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Year 2024, Volume: 14 Issue: 1, 17 - 38, 28.06.2024
https://doi.org/10.37094/adyujsci.1464592

Abstract

References

  • [1] Koshy, T., Fibonacci and Lucas numbers with applications, John Wiley and Sons, Proc. New York-Toronto, 2001.
  • [2] Hoggatt, J.R., Verner, E., Fibonacci and Lucas numbers, The Fibonacci Association, California, 1979.
  • [3] Basın, S.L., Hoggatt, V.E., A primer on the Fibonacci sequence, The Fibonacci Quaterly, 2, 61-68, 1963.
  • [4] Horadam, A.F., Jacobsthal representation numbers, Fibonacci Quart., 34, 40-54, 1996.
  • [5] Horadam, A.F., Pell identities, Fibonacci Quart., 9, 245 – 252, 1971.
  • [6] Horadam, A.F., Applications of modified Pell numbers to representations, Ulam Quaterly, 3(1), 1994.
  • [7] Horadam, A.F., Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly, 70(3), 289-291, 1963.
  • [8] Horadam, A.F., Quaternion recurrence relations, Ulam Quaterly, 2(2), 23-33, 1993.
  • [9] Horadam, A.F., Filipponi, P., Real Pell and Pell-Lucas numbers with real subscipts, The Fib. Quart., 33(5), 398-406, 1995.
  • [10] Cerin, Z., Gianella, G.M., On sums of Pell numbers, Acc. Sc. Torino-Atti Sc. Fis., 141, 23-31, 2007.
  • [11] Serkland, C.E., The Pell sequence and some generalizations, Master's Thesis, San Jose State University, San Jose, California, 1972.
  • [12] Daşdemir, A., A study on the Jacobsthal and Jacobsthal-Lucas numbers, DUFED, 3(1), 13-18, 2014.
  • [13] Szynal-Lianna, A., Wloch, I., A note on Jacobsthal quaternions, Adv. in Appl. Cliff. Algebr., 26(1), 441-447, 2016.
  • [14] Torunbalcı Aydın, F., Yüce, S., A new approach to Jacobsthal quaternions, Filomat 31(18), 5567-5579, 2017.
  • [15] Halıcı, S., On bicomplex Jacobsthal-Lucas numbers, Journal of Mathematical Sciences and Modelling, 3(3), 139-143, 2020.
  • [16] Çelik, S., Durukan, İ., Özkan, E., New recurrences on Pell numbers, Pell-Lucas numbers, Jacobsthal numbers and Jacobsthal-Lucas numbers, Chaos Solitons Fractals, 150, 111173, 2021.
  • [17] Arslan, H., On complex Gaussian Jacobsthal and Jacobsthal-Lucas quaternions, Cumhuriyet Sci. J., 41(1), 1-10, 2020.
  • [18] Özkan, E., Uysal, M., On quaternions with higher order Jacobsthal numbers components, GU J Sci., 36(1), 336-347, 2023.
  • [19] Bród, D., Szynal-Liana, A., On J(r, n)-Jacobsthal quaternions, Pure and Applied Mathematics Quarterly, 14(3–4), 579-590, 2018.
  • [20] Özkan, E., Taştan, M., A New Families of Gauss k-Jacobsthal Numbers and Gauss k-Jacobsthal-Lucas Numbers and Their Polynomials, Journal of Science and Arts, 4(53), 893-908, 2020.
  • [21] Özkan, E., Uysal, M., On Hyperbolic k-Jacobsthal and k-Jacobsthal-Lucas Octonions, Notes on Number Theory and Discrete Mathematics, 28(2), 318-330, 2022.
  • [22] Cartan, E., The theory of spinors, Dover Publications, New York, 1966.
  • [23] Vivarelli, M.D., Development of spinors descriptions of rotational mechanics from Euler's rigid body displacement theorem, Celestial Mechanics, 32, 193-207, 1984.
  • [24] Torres Del Castillo, G.F., Barrales, G.S., Spinor formulation of the differential geometry of curves, Revista Colombiana de Matematicas, 38, 27-34, 2004.
  • [25] Kişi, İ., Tosun, M., Spinor Darboux equations of curves in Euclidean 3-space, Math. Morav., 19(1), 87-93, 2015.
  • [26] Ünal, D., Kişi, İ., Tosun, M., Spinor Bishop equation of curves in Euclidean 3-space, Adv. in Appl. Cliff. Algebr., 23(3), 757-765, 2013.
  • [27] Erişir, T., Kardağ, N.C., Spinor representations of involute evolute curves in , Fundam. J. Math. Appl., 2(2), 148-155, 2019.
  • [28] Erişir, T., On spinor construction of Bertrand curves, AIMS Mathematics, 6(4), 3583-3591, 2021.
  • [29] Erişir, T., İsabeyoğlu, Z., The spinor expressions of Mannheim curves in Euclidean 3-space, Int. Electron. J. Geom., 16(1), 111-121, 2023.
  • [30] Erişir, T., Köse Öztaş, H., Spinor equations of successor curves, Univ. J. Math. Appl., 5(1), 32-41, 2022.
  • [31] Şenyurt, S., Çalışkan, A., Spinor Formulation of Sabban Frame of Curve on S2, Pure Mathematical Sciences, 4(1), 37-42, 2015.
  • [32] Okuyucu, O. Z., Yıldız, Ö. G., and Tosun, M., Spinor Frenet equations in three dimensional Lie Groups, Adv. in Appl. Cliff. Algebr., 26, 1341-1348, 2016.
  • [33] Ketenci, Z., Erişir, T., Güngör, M.A., A construction of hyperbolic spinors according to Frenet frame in Minkowski space, Journal of Dynamical Systems and Geometric Theories, 13(2), 179-193, 2015.
  • [34] Balcı, Y., Erişir, T., Güngör, M.A., Hyperbolic spinor Darboux equations of spacelike curves in Minkowski 3-space, Journal of the Chungcheong Mathematical Society, 28(4), 525-535, 2015.
  • [35] Erişir, T., Güngör, M.A., Tosun, M., Geometry of the hyperbolic spinors corresponding to alternative frame, Adv. in Appl. Cliff. Algebr., 25(4), 799-810, 2015.
  • [36] Tarakçıoğlu, M., Erişir, T., Güngör, M.A., Tosun, M., The hyperbolic spinor representation of transformations in by means of split quaternions, Adv. in Appl. Cliff. Algebr., 28(1), 26, 2018.
  • [37] Erişir, T., Güngör, M.A., Fibonacci spinors, International Journal of Geometric Methods in Modern Physics, 17(4), 2050065, 2020.
  • [38] Hacısalihoğlu, H.H., Geometry of motion and theory of quaternions, Science and Art Faculty of Gazi University Press, Ankara, 1983.
  • [39] Erişir, T., Yıldırım, E., On the fundamental spinor matrices of real quaternions, WSEAS Transactions on Mathematics, 22, 854-866, 2023.

