Araştırma Makalesi
BibTex RIS Kaynak Göster

On Dual Quaternions with $k-$Generalized Leonardo Components

Yıl 2023, Sayı: 44, 31 - 42, 30.09.2023
https://doi.org/10.53570/jnt.1328605

Öz

In this paper, we define a one-parameter generalization of Leonardo dual quaternions, namely $k-$generalized Leonardo-like dual quaternions. We introduce the properties of $k$-generalized Leonardo-like dual quaternions, including relations with Leonardo, Fibonacci, and Lucas dual quaternions. We investigate their characteristic relations, involving the Binet-like formula, the generating function, the summation formula, Catalan-like, Cassini-like, d'Ocagne-like, Tagiuri-like, and Hornsberger-like identities. The crucial part of the present paper is that one can reduce the calculations of Leonardo-like dual quaternions by considering $k$. For $k=1$, these results are generalizations of the ones for ordered Leonardo quadruple numbers. Finally, we discuss the need for further research.

Destekleyen Kurum

-

Proje Numarası

-

Teşekkür

-

Kaynakça

  • T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley \& Sons, New York, 2001.
  • P. Catarino, A. Borges, On Leonardo Numbers, Acta Mathematica Universitatis Comenianae 89 (1) (2020) 75--86.
  • E. W. Dijkstra, Fibonacci Numbers and Leonardo Numbers (1981), https://www.cs.utexas.edu/users/EWD/transcriptions/EWD07xx/EWD797.html, Accessed 10 July 2023.
  • K. Kuhapatanakul, J. Chobsorn, On the Generalized Leonardo Numbers, Integers (22) (2022) Article ID A48 7 pages.
  • P. Catarino, A. Borges, A Note on Incomplete Leonardo Numbers, Integers (20) (2020) Article ID A43 7 pages.
  • Y. Alp, E. G. Koçer, Hybrid Leonardo Numbers, Chaos, Solitons \& Fractals (150) (2021) Article ID 111128 5 pages.
  • Y. Alp, E. G. Koçer, Some Properties of Leonardo Numbers, Konuralp Journal of Mathematics 9 (1) (2021) 183--189.
  • A. Shannon, Ö. Deveci, A Note on Generalized and Extended Leonardo Sequences, Notes on Number Theory and Discrete Mathematics 28 (1) (2022) 109--114.
  • A. Karataş, On Complex Leonardo Numbers, Notes on Number Theory and Discrete Mathematics 28 (3) (2022) 458--465.
  • S. Ö. Karakuş, S. K. Nurkan, M. Turan, Hyper-Dual Leonardo Numbers, Konuralp Journal of Mathematics 10 (2) (2022) 269--275.
  • M. Shattuck, Combinatorial Proofs of Identities for the Generalized Leonardo Numbers, Notes on Number Theory and Discrete Mathematics 28 (4) (2022) 778--790.
  • Y. Soykan, Special Cases of Generalized Leonardo Numbers: Modified $p$-Leonardo, $p$-Leonardo-Lucas and $p$-Leonardo Numbers, Earthline Journal of Mathematical Sciences 11 (2) (2023) 317--342.
  • S. K. Nurkan, İ. A. Güven, Ordered Leonardo Quadruple Numbers, Symmetry 15 (1) (2023) Article ID 149 15 pages.
  • E. Tan, H. H. Leung, On Leonardo $p$-Numbers, Integers (23) (2023) Article ID A7 11 pages.
  • O. Dişkaya, H. Menken, P. M. M. C. Catarino, On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions, Journal of New Theory (42) (2023) 74--85.
  • A. F. Horadam, Complex Fibonacci Numbers and Fibonacci Quaternions, American Mathematical Monthly (70) (1963) 289--291.
  • W. R. Hamilton, Elements of Quaternions, Chelsea Publishing Company, New York, 1969.
  • W. R. Hamilton, Lectures on Quaternions, Hodges and Smith, Dublin, 1853.
  • W. R. Hamilton, On Quaternions; or On a New System of Imaginaries in Algebra, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science (3rd Series), xxv-xxxvi, (1844–1850), https://www.emis.de/classics/Hamilton/OnQuat.pdf, Accessed 10 July 2023.
  • S. Yüce, F. T. Aydın, A New Aspect of Dual Fibonacci Quaternions, Advances in Applied Clifford Algebras (26) (2016) 873--884.
  • W. K. Clifford, Preliminary Sketch of Bi-Quaternions, Proceedings of the London Mathematical Society s1–4 (1) (1873) 381--395.
  • J. D. Jr. Edmonds, Relativistic Reality: A Modern View, World Scientific, Singapore, 1997.
  • Z. Ercan, S. Yüce, On Properties of the Dual Quaternions, European Journal of Pure and Applied Mathematics 4 (2) (2011) 142--146.
  • V. Majernik, Quaternion Formulation of the Galilean Space-Time Transformation, Acta Physica Slovaca 56 (1) (2006) 9--14.
  • V. Majernik, M. Nagy, Quaternionic Form of Maxwell's Equations with Sources, Lettere al Nuovo Cimento (16) (1976) 165--169.
  • V. Majernik, Galilean Transformation Expressed by the Dual Four-Component Numbers, Acta Physica Polonica A (87) (1995) 919--923.
  • Y. Yaylı, E. E. Tutuncu, Generalized Galilean Transformations and Dual Quaternions, Scientia Magna 5 (1) (2009) 94--100.
Yıl 2023, Sayı: 44, 31 - 42, 30.09.2023
https://doi.org/10.53570/jnt.1328605

