On New Pell Spinor Sequences

Year 2024, Volume: 12 Issue: 4, 155 - 168

Tülay Erişir , Gökhan Mumcu , Mehmet Ali Güngör

https://doi.org/10.36753/mathenot.1451896

Abstract

Our motivation for this study is to define two new and particular sequences. The most essential feature of these sequences is that they are spinor sequences. In this study, these new spinor sequences obtained using spinor representations of Pell and Pell-Lucas quaternions are expressed. Moreover, some formulas such that Binet formulas, Cassini formulas and generating functions of these spinor sequences, which are called as Pell and Pell-Lucas spinor sequences, are given. Then, some relationships between Pell and Pell-Lucas spinor sequences are obtained. Therefore, an easier and more interesting representations of Pell and Pell-Lucas quaternions, which are a generalization of Pell and Pell-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.

Keywords

Pell, Pell-Lucas, Spinor

References

• [1] Koshy, T.: Fibonacci and Lucas numbers with applications, JohnWiley and Sons, Proc. New York-Toronto (2001).
• [2] Hoggatt, J.R., Verner, E.: Fibonacci And Lucas numbers, Houghton-Mifflin, 92P, Polo Alto, 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 modidifed 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] Çelik, S., Durukan, ˙I., Özkan, E.: New Recurrences On Pell Numbers, Pell-Lucas Numbers, Jacobsthal Numbers and Jacobsthal-Lucas Numbers. Chaos Solitons Fractals, 150, 111173 (2021).
• [10] Özkan, E., Uysal, M.: d-Gaussian Pell polynomials and their matrix representation Discret. Math. Algorithms Appl., 23, 2250138 (2022).
• [11] Halıcı, S.: On Bicomplex Jacobsthal-Lucas Numbers, Journal of Mathematical Sciences and Modelling, 3(3), 139–143, 2020 (2020).
• [12] Patel, N., Shrivastava, P.: Pell and Pell-Lucas Identities, Global Journal of Mathematical Sciences: Theory and Practical, 5(4), 229–236 (2013).
• [13] Koshy, T.: Pell and Pell-Lucas Numbers with Applications, Springer, New York (2014).
• [14] Halıcı, S., Da¸sdemir, A.: On Some Relationships Among Pell, Pell-Lucas And Modified Pell Sequences, Sakarya University Journal of Science, 14(2), 141–145 (2010).
• [15] Szynal-Lianna, A., Wloch, I.: Pell quaternions and Pell octonions, Adv. in Appl. Cliff. Algebr., 26(1), 435–440 (2016).
• [16] Catarino, P.: The Modified Pell and the Modified k-Pell Quaternions and Octonions, Adv. Appl. Clifford Algebras, 26, 577–590 (2016).
• [17] Çimen, C.B., ˙Ipek, A.: On pell quaternions and pell-lucas quaternions, Adv. in Appl. Cliff. Algebr., 26(1), 39–51 (2016).
• [18] Cartan, E.: The theory of spinors, Dover Publications, New York (1966).
• [19] Vivarelli, M.D.: Development of spinors descriptions of rotational mechanics from Euler’s rigid body displacement theorem, Celestial Mechanics, 32, 193–207 (1984).
• [20] Del Castillo, G.F.T., Barrales, G.S.: Spinor formulation of the differential geometry of curves, Revista Colombiana de Matematicas, 38, 27–34 (2004).
• [21] Kişi, İ., Tosun, M.: Spinor Darboux equations of curves in Euclidean 3-space, Math. Morav., 19(1), 87–93 (2015).
• [22] 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).
• [23] Erişir, T., Karadağ, N.C.: Spinor representations of involute evolute curves in E3, Fundam. J. Math. Appl., 2(2), 148–155 (2019).
• [24] Erişir, T.: On spinor construction of Bertrand curves, AIMS Mathematics, 6(4), 3583–3591 (2021).
• [25] Erişir, T., İsabeyoğlu, Z.: The Spinor Expressions of Mannheim Curves in Euclidean 3-Space, Int. Electron. J. Geom., 16(1), 111–121 (2023).
• [26] Erişir, T., Köse Öztaş, H.: Spinor Equations of Successor Curves, Univ. J. Math. Appl., 5(1), 32–41 (2022).
• [27] Okuyucu, Z., Yıldız, Ö.G. and Tosun, M.: Spinor Frenet equations in three dimensional Lie Groups, Adv. in Appl. Cliff. Algebr., 26, 1341–1348 (2016).
• [28] 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).
• [29] 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).
• [30] 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).
• [31] Tarakçıoğlu, M., Erişir, T., Güngör M.A., Tosun, M.: The hyperbolic spinor representation of transformations in R3_1 by means of split quaternions, Adv. in Appl. Cliff. Algebr., 28(1), 26 (2018).
• [32] Erişir, T., Güngör, M.A.: Fibonacci Spinors, International Journal of Geometric Methods in Modern Physics, 17(4), 2050065 (2020).
• [33] Hacısalihoğlu, H.H.: Geometry of motion and theory of quaternions, Science and Art Faculty of Gazi University Press, Ankara (1983).
• [34] Erişir, T., Yıldırım, E.: On the fundamental spinor matrices of real quaternions, WSEAS Transactions on Mathematics, 22, 854–866 (2023).
• [35] Horadam, A.F., Filipponi, P., Real Pell and Pell-Lucas numbers with real subscipts, The Fib. Quart., 33(5), 398–406 (1995).
• [36] Cerin, Z., Gianella, G.M.: On sums of Pell numbers, Acc. Sc. Torino-Atti Sc. Fis., 141, 23–31 (2007).
• [37] Serkland, C.E.: Pell Sequence and Some Generalizations, Master’s Thesis, San Jose State University, San Jose, California (1972).