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On Generalized Commutative Quaternions with Generalized Tetranacci Number Components

Year 2023, Volume: 13 Issue: 2, 309 - 315, 29.12.2023

Özge Arıbaş Mustafa Yıldız

Abstract

In this article, generalized commutative quaternions with generalized Tetranacci number components were introduced and studied. Then, we presented some algebraic properties of these quaternions, such as a Binet-like formula and the summation formula. Furthermore, the matrix representation of these generalized commutative quaternions was given.

Keywords

Generalized quaternions generalized Tetranacci numbers quaternions Tetranacci numbers

Ethical Statement

The authors declare that there is no conflict of interest.

Supporting Institution

This research received no external funding.

References

  • Akyiğit, M., Kosal, H. H., Tosun, M. 2014. Fibonacci generalized quaternions. Adv. Appl. Clifford Algebr., 24: 631-641. Doi: 10.1007/s00006-014-0458-0
  • Bród, D., Szynal-Liana, A., Włoch, I. 2022. On some combinatorial properties of generalized commutative Jacobsthal quaternions and generalized commutative Jacobsthal-Lucas quaternions. Czechoslov. Math. J., 72: 1239-1248. Doi: 10.21136/ CMJ.2022.0174-22
  • Bród, D., Szynal-Liana, A. 2023. Generalized commutative Jacobsthal quaternions and some matrices. Examples and Counterexamples, 3: 100102. Doi: 10.1016/j.exco.2023.100102
  • Cerda-Morales, G. 2017. On a generalization for Tribonacci quaternions. Mediterr. J. Math., 14: 239. Doi: 10.1007/s00009- 017-1042-3
  • Flaut, C. 2014. A Clifford algebra associated to generalized Fibonacci quaternions. Adv. Differ. Equ., 279. Doi: 10.1186/1687- 1847-2014-279
  • Flaut, C., Savin, D. 2015. Quaternion algebras and generalized Fibonacci–Lucas quaternions. Adv. Appl. Clifford Algebr., 25: 853-862. Doi: 10.1007/s00006-015-0542-0
  • Flaut, C., Shpakivskyi, V. 2013. On generalized Fibonacci quaternions and Fibonacci-Narayana quaternions. Adv. Appl. Clifford Algebr., 23: 673–688. Doi: 10.1007/s00006-013- 0388-2
  • Halici, S., Karataş, A. 2017. On a generalization for Fibonacci quaternions. Chaos, Solitons & Fractals, 98: 178-182. Doi: 10.1016/j.chaos.2017.03.03
  • Horadam, A. F. 1963. Complex Fibonacci numbers and Fibonacci quaternions. American Mathematical Monthly, 70: 289-291. Doi: 10.2307/2313129
  • Kızılateş, C. 2017. On the Quadra Lucas-Jacobsthal Numbers. Karaelmas Science and Engineering Journal, 7(2): 619-621.
  • Kızılateş, C. 2022. On quaternions with incomplete Fibonacci and Lucas numbers components. Util. Math., 110: 263-269.
  • Kızılateş, C., Catarino, P., Tuğlu, N. 2019. On the bicomplex generalized Tribonacci quaternions. Mathematics, 7(1): 80. Doi: 10.3390/math7010080
  • Kızılateş, C., Kone, T. 2021a. On higher order Fibonacci quaternions. J. Anal. 29: 1071-1082. Doi: 10.1007/s41478-020- 00295-1
  • Kızılateş, C., Kone, T. 2021b. On higher order Fibonacci hyper complex numbers. Chaos, Solitons & Fractals, 148, 111044. Doi: 10.1016/j.chaos.2021.111044
  • Kızılateş, C., Tuglu, N., Çekim, B. 2017. Binomial transform of quadrapell sequences and quadrapell matrix sequences. J. Sci. Arts, 1(38): 69-80.
  • Özkoç, A. 2015. Some algebraic identities on quadra Fibona-Pell integer sequence. Adv. Differ. Equ., 148(2015): 1-10. Doi: 10.1186/s13662-015-0486-7
  • Petroudi, S. H. J., Pirouz, M., Ozkoc, A. 2020. On some properties of particular Tetranacci sequences. J. Int. Math. Virtual Inst., 10(2): 361-376. Doi: 10.7251/JIMVI2002361P
  • Ramírez, J. L., Sirvent, V. F. 2015. A generalization of the k-bonacci sequence from Riordan arrays. Electron. J. Comb., 22(1), P1.38: 1-20.
  • Simsek, Y. 2023. Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 117, 130: 1-26. Doi: 10.1007/s13398-023- 01464-0
  • Soykan, Y. 2020. Bicomplex Tetranacci and Tetranacci-Lucas quaternions. Commun. Math. Appl., 11(1): 95-112. Doi: 10.26713/cma.v11i1.1212
  • Swamy, M. N. S. 1973. On generalized Fibonacci quaternions, Fibonacci Q., 11(5): 547-549.
  • Szynal-Liana, A., Włoch, I. 2022. Generalized commutative quaternions of the Fibonacci type. Bol. Soc. Mat. Mex. 28: 1. Doi: 10.1007/s40590-021-00386-4
  • Szynal-Liana, A., Włoch, I., Liana, M. 2023. Generalized commutative quaternion polynomials of the Fibonacci type. Ann. Univ. Mariae Curie-Skłodowska Lub.-Pol., A- Mathematica, 76(2): 33-44. Doi: 10.17951/a.2022.76.2.33-44
  • Taşcı, D. 2009. On Quadrapell numbers and Quadrapell polynomials. Hacet. J. Math. Stat., 38(3): 265-275.
  • Taşcı, D., Acar, H. 2017. Gaussian Tetranacci numbers. Commun. Math. Appl., 8(3): 379-386. Doi: 10.26713/cma.v8i3.615
  • Waddill, M. E. 1992. The Tetranacci sequence and generalizations. Fibonacci Q., 30(1): 9-20.
  • Yeşil Baran, F., Yetiş, T. 2019. On the norms of circulant matrices via generalized Tetranacci numbers. Bilecik Seyh Edebali University Journal of Science, 6(2): 444-454. Doi: 10.35193/ bseufbd.662239
  • Yeşil Baran, F. 2021. The eigenvalues of circulant matrices with generalized Tetranacci numbers. Gümüşhane Üniversitesi Fen Bilimleri Dergisi, 11(2): 417-423. Doi: 10.17714/gumusfenbil.830575