Öz

Proje Numarası

-

Kaynakça

  • T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley \& Sons, New York, 2001.
  • P. Catarino, A. Borges, On Leonardo Numbers, Acta Mathematica Universitatis Comenianae 89 (1) (2020) 75--86.
  • E. W. Dijkstra, Fibonacci Numbers and Leonardo Numbers (1981), https://www.cs.utexas.edu/users/EWD/transcriptions/EWD07xx/EWD797.html, Accessed 10 July 2023.
  • K. Kuhapatanakul, J. Chobsorn, On the Generalized Leonardo Numbers, Integers (22) (2022) Article ID A48 7 pages.
  • P. Catarino, A. Borges, A Note on Incomplete Leonardo Numbers, Integers (20) (2020) Article ID A43 7 pages.
  • Y. Alp, E. G. Koçer, Hybrid Leonardo Numbers, Chaos, Solitons \& Fractals (150) (2021) Article ID 111128 5 pages.
  • Y. Alp, E. G. Koçer, Some Properties of Leonardo Numbers, Konuralp Journal of Mathematics 9 (1) (2021) 183--189.
  • A. Shannon, Ö. Deveci, A Note on Generalized and Extended Leonardo Sequences, Notes on Number Theory and Discrete Mathematics 28 (1) (2022) 109--114.
  • A. Karataş, On Complex Leonardo Numbers, Notes on Number Theory and Discrete Mathematics 28 (3) (2022) 458--465.
  • S. Ö. Karakuş, S. K. Nurkan, M. Turan, Hyper-Dual Leonardo Numbers, Konuralp Journal of Mathematics 10 (2) (2022) 269--275.
  • M. Shattuck, Combinatorial Proofs of Identities for the Generalized Leonardo Numbers, Notes on Number Theory and Discrete Mathematics 28 (4) (2022) 778--790.
  • Y. Soykan, Special Cases of Generalized Leonardo Numbers: Modified $p$-Leonardo, $p$-Leonardo-Lucas and $p$-Leonardo Numbers, Earthline Journal of Mathematical Sciences 11 (2) (2023) 317--342.
  • S. K. Nurkan, İ. A. Güven, Ordered Leonardo Quadruple Numbers, Symmetry 15 (1) (2023) Article ID 149 15 pages.
  • E. Tan, H. H. Leung, On Leonardo $p$-Numbers, Integers (23) (2023) Article ID A7 11 pages.
  • O. Dişkaya, H. Menken, P. M. M. C. Catarino, On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions, Journal of New Theory (42) (2023) 74--85.
  • A. F. Horadam, Complex Fibonacci Numbers and Fibonacci Quaternions, American Mathematical Monthly (70) (1963) 289--291.
  • W. R. Hamilton, Elements of Quaternions, Chelsea Publishing Company, New York, 1969.
  • W. R. Hamilton, Lectures on Quaternions, Hodges and Smith, Dublin, 1853.
  • W. R. Hamilton, On Quaternions; or On a New System of Imaginaries in Algebra, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science (3rd Series), xxv-xxxvi, (1844–1850), https://www.emis.de/classics/Hamilton/OnQuat.pdf, Accessed 10 July 2023.
  • S. Yüce, F. T. Aydın, A New Aspect of Dual Fibonacci Quaternions, Advances in Applied Clifford Algebras (26) (2016) 873--884.
  • W. K. Clifford, Preliminary Sketch of Bi-Quaternions, Proceedings of the London Mathematical Society s1–4 (1) (1873) 381--395.
  • J. D. Jr. Edmonds, Relativistic Reality: A Modern View, World Scientific, Singapore, 1997.
  • Z. Ercan, S. Yüce, On Properties of the Dual Quaternions, European Journal of Pure and Applied Mathematics 4 (2) (2011) 142--146.
  • V. Majernik, Quaternion Formulation of the Galilean Space-Time Transformation, Acta Physica Slovaca 56 (1) (2006) 9--14.
  • V. Majernik, M. Nagy, Quaternionic Form of Maxwell's Equations with Sources, Lettere al Nuovo Cimento (16) (1976) 165--169.
  • V. Majernik, Galilean Transformation Expressed by the Dual Four-Component Numbers, Acta Physica Polonica A (87) (1995) 919--923.
  • Y. Yaylı, E. E. Tutuncu, Generalized Galilean Transformations and Dual Quaternions, Scientia Magna 5 (1) (2009) 94--100.
Toplam 27 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Cebir ve Sayı Teorisi
Bölüm Araştırma Makalesi
Yazarlar