On New Spinor Sequences of Jacobsthal and Jacobsthal-Lucas Quaternions

Year 2024, Volume: 14 Issue: 1, 17 - 38, 28.06.2024
https://doi.org/10.37094/adyujsci.1464592

Abstract

In this study, two new spinor sequences using spinor representations of Jacobsthal and Jacobsthal-Lucas quaternions are defined. Moreover, some formulas such that Binet, Cassini, sum formulas and generating functions of these spinor sequences, which are called as Jacobsthal and Jacobsthal-Lucas spinor sequences, are expressed. Then, the some relationships between Jacobsthal and Jacobsthal-Lucas spinors are obtained. Therefore, an easier and more interesting representation of Jacobsthal and Jacobsthal-Lucas quaternions, which are generalization of Jacobsthal and Jacobsthal-Lucas number sequences, are obtained. We believe that these new spinor sequences will be useful and advantageable in many branches of science, such as geometry, algebra and physics.

References

  • [1] Koshy, T., Fibonacci and Lucas numbers with applications, John Wiley and Sons, Proc. New York-Toronto, 2001.
  • [2] Hoggatt, J.R., Verner, E., Fibonacci and Lucas numbers, The Fibonacci Association, California, 1979.
  • [3] Basın, S.L., Hoggatt, V.E., A primer on the Fibonacci sequence, The Fibonacci Quaterly, 2, 61-68, 1963.
  • [4] Horadam, A.F., Jacobsthal representation numbers, Fibonacci Quart., 34, 40-54, 1996.
  • [5] Horadam, A.F., Pell identities, Fibonacci Quart., 9, 245 – 252, 1971.
  • [6] Horadam, A.F., Applications of modified Pell numbers to representations, Ulam Quaterly, 3(1), 1994.
  • [7] Horadam, A.F., Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly, 70(3), 289-291, 1963.
  • [8] Horadam, A.F., Quaternion recurrence relations, Ulam Quaterly, 2(2), 23-33, 1993.
  • [9] Horadam, A.F., Filipponi, P., Real Pell and Pell-Lucas numbers with real subscipts, The Fib. Quart., 33(5), 398-406, 1995.
  • [10] Cerin, Z., Gianella, G.M., On sums of Pell numbers, Acc. Sc. Torino-Atti Sc. Fis., 141, 23-31, 2007.
  • [11] Serkland, C.E., The Pell sequence and some generalizations, Master's Thesis, San Jose State University, San Jose, California, 1972.
  • [12] Daşdemir, A., A study on the Jacobsthal and Jacobsthal-Lucas numbers, DUFED, 3(1), 13-18, 2014.
  • [13] Szynal-Lianna, A., Wloch, I., A note on Jacobsthal quaternions, Adv. in Appl. Cliff. Algebr., 26(1), 441-447, 2016.
  • [14] Torunbalcı Aydın, F., Yüce, S., A new approach to Jacobsthal quaternions, Filomat 31(18), 5567-5579, 2017.
  • [15] Halıcı, S., On bicomplex Jacobsthal-Lucas numbers, Journal of Mathematical Sciences and Modelling, 3(3), 139-143, 2020.
  • [16] Çelik, S., Durukan, İ., Özkan, E., New recurrences on Pell numbers, Pell-Lucas numbers, Jacobsthal numbers and Jacobsthal-Lucas numbers, Chaos Solitons Fractals, 150, 111173, 2021.
  • [17] Arslan, H., On complex Gaussian Jacobsthal and Jacobsthal-Lucas quaternions, Cumhuriyet Sci. J., 41(1), 1-10, 2020.
  • [18] Özkan, E., Uysal, M., On quaternions with higher order Jacobsthal numbers components, GU J Sci., 36(1), 336-347, 2023.
  • [19] Bród, D., Szynal-Liana, A., On J(r, n)-Jacobsthal quaternions, Pure and Applied Mathematics Quarterly, 14(3–4), 579-590, 2018.