Genelleştirilmiş Tetranacci Sayı Bileşenleri ile Genelleştirilmiş Komutatif Kuaterniyonlar Üzerine

Year 2023, Volume: 13 Issue: 2, 309 - 315, 29.12.2023

Özge Arıbaş Mustafa Yıldız

Abstract

Bu çalışmada, genelleştirilmiş Tetranacci sayı bileşenleri ile genelleştirilmiş komutatif kuaterniyonlar tanımlanmış ve incelenmiştir. Bu kuarternionlara ait Binet-benzeri formül ve toplam formülü gibi bazı cebirsel özellikler sunulmuştur. Ayrıca, bu genelleştirilmiş komutatif kuaterniyonların matris temsilcisi verilmiştir.

Keywords

Genelleştirilmiş kuaterniyonlar genelleştirilmiş Tetranacci sayıları kuaterniyonlar Tetranacci sayıları

References

  • Akyiğit, M., Kosal, H. H., Tosun, M. 2014. Fibonacci generalized quaternions. Adv. Appl. Clifford Algebr., 24: 631-641. Doi: 10.1007/s00006-014-0458-0
  • Bród, D., Szynal-Liana, A., Włoch, I. 2022. On some combinatorial properties of generalized commutative Jacobsthal quaternions and generalized commutative Jacobsthal-Lucas quaternions. Czechoslov. Math. J., 72: 1239-1248. Doi: 10.21136/ CMJ.2022.0174-22
  • Bród, D., Szynal-Liana, A. 2023. Generalized commutative Jacobsthal quaternions and some matrices. Examples and Counterexamples, 3: 100102. Doi: 10.1016/j.exco.2023.100102
  • Cerda-Morales, G. 2017. On a generalization for Tribonacci quaternions. Mediterr. J. Math., 14: 239. Doi: 10.1007/s00009- 017-1042-3
  • Flaut, C. 2014. A Clifford algebra associated to generalized Fibonacci quaternions. Adv. Differ. Equ., 279. Doi: 10.1186/1687- 1847-2014-279
  • Flaut, C., Savin, D. 2015. Quaternion algebras and generalized Fibonacci–Lucas quaternions. Adv. Appl. Clifford Algebr., 25: 853-862. Doi: 10.1007/s00006-015-0542-0
  • Flaut, C., Shpakivskyi, V. 2013. On generalized Fibonacci quaternions and Fibonacci-Narayana quaternions. Adv. Appl. Clifford Algebr., 23: 673–688. Doi: 10.1007/s00006-013- 0388-2
  • Halici, S., Karataş, A. 2017. On a generalization for Fibonacci quaternions. Chaos, Solitons & Fractals, 98: 178-182. Doi: 10.1016/j.chaos.2017.03.03
  • Horadam, A. F. 1963. Complex Fibonacci numbers and Fibonacci quaternions. American Mathematical Monthly, 70: 289-291. Doi: 10.2307/2313129
  • Kızılateş, C. 2017. On the Quadra Lucas-Jacobsthal Numbers. Karaelmas Science and Engineering Journal, 7(2): 619-621.
  • Kızılateş, C. 2022. On quaternions with incomplete Fibonacci and Lucas numbers components. Util. Math., 110: 263-269.
  • Kızılateş, C., Catarino, P., Tuğlu, N. 2019. On the bicomplex generalized Tribonacci quaternions. Mathematics, 7(1): 80. Doi: 10.3390/math7010080
