Year 2025,
Volume: 17 Issue: 2, 527 - 533, 30.12.2025
Murat Turan
,
Sıddıka Özkaldı Karakuş
References
-
Adler, S.L., Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, New York, 88, 1995.
-
Alp, Y., Koçer, E.G., Some properties of Leonardo numbers, Konuralp J. Math., 9(1)(2021), 183–189.
-
Alp, Y., Koçer, E.G., Hybrid Leonardo numbers, Chaos, Solitons Fractals, 150(2021), 111–128.
-
Andrews, G.E., Askey, R., Roy, R., Special Functions, Cambridge University Press, Cambridge, 71, 1999.
-
Akkuş, İ., Kızılaslan, G., Quaternions: Quantum calculus approach with applications, Kuwait J. Sci., 46(4)(2019), 1–13.
-
Arfken, G.B., Weber, H.J., Mathematical Methods for Physicists, American Association of Physics Teachers, 1999.
-
Aydın Torunbalcı, F., Bicomplex Fibonacci quaternions, Chaos, Solitons Fractals, 106(2)(2018), 147–153.
-
Aydın Torunbalcı, F., q-Fibonacci bicomplex and q-Lucas bicomplex numbers, Notes Number Theory Discrete Math., 28(2)(2022), 261–275.
-
Aydın Torunbalcı, F., q-Leonardo bicomplex numbers, Konuralp J. Math., 11(2)(2023), 176–183.
-
da Fonseca, C.M., Kızılateş, C., Saraiva, P., Shannon, A.G., Generalised Leonardo numbers, Logic J. IGPL, (2025).
-
Catarino, P., Borges, A., On Leonardo numbers, Acta Math. Univ. Comenianae, 89(1)(2019), 75–86.
-
Clifford, W.K., Preliminary sketch of biquaternions, Proc. London Math. Soc., 4(64)(1873), 381–395.
-
Cohen, A., Shoham, M., Application of hyper-dual numbers to multi-body kinematics, J. Mech. Rob., 8(2015).
-
Cohen, A., Shoham, M., Application of hyper-dual numbers to rigid bodies equations of motion, Mech. Mach. Theory, 111(2017), 76–84.
-
Fike, J. A., Numerically exact derivative calculations using hyper-dual numbers, 3rd Annual Student Workshop in Simulation-Based Engineering and Design, (2009).
-
Fike, J.A., Alonso, J.J., The development of hyper-dual numbers for exact second-derivative calculations, 49th AIAA Aerospace Sciences Meeting, (2011).
-
Güven, İ.A., Nurkan, S.K., A new approach to Fibonacci, Lucas numbers and dual vectors, Adv. Appl. Clifford Algebras, 25(2015), 577–590.
-
Halıcı, S., On Fibonacci quaternions, Adv. Appl. Clifford Algebras, 22(2012), 321–327.
-
Halıcı, S., On complex Fibonacci quaternions, Adv. Appl. Clifford Algebras, 23(1)(2013), 105–112.
-
Hoggatt, V.E., Fibonacci and Lucas Numbers, Fibonacci Association, Houghton Mifflin, (1969).
-
Horadam, A.F., Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly, 70(3)(1963), 289–291.
-
Horadam, A.F., Basic properties of a certain generalized sequence of numbers, Fibonacci Q., 3(1965), 161–176.
-
Horadam, A.F., Quaternions recurrence relations, Ulam Quarterly, 2(2)(1993), 23–33.
-
Iyer, M.R., A note on Fibonacci quaternions, Fibonacci Q., 7(3)(1969), 225–229.
-
Iyer, M.R., Some results on Fibonacci quaternions, Fibonacci Q., 7(1969), 201–210.
-
Kürüz, F., Dağdeviren, A., Catarino, P., On Leonardo Pisano hybrinomials, Mathematics, 9(2021), 2923.
-
Kızılateş, C., Kone, T., On higher order Fibonacci hyper complex numbers, Chaos Solitons Fractals, 148(2021), 111044.
-
Kızılateş, C., On quaternions with incomplete Fibonacci and Lucas numbers components, Appl. Math. Comput. Sci. Stat., 110(2022).
-
Kızılateş, C., Du, W.-S., Terzioğlu, N., Chen, R.-C., New properties and matrix representations on higher-order generalized Fibonacci quaternions with q-integer components, Axioms, 13(2024), 677.
