In this paper, we introduce dual numbers with components including Leonardo number sequences. This novel approach facilitates our understanding of dual numbers and properties of Leonardo sequences. We also investigate fundamental properties and identities associated with Leonardo number sequences, such as Binet's formula and Catalan's, Cassini's and D'ocagne's identities. Furthermore, we also introduce a dual vector with components including Leonardo number sequences and dual angles. This extension not only deepens our understanding of dual numbers, it also highlights the interconnectedness between numerical sequences and geometric concepts. In the future it would be valuable to replicate a similar exploration and development of our findings on dual numbers with Leonardo number sequences.
Citation: Faik Babadağ, Ali Atasoy. A new approach to Leonardo number sequences with the dual vector and dual angle representation[J]. AIMS Mathematics, 2024, 9(6): 14062-14074. doi: 10.3934/math.2024684
In this paper, we introduce dual numbers with components including Leonardo number sequences. This novel approach facilitates our understanding of dual numbers and properties of Leonardo sequences. We also investigate fundamental properties and identities associated with Leonardo number sequences, such as Binet's formula and Catalan's, Cassini's and D'ocagne's identities. Furthermore, we also introduce a dual vector with components including Leonardo number sequences and dual angles. This extension not only deepens our understanding of dual numbers, it also highlights the interconnectedness between numerical sequences and geometric concepts. In the future it would be valuable to replicate a similar exploration and development of our findings on dual numbers with Leonardo number sequences.
| [1] | H. W. Guggenheimer, Differential geometry, New York: McGraw-Hill Book Company, 1963. |
| [2] | A. P. Kotelnikov, Screw calculus and some applications to geometry and mechanics, Annals of the Imperial University of Kazan, 1895. |
| [3] | E. Study, Geometrie der dynamen, Leipzig, 1903. |
| [4] |
G. R. Veldkamp, On the use of dual numbers, vectors and matrices in instantaneous, spatial kinematics, Mech. Mach. Theory, 11 (1976), 141–156. https://doi.org/10.1016/0094-114X(76)90006-9 doi: 10.1016/0094-114X(76)90006-9
|
| [5] | J. Wittenburg, Dual quaternions in kinematics of spatial mechanisms, In: Computer aided analysis and optimization of mechanical system dynamics, 9 (1984), 129–145. https://doi.org/10.1007/978-3-642-52465-3_4 |
| [6] |
A. T. Yang, F. Freudenstein, Application of dual number quaternions algebra to the analysis of spatial mechanisms, J. Appl. Mech., 31 (1964), 300–308. https://doi.org/10.1115/1.3629601 doi: 10.1115/1.3629601
|
| [7] |
P. Azariadis, N. Aspragathos, Computer graphics representation and transformation of geometric entities using dual unit vectors and line transformations, Comput. Graph., 25 (2001), 195–209. https://doi.org/10.1016/S0097-8493(00)00124-2 doi: 10.1016/S0097-8493(00)00124-2
|
| [8] | J. M. McCarthy, G. S. Soh, Geometric design of linkages, New York: Springer, 2011. https://doi.org/10.1007/978-1-4419-7892-9 |
| [9] |
Y. M. Moon, S. Kota, Automated synthesis of mechanisms using dual-vector algebra, Mech. Mach. Theory, 37 (2002), 143–166. https://doi.org/10.1016/S0094-114X(01)00073-8 doi: 10.1016/S0094-114X(01)00073-8
|
| [10] |
K. Teu, W. Kim, Estimation of the axis of a screw motion from noisy data new method based on Plucker lines, J. Biomech., 39 (2006), 2857–2862. https://doi.org/10.1016/j.jbiomech.2005.09.013 doi: 10.1016/j.jbiomech.2005.09.013
|
| [11] |
A. F. Horadam, A generalized Fibonacci sequence, Am. Math. Mon., 68 (1961), 455–459. https://doi.org/10.1080/00029890.1961.11989696 doi: 10.1080/00029890.1961.11989696
|
| [12] |
A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Am. Math. Mon., 70 (1963), 289–291. https://doi.org/10.2307/2313129 doi: 10.2307/2313129
|
| [13] | T. Koshy, Fibonacci and Lucas numbers with applications, A Wiley-Interscience Publication, 2001. https://doi.org/10.1002/9781118033067 |
| [14] | S. Vajda, Fibonacci and Lucas numbers, and the golden section: theory and applications, 1989. |
| [15] | V. E. Hoggatt, Fibonacci and Lucas numbers, Houghton Mifflin, 1969. |
| [16] |
I. A. Guven, S. K. Nurkan, A new approach to Fibonacci, Lucas numbers and dual vectors, Adv. Appl. Clifford Algebras, 25 (2015), 577–590. https://doi.org/10.1007/s00006-014-0516-7 doi: 10.1007/s00006-014-0516-7
|
| [17] | P. Catarino, A. Borges, On Leonardo numbers, Acta Math. Univ. Comen., 89 (2019), 75–86. |
| [18] | Y. Alp, E. G. Kocer, Some properties of Leonardo numbers, Konuralp J. Math., 9 (2021), 183–189. |
| [19] |
A. G. Shannon, O. Deveci, A note on generalized and extended Leonardo sequences, Notes Number Theory, 28 (2022), 109–114. https://doi.org/10.7546/nntdm.2022.28.1.109-114 doi: 10.7546/nntdm.2022.28.1.109-114
|
| [20] |
A. Karataş, On complex Leonardo numbers, Notes Number Theory, 28 (2022), 458–465. https://doi.org/10.7546/nntdm.2022.28.3.458-465 doi: 10.7546/nntdm.2022.28.3.458-465
|
| [21] |
M. Shattuck, Combinatorial proofs of identities for the generalized Leonardo numbers, Notes Number Theory, 28 (2022), 778–790. https://doi.org/10.7546/nntdm.2022.28.4.778-790 doi: 10.7546/nntdm.2022.28.4.778-790
|
| [22] |
E. Tan, D. Savin, S. Yılmaz, A new class of Leonardo hybrid numbers and some remarks on Leonardo quaternions over finite fields, Mathematics, 11 (2023), 4701. https://doi.org/10.3390/math11224701 doi: 10.3390/math11224701
|
| [23] |
S. K. Nurkan, İ. A. Güven, Ordered Leonardo quadruple numbers, Symmetry, 15 (2023), 149. https://doi.org/10.3390/sym15010149 doi: 10.3390/sym15010149
|
| [24] | K. Kuhapatanakul, J. Chobsorn, On the generalized Leonardo numbers, Integers Electron. J. Comb. Number Theory, 22 (2022), 48. |
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Faik Babadağ, Ali Atasoy. A new approach to Leonardo number sequences with the dual vector and dual angle representation[J]. AIMS Mathematics, 2024, 9(6): 14062-14074. doi: 10.3934/math.2024684
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