  • Kızılateş, C., Kone, T. 2021a. On higher order Fibonacci quaternions. J. Anal. 29: 1071-1082. Doi: 10.1007/s41478-020- 00295-1
  • Kızılateş, C., Kone, T. 2021b. On higher order Fibonacci hyper complex numbers. Chaos, Solitons & Fractals, 148, 111044. Doi: 10.1016/j.chaos.2021.111044
  • Kızılateş, C., Tuglu, N., Çekim, B. 2017. Binomial transform of quadrapell sequences and quadrapell matrix sequences. J. Sci. Arts, 1(38): 69-80.
  • Özkoç, A. 2015. Some algebraic identities on quadra Fibona-Pell integer sequence. Adv. Differ. Equ., 148(2015): 1-10. Doi: 10.1186/s13662-015-0486-7
  • Petroudi, S. H. J., Pirouz, M., Ozkoc, A. 2020. On some properties of particular Tetranacci sequences. J. Int. Math. Virtual Inst., 10(2): 361-376. Doi: 10.7251/JIMVI2002361P
  • Ramírez, J. L., Sirvent, V. F. 2015. A generalization of the k-bonacci sequence from Riordan arrays. Electron. J. Comb., 22(1), P1.38: 1-20.
  • Simsek, Y. 2023. Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 117, 130: 1-26. Doi: 10.1007/s13398-023- 01464-0
  • Soykan, Y. 2020. Bicomplex Tetranacci and Tetranacci-Lucas quaternions. Commun. Math. Appl., 11(1): 95-112. Doi: 10.26713/cma.v11i1.1212
  • Swamy, M. N. S. 1973. On generalized Fibonacci quaternions, Fibonacci Q., 11(5): 547-549.
  • Szynal-Liana, A., Włoch, I. 2022. Generalized commutative quaternions of the Fibonacci type. Bol. Soc. Mat. Mex. 28: 1. Doi: 10.1007/s40590-021-00386-4
  • Szynal-Liana, A., Włoch, I., Liana, M. 2023. Generalized commutative quaternion polynomials of the Fibonacci type. Ann. Univ. Mariae Curie-Skłodowska Lub.-Pol., A- Mathematica, 76(2): 33-44. Doi: 10.17951/a.2022.76.2.33-44
  • Taşcı, D. 2009. On Quadrapell numbers and Quadrapell polynomials. Hacet. J. Math. Stat., 38(3): 265-275.
  • Taşcı, D., Acar, H. 2017. Gaussian Tetranacci numbers. Commun. Math. Appl., 8(3): 379-386. Doi: 10.26713/cma.v8i3.615
  • Waddill, M. E. 1992. The Tetranacci sequence and generalizations. Fibonacci Q., 30(1): 9-20.
  • Yeşil Baran, F., Yetiş, T. 2019. On the norms of circulant matrices via generalized Tetranacci numbers. Bilecik Seyh Edebali University Journal of Science, 6(2): 444-454. Doi: 10.35193/ bseufbd.662239
  • Yeşil Baran, F. 2021. The eigenvalues of circulant matrices with generalized Tetranacci numbers. Gümüşhane Üniversitesi Fen Bilimleri Dergisi, 11(2): 417-423. Doi: 10.17714/gumusfenbil.830575
There are 28 citations in total.

