In this paper, we introduce two types of hyper-dual numbers with components including Pell and Pell-Lucas numbers. This novel approach facilitates our understanding of hyper-dual numbers and properties of Pell and Pell-Lucas numbers. We also investigate fundamental properties and identities associated with Pell and Pell-Lucas numbers, such as the Binet-like formulas, Vajda-like, Catalan-like, Cassini-like, and d'Ocagne-like identities. Furthermore, we also define hyper-dual vectors by using Pell and Pell-Lucas vectors and discuse hyper-dual angles. This extensionis not only dependent on our understanding of dual numbers, it also highlights the interconnectedness between integer sequences and geometric concepts.
Citation: Faik Babadağ, Ali Atasoy. On hyper-dual vectors and angles with Pell, Pell-Lucas numbers[J]. AIMS Mathematics, 2024, 9(11): 30655-30666. doi: 10.3934/math.20241480
In this paper, we introduce two types of hyper-dual numbers with components including Pell and Pell-Lucas numbers. This novel approach facilitates our understanding of hyper-dual numbers and properties of Pell and Pell-Lucas numbers. We also investigate fundamental properties and identities associated with Pell and Pell-Lucas numbers, such as the Binet-like formulas, Vajda-like, Catalan-like, Cassini-like, and d'Ocagne-like identities. Furthermore, we also define hyper-dual vectors by using Pell and Pell-Lucas vectors and discuse hyper-dual angles. This extensionis not only dependent on our understanding of dual numbers, it also highlights the interconnectedness between integer sequences and geometric concepts.
[1] | A. P. Kotelnikov, Screw calculus and some applications to geometry and mechanics, In: Annals of the Imperial university of Kazan, 1895. |
[2] | E. Study, Geometry der dynamen, Leipzig, 1901. |
[3] | A. M. Frydryszak, Dual numbers and supersymmetric mechanics, Czech. J. Phys., 55 (2005), 1409–1414. https://doi.org/10.1007/s10582-006-0018-5 doi: 10.1007/s10582-006-0018-5 |
[4] | R. M. Wald, A new type of gauge invariance for a collection of massles spin-2 fields. II. Geometrical interpretation, Class. Quantum Grav., 5 (1987), 1279. https://doi.org/10.1088/0264-9381/4/5/025 doi: 10.1088/0264-9381/4/5/025 |
[5] | I. Fischer, Dual-number methods in kinematics, statics and dynamics, New York: CRC Press, 1998. https://doi.org/10.1201/9781315141473 |
[6] | W. B. V. Kandasamy, F. Smarandache, Dual numbers, Ohio, USA: ZIP Publishing, 2012. |
[7] | 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 |
[8] | J. Fike, J. Alonso, The development of hyper-dual numbers for exact second-derivative calculations, 49th AIAA Aerodpace sciences meeting including the new horizons forum and aerospace exposition, 2012. https://doi.org/10.2514/6.2011-886 |
[9] | J. A. Fike, J. J. Alonso, Automatic differentiation through the use of hyper-dual numbers for second derivatives, In: Recent advances in algorithmic differentiation, Heidelberg: Springer, 87 (2011), 163–173. https://doi.org/10.1007/978-3-642-30023-3_15 |
[10] | A. Cohen, M. Shoham, Application of hyper-dual numbers to rigid bodies equations of motion, Mech. Mach. Theory, 111 (2017), 76–84. https://doi.org/10.1016/j.mechmachtheory.2017.01.013 doi: 10.1016/j.mechmachtheory.2017.01.013 |
[11] | A. F. Horadam, A generalized Fibonacci sequence, Amer. Math. Monthly, 69 (1961), 455–459. https://doi.org/10.1080/00029890.1961.11989696 doi: 10.1080/00029890.1961.11989696 |
[12] | T. Koshy, Fibonacci and Lucas numbers with applications, John Wiley & Sons, Inc., 2017. https://doi.org/10.1002/9781118742327 |
[13] | E. Verner, J. Hoggatt, Fibonacci and Lucas numbers, Houghton Mifflin, 1969. |
[14] | Y. Panwar, A note on the generalized k-Fibonacci sequence, Naturengs, 2 (2021), 29–39. https://doi.org/10.46572/naturengs.937010 doi: 10.46572/naturengs.937010 |
[15] | W. M. Abd-Elhameed, A. Napoli, New formulas of convolved Pell polynomials, AIMS Mathematics, 9 (2023), 565–593. http://dx.doi.org/10.3934/math.2024030 doi: 10.3934/math.2024030 |
[16] | M. Bicknell, A primer on the Pell sequence and related sequence, Fibonacci Quart., 13 (1975), 345–349. https://doi.org/10.1080/00150517.1975.12430627 doi: 10.1080/00150517.1975.12430627 |
[17] | A. F. Horadam, Pell identities, Fibonacci Quart., 9 (1971), 245–263. https://doi.org/10.1080/00150517.1971.12431004 |
[18] | T. Koshy, Pell and Pell-Lucas numbers with applications, New York: Springer, 2014. https://doi.org/10.1007/978-1-4614-8489-9 |