-
Kızılateş, C., Polatlı, E., Terzioğlu, N., Du, W.-S., On higher-order generalized Fibonacci hybrid numbers with q-integer components, Symmetry, 17(2025), 584.
-
Koshy, T., Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, Hoboken, NJ, (2019).
-
Turan, M., Özkaldı Karakuş, S., Kaya Nurkan, S., A new perspective on bicomplex numbers with Leonardo number components, Commun. Fac. Sci. Univ. Ankara Ser. A1 Math. Stat., 72(2)(2023), 340–351.
-
Nurkan, S.K., Güven, İ.A., Dual Fibonacci quaternions, Adv. Appl. Clifford Algebras, 25(2)(2015), 403–414.
-
Nurkan, S.K., Güven, İ.A., A note on bicomplex Fibonacci and Lucas numbers, Int. J. Pure Appl. Math., 120(3)(2015), 365–377.
-
Nurkan, S.K., Güven, İ.A., Ordered Leonardo quadruple numbers, Symmetry, 15(2023), 149.
-
Omur, N., Koparal, S., On hyper-dual generalized Fibonacci numbers, Notes Number Theory Discrete Math., 26(1)(2020), 191–198.
-
Shannon, A.G., A note on generalized Leonardo numbers, Notes Number Theory Discrete Math., 25(3)(2019), 97–101.
-
Sloane, N.J.A., The On-line Encyclopedia of Integer Sequences, (1964), http://oeis.org.
-
Vajda, S., Fibonacci and Lucas Numbers and the Golden Section, Ellis Horwood Ltd., England, 1989.
-
Karakuş, S.Ö., Nurkan, S.K., Turan, M., Hyper-dual Leonardo numbers, Konuralp J. Math., 10(2)(2022), 269–275.
-
Vajda, S.,Fibonacci and Lucas Numbers and the Golden Section, Ellis Horwood Limited Publ., England, 1989.
A New Perspective On Hyper Dual Fibonnaci And Hyper Dual Lucas Numbers
Year 2025,
Volume: 17 Issue: 2, 527 - 533, 30.12.2025
Murat Turan
,
Sıddıka Özkaldı Karakuş
Abstract
In this paper, we introduction hyper dual numbers with hyper dual Fibonacci and Lucas number coefficients . Firstly, we obtained for these new number recurrence relation and Binet’s formula. Using Binet’s formula, we obtained some sum formulas and specific identities.
References
-
Adler, S.L., Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, New York, 88, 1995.
-
Alp, Y., Koçer, E.G., Some properties of Leonardo numbers, Konuralp J. Math., 9(1)(2021), 183–189.
-
Alp, Y., Koçer, E.G., Hybrid Leonardo numbers, Chaos, Solitons Fractals, 150(2021), 111–128.
-
Andrews, G.E., Askey, R., Roy, R., Special Functions, Cambridge University Press, Cambridge, 71, 1999.
-
Akkuş, İ., Kızılaslan, G., Quaternions: Quantum calculus approach with applications, Kuwait J. Sci., 46(4)(2019), 1–13.
-
Arfken, G.B., Weber, H.J., Mathematical Methods for Physicists, American Association of Physics Teachers, 1999.
-
Aydın Torunbalcı, F., Bicomplex Fibonacci quaternions, Chaos, Solitons Fractals, 106(2)(2018), 147–153.
-
Aydın Torunbalcı, F., q-Fibonacci bicomplex and q-Lucas bicomplex numbers, Notes Number Theory Discrete Math., 28(2)(2022), 261–275.
-
Aydın Torunbalcı, F., q-Leonardo bicomplex numbers, Konuralp J. Math., 11(2)(2023), 176–183.
-
da Fonseca, C.M., Kızılateş, C., Saraiva, P., Shannon, A.G., Generalised Leonardo numbers, Logic J. IGPL, (2025).
-
Catarino, P., Borges, A., On Leonardo numbers, Acta Math. Univ. Comenianae, 89(1)(2019), 75–86.
-
Clifford, W.K., Preliminary sketch of biquaternions, Proc. London Math. Soc., 4(64)(1873), 381–395.
-
Cohen, A., Shoham, M., Application of hyper-dual numbers to multi-body kinematics, J. Mech. Rob., 8(2015).
