Hyper-Dual Leonardo Quaternions

Year 2024, , 78 - 89, 30.09.2024

Tülay Yağmur

https://doi.org/10.53570/jnt.1525070

Abstract

In this paper, hyper-dual Leonardo quaternions are defined and studied. Some basic properties of the hyper-dual Leonardo quaternions, including their relationships with the hyper-dual Fibonacci quaternions and hyper-dual Lucas quaternions, are analyzed. In addition, some formulae and identities, such as the recurrence relations, Binet's formula, generating functions, Vajda's identity, certain sum formulae, and some binomial-sum formulae, are investigated for hyper-dual Leonardo quaternions.

Keywords

Dual numbers, hyper-dual numbers, quaternions, Fibonacci numbers, Leonardo numbers

References

• W. K. Clifford, Preliminary sketch of biquaternions, Proceeding of the London Mathematical Society s1-4 (1) (1871) 381-395.
• J. A. Fike, J. J. Alonso, The development of hyper-dual numbers for exact second-derivative calculations, in: 49th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, 2011, 17 pages.
• Y. L. Gu, J. Y. S. Luh, Dual-number transformation and its applications to robotics, IEEE Journal of Robotics and Automation RA-3 (6) (1987) 615-623.
• H. H. Cheng, Programming with dual numbers and its applications in mechanisms design, Engineering with Computers 10 (1994) 212-229.
• V. Brodsky, M. Shoham, Dual numbers representation of rigid body dynamics, Mechanism and Machine Theory 34 (1999) 693-718.
• E. Pennestri, R. Stefanelli, Linear algebra and numerical algorithms using dual numbers, Multibody Systems Dynamics 18 (2007) 323-344.
• A. Cohen, M. Shoham, Application of hyper-dual numbers to multibody kinematics, Journal of Mechanisms and Robotics 8 (2016) Article ID 011015 4 pages.
• N. Behr, G. Dattoli, A. Lattanzi, S. Licciardi, Dual numbers and operational umbral methods, Axioms 8 (2019) 77 11 pages.
• A. Cohen, M. Shoham, Hyper dual quaternions representation of rigid bodies kinematics, Mechanism and Machine Theory 150 (2020) Article ID 103861 9 pages.
• S. Aslan, M. Bekar, Y. Yaylı, Hyper-dual split quaternions and rigid body motion, Journal of Geometry and Physics 158 (2020) Article ID 103876 12 pages.
• S. Aslan, Kinematic applications of hyper-dual numbers, International Electronic Journal of Geometry 14 (2) (2021) 292-304.
• M. Fujikawa, M. Tanaka, N. Mitsume, Y. Imoto, Hyper-dual number-based numerical differentiation of eigensystems, Computer Methods in Applied Mechanics and Engineering 390 (2022) Article ID 114452 21 pages.
• B. Aktaş, O. Durmaz, H. Gündoğan, The inequalities on dual numbers and their topological structures, Turkish Journal of Mathematics 47 (5) (2023) 1318-1334.
• W. R. Hamilton, Lectures on quaternions, Hodges and Smith, Dublin, 1853.
• T. Koshy, Fibonacci and Lucas numbers with applications, John Wiley and Sons, New York, 2001.
• V. E. Hoggatt Jr., Fibonacci and Lucas numbers, Houghton Mifflin Company, Boston, 1969.
• S. Vajda, Fibonacci and Lucas numbers, and the golden section, Theory and Applications, Ellis Horwood Limited, Chichester, 1989.
• P. Catarino, A. Borges, On Leonardo numbers, Acta Mathematica Universitatis Comenianae 89 (1) (2020) 75-86.
• Y. Alp, E. G. Koçer, Some properties of Leonardo numbers, Konuralp Journal of Mathematics 9 (1) (2021) 183-189.
• A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, The American Mathematical Monthly 70 (3) (1963) 289-291.
• M. R. Iyer, A note on Fibonacci quaternions, The Fibonacci Quarterly 7 (3) (1969) 225-229.
• S. Halıcı, On Fibonacci quaternions, Advances in Applied Clifford Algebras 22 (2012) 321-327.
• P. D. Beites, P. Catarino, On the Leonardo quaternions sequence, Hacettepe Journal of Mathematics and Statistics 53 (4) (2024) 1001-1023.
• N. Ömür, S. Koparal, On hyper-dual generalized Fibonacci numbers, Notes on Number Theory and Discrete Mathematics 26 (1) (2020) 191-198.
• S. Ö. Karakuş, S. K. Nurkan, M. Turan, Hyper-dual Leonardo numbers, Konuralp Journal of Mathematics 10 (2) (2022) 269-275.
• J. P. Ward, Quaternions and Cayley numbers: Algebra and applications, Kluwer, London, 1997.
• N. R. Ait-Amrane, İ. Gök, E. Tan, Hyper-dual Horadam quaternions, Miskolc Mathematical Notes 22 (2) (2021) 903-913.