Research Article
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Year 2025, Volume: 17 Issue: 2, 527 - 533, 30.12.2025
https://doi.org/10.47000/tjmcs.1564888
https://izlik.org/JA36SP58MC

Abstract

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
https://doi.org/10.47000/tjmcs.1564888
https://izlik.org/JA36SP58MC

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.
There are 41 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory, Algebraic and Differential Geometry
Journal Section Research Article
Authors

Murat Turan 0000-0001-9684-7924

Sıddıka Özkaldı Karakuş 0000-0002-2699-4109

Submission Date October 10, 2024
Acceptance Date September 15, 2025
Publication Date December 30, 2025
DOI https://doi.org/10.47000/tjmcs.1564888
IZ https://izlik.org/JA36SP58MC
Published in Issue Year 2025 Volume: 17 Issue: 2

Cite

APA Turan, M., & Özkaldı Karakuş, S. (2025). A New Perspective On Hyper Dual Fibonnaci And Hyper Dual Lucas Numbers. Turkish Journal of Mathematics and Computer Science, 17(2), 527-533. https://doi.org/10.47000/tjmcs.1564888
AMA 1.Turan M, Özkaldı Karakuş S. A New Perspective On Hyper Dual Fibonnaci And Hyper Dual Lucas Numbers. TJMCS. 2025;17(2):527-533. doi:10.47000/tjmcs.1564888
Chicago Turan, Murat, and Sıddıka Özkaldı Karakuş. 2025. “A New Perspective On Hyper Dual Fibonnaci And Hyper Dual Lucas Numbers”. Turkish Journal of Mathematics and Computer Science 17 (2): 527-33. https://doi.org/10.47000/tjmcs.1564888.
EndNote Turan M, Özkaldı Karakuş S (December 1, 2025) A New Perspective On Hyper Dual Fibonnaci And Hyper Dual Lucas Numbers. Turkish Journal of Mathematics and Computer Science 17 2 527–533.
IEEE [1]M. Turan and S. Özkaldı Karakuş, “A New Perspective On Hyper Dual Fibonnaci And Hyper Dual Lucas Numbers”, TJMCS, vol. 17, no. 2, pp. 527–533, Dec. 2025, doi: 10.47000/tjmcs.1564888.
ISNAD Turan, Murat - Özkaldı Karakuş, Sıddıka. “A New Perspective On Hyper Dual Fibonnaci And Hyper Dual Lucas Numbers”. Turkish Journal of Mathematics and Computer Science 17/2 (December 1, 2025): 527-533. https://doi.org/10.47000/tjmcs.1564888.
JAMA 1.Turan M, Özkaldı Karakuş S. A New Perspective On Hyper Dual Fibonnaci And Hyper Dual Lucas Numbers. TJMCS. 2025;17:527–533.
MLA Turan, Murat, and Sıddıka Özkaldı Karakuş. “A New Perspective On Hyper Dual Fibonnaci And Hyper Dual Lucas Numbers”. Turkish Journal of Mathematics and Computer Science, vol. 17, no. 2, Dec. 2025, pp. 527-33, doi:10.47000/tjmcs.1564888.
Vancouver 1.Murat Turan, Sıddıka Özkaldı Karakuş. A New Perspective On Hyper Dual Fibonnaci And Hyper Dual Lucas Numbers. TJMCS. 2025 Dec. 1;17(2):527-33. doi:10.47000/tjmcs.1564888