Year 2022,
Volume: 10 Issue: 2, 269 - 275, 31.10.2022
Sıddıka Özkaldı Karakuş
,
Semra Kaya Nurkan
,
Murat Turan
References
- 1] Y. Alp and E. G Kocer, Some properties of Leonardo numbers, Konuralp J. Math. 9(1) (2021), 183-189.
- [2] Y. Alp and E. G Kocer, Hybrid Leonardo numbers, Chaos, Solitons and Fractals 150 (2021), 111-128.
- [3] F. R. V. Alves and P. Catarino, A forma matricial dos n ́umeros de Leonardo, Ciencia e natura, 42, (2020).
- [4] P. Catarino and A. Borges, On Leonardo Numbers, Acta Mathematica Universitatis Comenianae, Vol:89, No.1 (2019), 75-86.
- [5] P. Catarino and A. Borges, A Note on Incomplete Leonardo Numbers, Integers, Vol:20 (2020).
- [6] W. K Clifford, Preliminary sketch of biquaternions, Proc London Mathematical Society. 4(64), (1873), 381-395.
- [7] A. Cohen and M. Shoham, Application of hyper-dual numbers to multi-body kinematics. J. Mech. Rob. 8, (2015). doi: 10.1115/1.4030588.
- [8] A. Cohen and M. Shoham, Application of hyper-dual numbers to rigid bodies equations of motion. J. Mech. Mach. Theory. 111, (2017), 76-84 .
- [9] J. A. Fike, Numerically exact derivative calculations using hyper-dual numbers, 3rd Annual Student Joint Workshop in Simulation-Based Engineering and Design, (2009).
- [10] J. A. Fike and J. 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. (2011), 4-7.
- [11] S. Halici, On Fibonacci quaternions, Adv. Appl. Clifford Algebras, 22, (2012) 321-327.
- [12] Hoggatt V. E., Fibonacci and Lucas Numbers, A publication of the Fibonacci Association University of Santa Clara, Santa Clara, Houghton Mifflin Company, 1969.
- [13] A. F. Horadam, Basic properties of a certain generalized sequence of numbers. Fibonacci Q., 3 (1965), 161-176.
- [14] C. Kızılates and T. Kone, On higher order Fibonacci quaternions, J. Anal. (2021), DOI: 10.1007/s41478-020-00295-1.
- [15] Kızılates C, Kone T, On higher order Fibonacci hyper complex numbers, Chaos Solitons Fractals 148 (2021), 111044.
- [16] Koshy T., Fibonacci and Lucas Numbers with Applications, John Wiley and Sons: Hoboken, NJ, USA, 2019.
- [17] F. Kuruz, A. Dagdeviren and P. Catarino, On Leonardo Pisano Hybrinomials, Mathematics, 9 (2021), 2923, https:/doi.org/10.3390/math9222923
- [18] S. K. Nurkan, I. A. Guven, Dual Fibonacci quaternions, Adv. Appl. Clifford Algebras, 25(2) (2015), 403-414.
- [19] N. Omur, S. Koparal, On hyper-dual generalized Fibonacci numbers, Notes on Number Theory and Discrete Mathematics, 26(1) (2020), 191-198.
- [20] A. G. Shannon, A note on generalized Leonardo numbers, Notes Number Theory Discrete Math 25(3) (2019), 97-101. doi: 10.7546/nntdm.2019.25.3.97-101.
- [21] Sloane N. J. A., The on-line encyclopedia of integers sequences, Http.//oeis.org , 1964.
- [22] Vajda S., Fibonacci and Lucas Numbers and the golden section, Ellis Horwood Limited Publ., England, 1989.
Hyper-Dual Leonardo Numbers
Year 2022,
Volume: 10 Issue: 2, 269 - 275, 31.10.2022
Sıddıka Özkaldı Karakuş
,
Semra Kaya Nurkan
,
Murat Turan
Abstract
In the present paper, the hyper-dual Leonardo numbers will be introduced with the use of Leonardo numbers. Some algebraic properties of these numbers such as recurrence relation, generating function, Catalan's and Cassini's identity, Binet's formula, sum formulas will also be obtained.
References
- 1] Y. Alp and E. G Kocer, Some properties of Leonardo numbers, Konuralp J. Math. 9(1) (2021), 183-189.
- [2] Y. Alp and E. G Kocer, Hybrid Leonardo numbers, Chaos, Solitons and Fractals 150 (2021), 111-128.
- [3] F. R. V. Alves and P. Catarino, A forma matricial dos n ́umeros de Leonardo, Ciencia e natura, 42, (2020).
- [4] P. Catarino and A. Borges, On Leonardo Numbers, Acta Mathematica Universitatis Comenianae, Vol:89, No.1 (2019), 75-86.
- [5] P. Catarino and A. Borges, A Note on Incomplete Leonardo Numbers, Integers, Vol:20 (2020).
- [6] W. K Clifford, Preliminary sketch of biquaternions, Proc London Mathematical Society. 4(64), (1873), 381-395.
- [7] A. Cohen and M. Shoham, Application of hyper-dual numbers to multi-body kinematics. J. Mech. Rob. 8, (2015). doi: 10.1115/1.4030588.
- [8] A. Cohen and M. Shoham, Application of hyper-dual numbers to rigid bodies equations of motion. J. Mech. Mach. Theory. 111, (2017), 76-84 .
- [9] J. A. Fike, Numerically exact derivative calculations using hyper-dual numbers, 3rd Annual Student Joint Workshop in Simulation-Based Engineering and Design, (2009).
- [10] J. A. Fike and J. 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. (2011), 4-7.
- [11] S. Halici, On Fibonacci quaternions, Adv. Appl. Clifford Algebras, 22, (2012) 321-327.
- [12] Hoggatt V. E., Fibonacci and Lucas Numbers, A publication of the Fibonacci Association University of Santa Clara, Santa Clara, Houghton Mifflin Company, 1969.
- [13] A. F. Horadam, Basic properties of a certain generalized sequence of numbers. Fibonacci Q., 3 (1965), 161-176.
- [14] C. Kızılates and T. Kone, On higher order Fibonacci quaternions, J. Anal. (2021), DOI: 10.1007/s41478-020-00295-1.
- [15] Kızılates C, Kone T, On higher order Fibonacci hyper complex numbers, Chaos Solitons Fractals 148 (2021), 111044.
- [16] Koshy T., Fibonacci and Lucas Numbers with Applications, John Wiley and Sons: Hoboken, NJ, USA, 2019.
- [17] F. Kuruz, A. Dagdeviren and P. Catarino, On Leonardo Pisano Hybrinomials, Mathematics, 9 (2021), 2923, https:/doi.org/10.3390/math9222923
- [18] S. K. Nurkan, I. A. Guven, Dual Fibonacci quaternions, Adv. Appl. Clifford Algebras, 25(2) (2015), 403-414.
- [19] N. Omur, S. Koparal, On hyper-dual generalized Fibonacci numbers, Notes on Number Theory and Discrete Mathematics, 26(1) (2020), 191-198.
- [20] A. G. Shannon, A note on generalized Leonardo numbers, Notes Number Theory Discrete Math 25(3) (2019), 97-101. doi: 10.7546/nntdm.2019.25.3.97-101.
- [21] Sloane N. J. A., The on-line encyclopedia of integers sequences, Http.//oeis.org , 1964.
- [22] Vajda S., Fibonacci and Lucas Numbers and the golden section, Ellis Horwood Limited Publ., England, 1989.