Research Article
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Year 2020, Volume: 41 Issue: 1, 1 - 10, 22.03.2020

Hasan Arslan

https://doi.org/10.17776/csj.613058
https://izlik.org/JA85FS65CT

Abstract

References

  • REFERENCES
  • [1] Hamilton, W. R., Elements of Quaternions, Longmans, Gren and Co., London, 1866.
  • [2] Chou, J. C. K., Quaternion Kinematics and Dynamic Differantial Equation, IEEE Transaction on Robotics and Automation, 8(1) (1992) 53-63.
  • [3] Conte, E., Biquaternion Quantum Mechanics, Pitagora Editrice, Via del Legatore, Blogna, Italy, 2000.
  • [4] Conway, A. W., The Quaternionic Form of Relativity, Phil. Mag., 24 (1912) 208-211.
  • [5] Gurlebeck, K., Sprossig, W., Quaternionic and Clifford Calculus for Physicists and Engineers, John Wiley & Sons, Chichester, New York, 1997.
  • [6] Jolly, D. C., Isomorphism between matrices and quaternions, Lett. Nuovo Cimento., 44(2) (1985) 80-82.
  • [7] Negi, O. P. S., Bisht, S., Bisht, P. S., Revisiting Quaternion Formulation and Electromagnetism, I1 Nuovo Cimento, 113B(12) (1998) 1449-1467.
  • [8] Tanişli, M., Özdaş, K., Application of Quaternion Representation to Stanford Manipulator, Balkan Physics Letters, 5(2) (1997) 65-68.
  • [9] Horadam, A. F., Jacobsthal and Pell Curves, Fibonacci Quart., 26 (1988) 79-83.
  • [10] Horadam, A. F., Jacobsthal Representation Numbers, Fibonacci Quart., 34 (1996) 40-54.
  • [11] Aydın, Torunbalcı F., Yüce, S., A New Approach to Jacobsthal Quaternions, Filomat, 31(18) (2017) 5567-5579.
  • [12] Aydın, Torunbalcı F., On Generalizations of the Jacobsthal Sequence, Notes on Number Theory and Discrete Mathematics, 24(1) (2018) 120-135.
  • [13] Catarino, P., The Modified Pell and Modified k-Pell Quaternions and Octonions, Adv. Appl. Clifford Algebras, 26(2) (2016) 577-590.
  • [14] Çimen, C. B., İpek, A., On Pell quaternions and Pell-Lucas quaternions, Adv. Appl. Clifford Algebras, 26(1) (2016) 39-51.
  • [15] Halici, S., On Fibonacci Quaternions, Adv. Appl. Clifford Algebras, 22(2) (2012) 321-327.
  • [16] Halici, S., On complex Fibonacci Quaternions, Adv. Appl. Clifford Algebras, 23 (2013) 105-112.
  • [17] Horadam, A. F., Complex Fibonacci Numbers and Fibonacci Quaternions, American Math. Monthly, 70(3) (1963) 289-291.
  • [18] Horadam, A. F., Quaternion recurrence relations, Ulam Quart., 2(2) (1993) 23-33.
  • [19] Szynal-Liana, A., Wloch, I., A Note on Jacobsthal Quaternions, Adv. Appl. Clifford Algebr., 26(1) (2016) 441-447.
  • [20] Aşçı, M., Gürel, E., Gaussian Jacobsthal and Gaussian Jacobsthal-Lucas polynomials, Note on Number Theory and Discrete Mathematics, 19(1) (2013) 25-36.
  • [21] Aşçı, M., Gürel, E., Gaussian Jacobsthal and Gaussian Jacobsthal-Lucas numbers, Ars Combin., 111 (2013) 53-63.
  • [22] Cerin, Z., Formulae for Sums of Jacobsthal-Lucas Numbers, International Mathematical Forum, 2(40) (2007) 1969–1984.
  • [23] Daşdemir, A., On the Jacobsthal Numbers by Matrix Method, SDU Journal of Science (E- Journal), 71 (2012) 69–76.
  • [24] Koshy, T., Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience Publication, 2001.
  • [25] Köken, F., Bozkurt, D., On the Jacobsthal Numbers by Matrix Methods, International Journal of Contemporary Mathematical Sciences, 3(13) (2008) 605–614.
  • [26] Jordan, J. H., Gaussian Fibonacci and Lucas Numbers, Fibonacci Quart., 3 (1965) 315-318.
  • [27] Pethe, S., Horadam, A. F., Generalized Gaussian Fibonacci Numbers, Bull. Austral. Math. Soc., 33(1) (1986) 37-48.
  • [28] Stephan, J. S., Fundamental Representations and Algebraic Properties of Biquaternions or Complexified Quaternions, Adv. Appl. Clifford Algebras, 21 (2011) 607-636.

On complex gaussian jacobsthal and jacobsthal-lucas quaternions

Year 2020, Volume: 41 Issue: 1, 1 - 10, 22.03.2020

Hasan Arslan

https://doi.org/10.17776/csj.613058
https://izlik.org/JA85FS65CT

Abstract

The main aim of this work is to introduce the complex Gaussian Jacobsthal and Jacobsthal-Lucas quaternions and investigate their structures. We obtain the recurrence relations, Binet formulas and generating functions for these quaternions. We also give their Cassini identities by using Binet formulas. Furthermore, we prove some results for these quaternions such as summation formulas. 

