaternions. FUJMA. 2021;4(1):10-6.
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