  • [20] Özkan, E., Taştan, M., A New Families of Gauss k-Jacobsthal Numbers and Gauss k-Jacobsthal-Lucas Numbers and Their Polynomials, Journal of Science and Arts, 4(53), 893-908, 2020.
  • [21] Özkan, E., Uysal, M., On Hyperbolic k-Jacobsthal and k-Jacobsthal-Lucas Octonions, Notes on Number Theory and Discrete Mathematics, 28(2), 318-330, 2022.
  • [22] Cartan, E., The theory of spinors, Dover Publications, New York, 1966.
  • [23] Vivarelli, M.D., Development of spinors descriptions of rotational mechanics from Euler's rigid body displacement theorem, Celestial Mechanics, 32, 193-207, 1984.
  • [24] Torres Del Castillo, G.F., Barrales, G.S., Spinor formulation of the differential geometry of curves, Revista Colombiana de Matematicas, 38, 27-34, 2004.
  • [25] Kişi, İ., Tosun, M., Spinor Darboux equations of curves in Euclidean 3-space, Math. Morav., 19(1), 87-93, 2015.
  • [26] Ünal, D., Kişi, İ., Tosun, M., Spinor Bishop equation of curves in Euclidean 3-space, Adv. in Appl. Cliff. Algebr., 23(3), 757-765, 2013.
  • [27] Erişir, T., Kardağ, N.C., Spinor representations of involute evolute curves in , Fundam. J. Math. Appl., 2(2), 148-155, 2019.
  • [28] Erişir, T., On spinor construction of Bertrand curves, AIMS Mathematics, 6(4), 3583-3591, 2021.
  • [29] Erişir, T., İsabeyoğlu, Z., The spinor expressions of Mannheim curves in Euclidean 3-space, Int. Electron. J. Geom., 16(1), 111-121, 2023.
  • [30] Erişir, T., Köse Öztaş, H., Spinor equations of successor curves, Univ. J. Math. Appl., 5(1), 32-41, 2022.
  • [31] Şenyurt, S., Çalışkan, A., Spinor Formulation of Sabban Frame of Curve on S2, Pure Mathematical Sciences, 4(1), 37-42, 2015.
  • [32] Okuyucu, O. Z., Yıldız, Ö. G., and Tosun, M., Spinor Frenet equations in three dimensional Lie Groups, Adv. in Appl. Cliff. Algebr., 26, 1341-1348, 2016.
  • [33] Ketenci, Z., Erişir, T., Güngör, M.A., A construction of hyperbolic spinors according to Frenet frame in Minkowski space, Journal of Dynamical Systems and Geometric Theories, 13(2), 179-193, 2015.
  • [34] Balcı, Y., Erişir, T., Güngör, M.A., Hyperbolic spinor Darboux equations of spacelike curves in Minkowski 3-space, Journal of the Chungcheong Mathematical Society, 28(4), 525-535, 2015.
  • [35] Erişir, T., Güngör, M.A., Tosun, M., Geometry of the hyperbolic spinors corresponding to alternative frame, Adv. in Appl. Cliff. Algebr., 25(4), 799-810, 2015.
  • [36] Tarakçıoğlu, M., Erişir, T., Güngör, M.A., Tosun, M., The hyperbolic spinor representation of transformations in by means of split quaternions, Adv. in Appl. Cliff. Algebr., 28(1), 26, 2018.
  • [37] Erişir, T., Güngör, M.A., Fibonacci spinors, International Journal of Geometric Methods in Modern Physics, 17(4), 2050065, 2020.
  • [38] Hacısalihoğlu, H.H., Geometry of motion and theory of quaternions, Science and Art Faculty of Gazi University Press, Ankara, 1983.
  • [39] Erişir, T., Yıldırım, E., On the fundamental spinor matrices of real quaternions, WSEAS Transactions on Mathematics, 22, 854-866, 2023.
There are 39 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Mathematics
Authors