Keywords

Recurrence relation Jacobsthal and Jacobsthal-Lucas numbers Gaussian numbers Quaternions Quaternions

References

  • REFERENCES
  • [1] Hamilton, W. R., Elements of Quaternions, Longmans, Gren and Co., London, 1866.
  • [2] Chou, J. C. K., Quaternion Kinematics and Dynamic Differantial Equation, IEEE Transaction on Robotics and Automation, 8(1) (1992) 53-63.
  • [3] Conte, E., Biquaternion Quantum Mechanics, Pitagora Editrice, Via del Legatore, Blogna, Italy, 2000.
  • [4] Conway, A. W., The Quaternionic Form of Relativity, Phil. Mag., 24 (1912) 208-211.
  • [5] Gurlebeck, K., Sprossig, W., Quaternionic and Clifford Calculus for Physicists and Engineers, John Wiley & Sons, Chichester, New York, 1997.
  • [6] Jolly, D. C., Isomorphism between matrices and quaternions, Lett. Nuovo Cimento., 44(2) (1985) 80-82.
  • [7] Negi, O. P. S., Bisht, S., Bisht, P. S., Revisiting Quaternion Formulation and Electromagnetism, I1 Nuovo Cimento, 113B(12) (1998) 1449-1467.
  • [8] Tanişli, M., Özdaş, K., Application of Quaternion Representation to Stanford Manipulator, Balkan Physics Letters, 5(2) (1997) 65-68.
  • [9] Horadam, A. F., Jacobsthal and Pell Curves, Fibonacci Quart., 26 (1988) 79-83.
  • [10] Horadam, A. F., Jacobsthal Representation Numbers, Fibonacci Quart., 34 (1996) 40-54.
  • [11] Aydın, Torunbalcı F., Yüce, S., A New Approach to Jacobsthal Quaternions, Filomat, 31(18) (2017) 5567-5579.
  • [12] Aydın, Torunbalcı F., On Generalizations of the Jacobsthal Sequence, Notes on Number Theory and Discrete Mathematics, 24(1) (2018) 120-135.
  • [13] Catarino, P., The Modified Pell and Modified k-Pell Quaternions and Octonions, Adv. Appl. Clifford Algebras, 26(2) (2016) 577-590.
  • [14] Çimen, C. B., İpek, A., On Pell quaternions and Pell-Lucas quaternions, Adv. Appl. Clifford Algebras, 26(1) (2016) 39-51.
  • [15] Halici, S., On Fibonacci Quaternions, Adv. Appl. Clifford Algebras, 22(2) (2012) 321-327.
  • [16] Halici, S., On complex Fibonacci Quaternions, Adv. Appl. Clifford Algebras, 23 (2013) 105-112.
  • [17] Horadam, A. F., Complex Fibonacci Numbers and Fibonacci Quaternions, American Math. Monthly, 70(3) (1963) 289-291.
  • [18] Horadam, A. F., Quaternion recurrence relations, Ulam Quart., 2(2) (1993) 23-33.
  • [19] Szynal-Liana, A., Wloch, I., A Note on Jacobsthal Quaternions, Adv. Appl. Clifford Algebr., 26(1) (2016) 441-447.
  • [20] Aşçı, M., Gürel, E., Gaussian Jacobsthal and Gaussian Jacobsthal-Lucas polynomials, Note on Number Theory and Discrete Mathematics, 19(1) (2013) 25-36.
  • [21] Aşçı, M., Gürel, E., Gaussian Jacobsthal and Gaussian Jacobsthal-Lucas numbers, Ars Combin., 111 (2013) 53-63.
  • [22] Cerin, Z., Formulae for Sums of Jacobsthal-Lucas Numbers, International Mathematical Forum, 2(40) (2007) 1969–1984.
  • [23] Daşdemir, A., On the Jacobsthal Numbers by Matrix Method, SDU Journal of Science (E- Journal), 71 (2012) 69–76.
  • [24] Koshy, T., Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience Publication, 2001.
  • [25] Köken, F., Bozkurt, D., On the Jacobsthal Numbers by Matrix Methods, International Journal of Contemporary Mathematical Sciences, 3(13) (2008) 605–614.
  • [26] Jordan, J. H., Gaussian Fibonacci and Lucas Numbers, Fibonacci Quart., 3 (1965) 315-318.
  • [27] Pethe, S., Horadam, A. F., Generalized Gaussian Fibonacci Numbers, Bull. Austral. Math. Soc., 33(1) (1986) 37-48.
  • [28] Stephan, J. S., Fundamental Representations and Algebraic Properties of Biquaternions or Complexified Quaternions, Adv. Appl. Clifford Algebras, 21 (2011) 607-636.
There are 29 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Hasan Arslan 0000-0002-0430-8737

Submission Date August 29, 2019
Acceptance Date March 9, 2020
Publication Date March 22, 2020
DOI https://doi.org/10.17776/csj.613058
IZ https://izlik.org/JA85FS65CT
Published in Issue Year 2020 Volume: 41 Issue: 1

Cite

APA Arslan, H. (2020). On complex gaussian jacobsthal and jacobsthal-lucas quaternions. Cumhuriyet Science Journal, 41(1), 1-10. https://doi.org/10.17776/csj.613058

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