-
Cohen, A., Shoham, M., Application of hyper-dual numbers to rigid bodies equations of motion, Mech. Mach. Theory, 111(2017), 76–84.
-
Fike, J. A., Numerically exact derivative calculations using hyper-dual numbers, 3rd Annual Student Workshop in Simulation-Based Engineering and Design, (2009).
-
Fike, J.A., Alonso, J.J., The development of hyper-dual numbers for exact second-derivative calculations, 49th AIAA Aerospace Sciences Meeting, (2011).
-
Güven, İ.A., Nurkan, S.K., A new approach to Fibonacci, Lucas numbers and dual vectors, Adv. Appl. Clifford Algebras, 25(2015), 577–590.
-
Halıcı, S., On Fibonacci quaternions, Adv. Appl. Clifford Algebras, 22(2012), 321–327.
-
Halıcı, S., On complex Fibonacci quaternions, Adv. Appl. Clifford Algebras, 23(1)(2013), 105–112.
-
Hoggatt, V.E., Fibonacci and Lucas Numbers, Fibonacci Association, Houghton Mifflin, (1969).
-
Horadam, A.F., Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly, 70(3)(1963), 289–291.
-
Horadam, A.F., Basic properties of a certain generalized sequence of numbers, Fibonacci Q., 3(1965), 161–176.
-
Horadam, A.F., Quaternions recurrence relations, Ulam Quarterly, 2(2)(1993), 23–33.
-
Iyer, M.R., A note on Fibonacci quaternions, Fibonacci Q., 7(3)(1969), 225–229.
-
Iyer, M.R., Some results on Fibonacci quaternions, Fibonacci Q., 7(1969), 201–210.
-
Kürüz, F., Dağdeviren, A., Catarino, P., On Leonardo Pisano hybrinomials, Mathematics, 9(2021), 2923.
-
Kızılateş, C., Kone, T., On higher order Fibonacci hyper complex numbers, Chaos Solitons Fractals, 148(2021), 111044.
-
Kızılateş, C., On quaternions with incomplete Fibonacci and Lucas numbers components, Appl. Math. Comput. Sci. Stat., 110(2022).
-
Kızılateş, C., Du, W.-S., Terzioğlu, N., Chen, R.-C., New properties and matrix representations on higher-order generalized Fibonacci quaternions with q-integer components, Axioms, 13(2024), 677.
-
Kızılateş, C., Polatlı, E., Terzioğlu, N., Du, W.-S., On higher-order generalized Fibonacci hybrid numbers with q-integer components, Symmetry, 17(2025), 584.
-
Koshy, T., Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, Hoboken, NJ, (2019).
-
Turan, M., Özkaldı Karakuş, S., Kaya Nurkan, S., A new perspective on bicomplex numbers with Leonardo number components, Commun. Fac. Sci. Univ. Ankara Ser. A1 Math. Stat., 72(2)(2023), 340–351.
-
Nurkan, S.K., Güven, İ.A., Dual Fibonacci quaternions, Adv. Appl. Clifford Algebras, 25(2)(2015), 403–414.
-
Nurkan, S.K., Güven, İ.A., A note on bicomplex Fibonacci and Lucas numbers, Int. J. Pure Appl. Math., 120(3)(2015), 365–377.
-
Nurkan, S.K., Güven, İ.A., Ordered Leonardo quadruple numbers, Symmetry, 15(2023), 149.
-
Omur, N., Koparal, S., On hyper-dual generalized Fibonacci numbers, Notes Number Theory Discrete Math., 26(1)(2020), 191–198.
-
Shannon, A.G., A note on generalized Leonardo numbers, Notes Number Theory Discrete Math., 25(3)(2019), 97–101.
-
Sloane, N.J.A., The On-line Encyclopedia of Integer Sequences, (1964), http://oeis.org.
-
Vajda, S., Fibonacci and Lucas Numbers and the Golden Section, Ellis Horwood Ltd., England, 1989.
-
Karakuş, S.Ö., Nurkan, S.K., Turan, M., Hyper-dual Leonardo numbers, Konuralp J. Math., 10(2)(2022), 269–275.
-
Vajda, S.,Fibonacci and Lucas Numbers and the Golden Section, Ellis Horwood Limited Publ., England, 1989.