Algebraic and geometric properties of Fibonacci Cartan numbers

Bahar Kuloğlu; Hasan Gökbaş; William Ramirez; Clemente Cesarano; Bahar Kuloğlu; Hasan Gökbaş; William Ramirez; Clemente Cesarano

doi:10.3934/nhm.2026087

Networks and Heterogeneous Media

2026, Volume 21, Issue 6: 2329-2367. doi: 10.3934/nhm.2026087

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Algebraic and geometric properties of Fibonacci Cartan numbers

1.
Department of Engineering Basic Sciences, Sivas University of Science and Technology, Sivas, Türkiye
2.
Bitlis Eren University, Science and Arts Faculty, Bitlis, Türkiye
3.
Department of Natural and Exact Sciences, Universidad de la Costa, Barranquilla, Colombia
4.
Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele Ⅱ, Rome, Italy

Received: 22 July 2026 Revised: 14 August 2026 Accepted: 03 October 2026 Published: 02 November 2026

In this paper we developed an algebraic and geometric study of Fibonacci – Cartan numbers and Fibonacci Pauli–Cartan numbers by integrating Fibonacci sequences with the algebraic structure of Cartan numbers in pseudo–Euclidean spaces. Building on the concept of Fibonacci–Cartan numbers, we investigated their fundamental algebraic and geometric properties, including conjugation, character, norm, and causal classification, proving that they are spacelike elements and establishing their geometric interpretation within a Minkowski–type framework. Matrix, polar, and exponential representations were obtained, enabling the derivation of De Moivre-type identities, formulas for powers and roots, and characterizations in terms of the determinant and trace of real $ 2\times 2 $ matrices. In addition, the powers of Fibonacci–Cartan numbers were represented in terms of Chebyshev polynomials of the first and second kinds, yielding explicit polynomial expressions for their scalar and vector parts. Scalar and vector products were also analyzed, providing a consistent geometric interpretation in a three–dimensional Cartan space. Furthermore, Fibonacci Pauli–Cartan numbers were defined and studied through recurrence relations, Binet formulas, ordinary and exponential generating functions, summation formulas, and real matrix representations. Finally, rotational properties associated with lightlike directions were examined, showing that parabolic rotations degenerate into linear trajectories within the Cartan geometric framework.
- Fibonacci–Cartan numbers,
- Fibonacci Pauli–Cartan numbers,
- Cartan geometry,
- Pauli matrices,
- De Moivre's formula,
- parabolic rotations,
- Chebyshev polynomials
Citation: Bahar Kuloğlu, Hasan Gökbaş, William Ramirez, Clemente Cesarano. Algebraic and geometric properties of Fibonacci Cartan numbers[J]. Networks and Heterogeneous Media, 2026, 21(6): 2329-2367. doi: 10.3934/nhm.2026087

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Abstract

In this paper we developed an algebraic and geometric study of Fibonacci – Cartan numbers and Fibonacci Pauli–Cartan numbers by integrating Fibonacci sequences with the algebraic structure of Cartan numbers in pseudo–Euclidean spaces. Building on the concept of Fibonacci–Cartan numbers, we investigated their fundamental algebraic and geometric properties, including conjugation, character, norm, and causal classification, proving that they are spacelike elements and establishing their geometric interpretation within a Minkowski–type framework. Matrix, polar, and exponential representations were obtained, enabling the derivation of De Moivre-type identities, formulas for powers and roots, and characterizations in terms of the determinant and trace of real $ 2\times 2 $ matrices. In addition, the powers of Fibonacci–Cartan numbers were represented in terms of Chebyshev polynomials of the first and second kinds, yielding explicit polynomial expressions for their scalar and vector parts. Scalar and vector products were also analyzed, providing a consistent geometric interpretation in a three–dimensional Cartan space. Furthermore, Fibonacci Pauli–Cartan numbers were defined and studied through recurrence relations, Binet formulas, ordinary and exponential generating functions, summation formulas, and real matrix representations. Finally, rotational properties associated with lightlike directions were examined, showing that parabolic rotations degenerate into linear trajectories within the Cartan geometric framework.

