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Yilmaz, Nazmiye

Split complex bi-periodic Fibonacci and Lucas numbers. (English) Zbl 1491.11026

Commun. Fac. Sci. Univ. Ank., Sér. A1, Math. Stat. 71, No. 1, 153-164 (2022).
Summary: The initial idea of this paper is to investigate the split complex bi-periodic Fibonacci and Lucas numbers by using SCFLN now on. We try to show some properties of SCFLN by taking into account the properties of the split complex numbers. Then, we present interesting relationships between SCFLN.

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
17A45 Quadratic algebras (but not quadratic Jordan algebras)

Keywords:

bi-periodic Fibonacci number; bi-periodic Lucas number; generalized Fibonacci number; split complex bi-periodic Fibonacci and Lucas numbers
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\textit{N. Yilmaz}, Commun. Fac. Sci. Univ. Ank., Sér. A1, Math. Stat. 71, No. 1, 153--164 (2022; Zbl 1491.11026)
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References:

[1] Aydin, F. T., Hyperbolic Fibonacci sequence, Universal Journal of Mathematics and Applications, 2(2) (2019), 59-64. https://doi.org/10.32323/ujma.473514
[2] Bala, A., Verma, V., Some properties of bi-variate bi-periodic Lucas polynomials, Annals of the Romanian Society for Cell Biology, 25(4) (2021), 8778-8784. https://www.annalsofrscb.ro/index.php/journal/article/view/3598.
[3] Bilgici, G., Two generalizations of Lucas sequence, Applied Mathematics and Computation, 245 (2014), 526-538. https://doi.org/10.1016/j.amc.2014.07.111. · Zbl 1335.40002
[4] Catoni, F., Boccaletti, R., Cannata, R., Catoni, V., Nichelatti, E., Zampatti, P., The Mathematics of Minkowski Space-Time, Birkhauser, Basel, 2008. · Zbl 1151.53001
[5] Dikmen, C.M., Hyperbolic Jacobsthal Numbers, Asian Research Journal of Mathematics, (2019) 1-9. https://doi.org/10.9734/arjom/2019/v15i430153
[6] Edson, M., Yayenie, O., A new generalization of Fibonacci sequences and extended Binet’s formula, Integers, 9(A48) (2009), 639-654. https://doi.org/10.1515/INTEG.2009.051 · Zbl 1248.11009
[7] Gargoubi, H., Kossentini, S., f-algebra structure on hyperbolic numbers, Adv. Appl. Clifford Algebr., 26(4) (2016), 1211-1233. https://doi.org/10.1007/s00006-016-0644-3 · Zbl 1403.06028
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[9] Khadjiev, D., Goksal Y., Applications of hyperbolic numbers to the invariant theory in two-dimensional pseudo-Euclidean space, Adv. Appl. Clifford Algebr., 26 (2016), 645-668. https://doi.org/10.1007/s00006-015-0627-9 · Zbl 1342.51016
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[11] Motter, A. E, Rosa, A. F., Hyperbolic calculus, Adv. Appl. Clifford Algebr., 8(1) (1998), 109-128. · Zbl 0922.30035
[12] Sobczyk, G., The hyperbolic number plane, The College Mathematics Journal, 26(4) (1995), 268-280.
[13] Soykan, Y., On hyperbolic numbers with generalized Fibonacci numbers components, preprint, (2019). · Zbl 1465.11066
[14] Soykan, Y., Gocen, M., Properties of hyperbolic generalized Pell numbers, Notes on Number Theory and Discrete Mathematics, 26(4) (2020), 136-153. https://doi.org/10.7546/nntdm.2020.26.4.136-153
[15] Tan, E., Leung, H.H., Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences, Advances in Difference Equations, 2020.1 (2020), 1-11. https://doi.org/10.1186/s13662-020-2507-4 · Zbl 1487.11018
[16] Tasyurdu, Y., Hyperbolic Tribonacci and Tribonacci-Lucas sequences, International Journal of Mathematical Analysis, 13(12) (2019), 565-572. https://doi.org/10.12988/ijma.2019.91167