Details

Primary Language English
Subjects Applied Mathematics (Other)
Journal Section Research Articles
Authors

Özge Arıbaş 0000-0002-8711-2413

Mustafa Yıldız 0000-0003-3367-7176

Publication Date December 29, 2023
Published in Issue Year 2023 Volume: 13 Issue: 2

Cite

APA Arıbaş, Ö., & Yıldız, M. (2023). On Generalized Commutative Quaternions with Generalized Tetranacci Number Components. Karaelmas Fen Ve Mühendislik Dergisi, 13(2), 309-315. https://doi.org/10.7212/karaelmasfen.1359322
AMA Arıbaş Ö, Yıldız M. On Generalized Commutative Quaternions with Generalized Tetranacci Number Components. Karaelmas Fen ve Mühendislik Dergisi. December 2023;13(2):309-315. doi:10.7212/karaelmasfen.1359322
Chicago Arıbaş, Özge, and Mustafa Yıldız. “On Generalized Commutative Quaternions With Generalized Tetranacci Number Components”. Karaelmas Fen Ve Mühendislik Dergisi 13, no. 2 (December 2023): 309-15. https://doi.org/10.7212/karaelmasfen.1359322.
EndNote Arıbaş Ö, Yıldız M (December 1, 2023) On Generalized Commutative Quaternions with Generalized Tetranacci Number Components. Karaelmas Fen ve Mühendislik Dergisi 13 2 309–315.
IEEE Ö. Arıbaş and M. Yıldız, “On Generalized Commutative Quaternions with Generalized Tetranacci Number Components”, Karaelmas Fen ve Mühendislik Dergisi, vol. 13, no. 2, pp. 309–315, 2023, doi: 10.7212/karaelmasfen.1359322.
ISNAD Arıbaş, Özge - Yıldız, Mustafa. “On Generalized Commutative Quaternions With Generalized Tetranacci Number Components”. Karaelmas Fen ve Mühendislik Dergisi 13/2 (December 2023), 309-315. https://doi.org/10.7212/karaelmasfen.1359322.
JAMA Arıbaş Ö, Yıldız M. On Generalized Commutative Quaternions with Generalized Tetranacci Number Components. Karaelmas Fen ve Mühendislik Dergisi. 2023;13:309–315.
MLA Arıbaş, Özge and Mustafa Yıldız. “On Generalized Commutative Quaternions With Generalized Tetranacci Number Components”. Karaelmas Fen Ve Mühendislik Dergisi, vol. 13, no. 2, 2023, pp. 309-15, doi:10.7212/karaelmasfen.1359322.
Vancouver Arıbaş Ö, Yıldız M. On Generalized Commutative Quaternions with Generalized Tetranacci Number Components. Karaelmas Fen ve Mühendislik Dergisi. 2023;13(2):309-15.
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