Tülay Erişir 0000-0001-6444-1460

Mehmet Ali Güngör 0000-0003-1863-3183

Publication Date June 28, 2024
Submission Date April 4, 2024
Acceptance Date June 14, 2024
Published in Issue Year 2024 Volume: 14 Issue: 1

Cite

APA Erişir, T., & Güngör, M. A. (2024). On New Spinor Sequences of Jacobsthal and Jacobsthal-Lucas Quaternions. Adıyaman University Journal of Science, 14(1), 17-38. https://doi.org/10.37094/adyujsci.1464592
AMA Erişir T, Güngör MA. On New Spinor Sequences of Jacobsthal and Jacobsthal-Lucas Quaternions. ADYU J SCI. June 2024;14(1):17-38. doi:10.37094/adyujsci.1464592
Chicago Erişir, Tülay, and Mehmet Ali Güngör. “On New Spinor Sequences of Jacobsthal and Jacobsthal-Lucas Quaternions”. Adıyaman University Journal of Science 14, no. 1 (June 2024): 17-38. https://doi.org/10.37094/adyujsci.1464592.
EndNote Erişir T, Güngör MA (June 1, 2024) On New Spinor Sequences of Jacobsthal and Jacobsthal-Lucas Quaternions. Adıyaman University Journal of Science 14 1 17–38.
IEEE T. Erişir and M. A. Güngör, “On New Spinor Sequences of Jacobsthal and Jacobsthal-Lucas Quaternions”, ADYU J SCI, vol. 14, no. 1, pp. 17–38, 2024, doi: 10.37094/adyujsci.1464592.
ISNAD Erişir, Tülay - Güngör, Mehmet Ali. “On New Spinor Sequences of Jacobsthal and Jacobsthal-Lucas Quaternions”. Adıyaman University Journal of Science 14/1 (June 2024), 17-38. https://doi.org/10.37094/adyujsci.1464592.
JAMA Erişir T, Güngör MA. On New Spinor Sequences of Jacobsthal and Jacobsthal-Lucas Quaternions. ADYU J SCI. 2024;14:17–38.
MLA Erişir, Tülay and Mehmet Ali Güngör. “On New Spinor Sequences of Jacobsthal and Jacobsthal-Lucas Quaternions”. Adıyaman University Journal of Science, vol. 14, no. 1, 2024, pp. 17-38, doi:10.37094/adyujsci.1464592.
Vancouver Erişir T, Güngör MA. On New Spinor Sequences of Jacobsthal and Jacobsthal-Lucas Quaternions. ADYU J SCI. 2024;14(1):17-38.

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