[17] Verma, V., Bala, A., On properties of generalized bi-variate bi-periodic Fibonacci polynomials, International Journal of Advanced Science and Technology, 29(3) (2020), 8065-8072.
[18] Yayenie, O., A note on generalized Fibonacci sequence, Applied Mathematics and Computation, 217(12) (2011), 5603-5611. https://doi.org/10.1016/j.amc.2010.12.038 · Zbl 1226.11023
[19] Aydin, F. T., Hyperbolic Fibonacci sequence, Universal Journal of Mathematics and Applications, 2(2) (2019), 59-64. https://doi.org/10.32323/ujma.473514
[20] Bala, A., Verma, V., Some properties of bi-variate bi-periodic Lucas polynomials, Annals of the Romanian Society for Cell Biology, 25(4) (2021), 8778-8784. https://www.annalsofrscb.ro/index.php/journal/article/view/3598.
[21] Bilgici, G., Two generalizations of Lucas sequence, Applied Mathematics and Computation, 245 (2014), 526-538. https://doi.org/10.1016/j.amc.2014.07.111. · Zbl 1335.40002
[22] Catoni, F., Boccaletti, R., Cannata, R., Catoni, V., Nichelatti, E., Zampatti, P., The Mathematics of Minkowski Space-Time, Birkhauser, Basel, 2008. · Zbl 1151.53001
[23] Dikmen, C.M., Hyperbolic Jacobsthal Numbers, Asian Research Journal of Mathematics, (2019) 1-9. https://doi.org/10.9734/arjom/2019/v15i430153
[24] Edson, M., Yayenie, O., A new generalization of Fibonacci sequences and extended Binet’s formula, Integers, 9(A48) (2009), 639-654. https://doi.org/10.1515/INTEG.2009.051 · Zbl 1248.11009
[25] Gargoubi, H., Kossentini, S., f-algebra structure on hyperbolic numbers, Adv. Appl. Clifford Algebr., 26(4) (2016), 1211-1233. https://doi.org/10.1007/s00006-016-0644-3 · Zbl 1403.06028
[26] Koshy, T., Fibonacci and Lucas Numbers with Applications, John Wiley and Sons Inc, NY, 2001. · Zbl 0984.11010
[27] Khadjiev, D., Goksal Y., Applications of hyperbolic numbers to the invariant theory in two-dimensional pseudo-Euclidean space, Adv. Appl. Clifford Algebr., 26 (2016), 645-668. https://doi.org/10.1007/s00006-015-0627-9 · Zbl 1342.51016
[28] Khrennikov, A., Segre, G., An Introduction to Hyperbolic Analysis, arxiv, 2005. http://arxiv.org/abs/math-ph/0507053v2.
[29] Motter, A. E, Rosa, A. F., Hyperbolic calculus, Adv. Appl. Clifford Algebr., 8(1) (1998), 109-128. · Zbl 0922.30035
[30] Sobczyk, G., The hyperbolic number plane, The College Mathematics Journal, 26(4) (1995), 268-280.
[31] Soykan, Y., On hyperbolic numbers with generalized Fibonacci numbers components, preprint, (2019). · Zbl 1465.11066
[32] Soykan, Y., Gocen, M., Properties of hyperbolic generalized Pell numbers, Notes on Number Theory and Discrete Mathematics, 26(4) (2020), 136-153. https://doi.org/10.7546/nntdm.2020.26.4.136-153
[33] Tan, E., Leung, H.H., Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences, Advances in Difference Equations, 2020.1 (2020), 1-11. https://doi.org/10.1186/s13662-020-2507-4 · Zbl 1487.11018
[34] Tasyurdu, Y., Hyperbolic Tribonacci and Tribonacci-Lucas sequences, International Journal of Mathematical Analysis, 13(12) (2019), 565-572. https://doi.org/10.12988/ijma.2019.91167
[35] Verma, V., Bala, A., On properties of generalized bi-variate bi-periodic Fibonacci polynomials, International Journal of Advanced Science and Technology, 29(3) (2020), 8065-8072.
[36] Yayenie, O., A note on generalized Fibonacci sequence, Applied Mathematics and Computation, 217(12) (2011), 5603-5611. https://doi.org/10.1016/j.amc.2010.12.038 · Zbl 1226.11023
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