Çiğdem Zeynep Yılmaz 0000-0002-1598-9902

Gülsüm Yeliz Saçlı 0000-0002-8647-1801

Proje Numarası -
Yayımlanma Tarihi 30 Eylül 2023
Gönderilme Tarihi 17 Temmuz 2023
Yayımlandığı Sayı Yıl 2023 Sayı: 44

Kaynak Göster

APA Yılmaz, Ç. Z., & Saçlı, G. Y. (2023). On Dual Quaternions with $k-$Generalized Leonardo Components. Journal of New Theory(44), 31-42. https://doi.org/10.53570/jnt.1328605
AMA Yılmaz ÇZ, Saçlı GY. On Dual Quaternions with $k-$Generalized Leonardo Components. JNT. Eylül 2023;(44):31-42. doi:10.53570/jnt.1328605
Chicago Yılmaz, Çiğdem Zeynep, ve Gülsüm Yeliz Saçlı. “On Dual Quaternions With $k-$Generalized Leonardo Components”. Journal of New Theory, sy. 44 (Eylül 2023): 31-42. https://doi.org/10.53570/jnt.1328605.
EndNote Yılmaz ÇZ, Saçlı GY (01 Eylül 2023) On Dual Quaternions with $k-$Generalized Leonardo Components. Journal of New Theory 44 31–42.
IEEE Ç. Z. Yılmaz ve G. Y. Saçlı, “On Dual Quaternions with $k-$Generalized Leonardo Components”, JNT, sy. 44, ss. 31–42, Eylül 2023, doi: 10.53570/jnt.1328605.
ISNAD Yılmaz, Çiğdem Zeynep - Saçlı, Gülsüm Yeliz. “On Dual Quaternions With $k-$Generalized Leonardo Components”. Journal of New Theory 44 (Eylül 2023), 31-42. https://doi.org/10.53570/jnt.1328605.
JAMA Yılmaz ÇZ, Saçlı GY. On Dual Quaternions with $k-$Generalized Leonardo Components. JNT. 2023;:31–42.
MLA Yılmaz, Çiğdem Zeynep ve Gülsüm Yeliz Saçlı. “On Dual Quaternions With $k-$Generalized Leonardo Components”. Journal of New Theory, sy. 44, 2023, ss. 31-42, doi:10.53570/jnt.1328605.
Vancouver Yılmaz ÇZ, Saçlı GY. On Dual Quaternions with $k-$Generalized Leonardo Components. JNT. 2023(44):31-42.


TR Dizin 26024

Electronic Journals Library (EZB) 13651



Academindex 28993

SOBİAD 30256                                                   

Scilit 20865                                                  


29324 As of 2021, JNT is licensed under a Creative Commons Attribution-NonCommercial 4.0 International Licence (CC BY-NC).