References

[1]	B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983. Available from: https://shop.elsevier.com/books/semi-riemannian-geometry-with-applications-to-relativity/oneill/978-0-12-526740-3.
[2]	H. Busemann, The Geometry of Geodesics, Academic Press, New York, 1955. Available from: https://ci.nii.ac.jp/ncid/BA0358873X.
[3]	E. Cartan, Lectures on the Geometry of Riemannian Spaces [Leçons sur la geometrie des espaces de Riemann], Gauthier-Villars, Paris, 1928. Available from: https://rcin.org.pl/dlibra/publication/11909/edition/45404/lecons-sur-la-geometrie-des-espaces-de-riemann-cartan-elie-joseph-1869-1951?language = en.
[4]	R. Lopez, Differential geometry of curves and surfaces in Lorentz–Minkowski space, Int. Electron. J. Geom. 7 (2014), 44–107. https://doi.org/10.36890/iejg.594497 doi: 10.36890/iejg.594497
[5]	K. L. Duggal, A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publishers, Dordrecht, 1996. https://doi.org/10.1007/978-94-017-2089-2
[6]	K. L. Duggal, B. Sahin, Differential Geometry of Lightlike Submanifolds, Birkhäuser, Basel, 2010. https://doi.org/10.1007/978-3-0346-0251-8
[7]	J. Cockle, On systems of algebra involving more than one imaginary; and on equations of the fifth degree, Philos. Mag., 35 (1849), 434–437. https://doi.org/10.1080/14786444908646384 doi: 10.1080/14786444908646384
[8]	A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly, 70 (1963), 289–291. https://doi.org/10.2307/2313129 doi: 10.2307/2313129
[9]	I. M. Yaglom, A Simple Non-Euclidean Geometry and Its Physical Basis, Springer, New York, 1979. https://doi.org/10.1007/978-1-4612-6135-3
[10]	G. Sobczyk, The hyperbolic number plane, College Math. J., 26 (1995), 268–280. https://doi.org/10.1080/07468342.1995.11973712 doi: 10.1080/07468342.1995.11973712
[11]	A. Ferrandez, A. Gimenez, P. Lucas, Null helices in Lorentzian space forms, Int. J. Mod. Phys. A, 16 (2001), 4845–4863. https://doi.org/10.1142/S0217751X01005821 doi: 10.1142/S0217751X01005821
[12]	K. L. Duggal, D. H. Jin, Null Curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scientific, Singapore, 2007. https://doi.org/10.1142/6449
[13]	J. P. Ward, Quaternions and Cayley Numbers, Springer, Dordrecht, 1997. https://doi.org/10.1007/978-94-011-5768-1
[14]	I. Ozturk, Introduction to Cartan numbers and some geometric applications, Math. Methods Appl. Sci., 46 (2023), 7514–7542. https://doi.org/10.1002/mma.8982 doi: 10.1002/mma.8982
[15]	R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, 2012. https://doi.org/10.1017/CBO9781139020411
[16]	I. Ozturk, H. Cakir, Fibonacci Cartan and Lucas Cartan numbers, Open Math., 23 (2025), 20250204. https://doi.org/10.1515/math-2025-0204 doi: 10.1515/math-2025-0204
[17]	I. Ozturk, Dual Cartan numbers and pseudo-null lines, Math. Methods Appl. Sci., 48 (2025), 207–217. https://doi.org/10.1002/mma.10323 doi: 10.1002/mma.10323
[18]	J. E. Kim, A representation of de Moivre's formula over Pauli quaternions, Ann. Acad. Rom. Sci. Ser. Math. Appl., 9 (2017), 145–151. Available from: https://aos.ro/wp-content/anale/MVol9Nr2Art.3.pdf.
[19]	F. T. Aydin, Pauli-Fibonacci quaternions, Notes Number Theory Discret. Math., 27 (2021), 184–193. https://doi.org/10.7546/nntdm.2021.27.3.184-193 doi: 10.7546/nntdm.2021.27.3.184-193
[20]	Z. Işbilir, M. Akyigit, M. Tosun, Pauli-Leonardo quaternions, Notes Number Theory Discret. Math., 29 (2023), 1–16. https://doi.org/10.7546/nntdm.2023.29.1.1-16 doi: 10.7546/nntdm.2023.29.1.1-16
[21]	A. Z. Azak, Pauli Gaussian Fibonacci and Pauli Gaussian Lucas quaternions, Mathematics, 10 (2022), 4655. https://doi.org/10.3390/math10244655 doi: 10.3390/math10244655
[22]	B. Yılmaz, N. G. Bilgin, Y. Soykan, Generalized Pauli Fibonacci polynomial quaternions, Axioms, 14 (2025), 449. https://doi.org/10.3390/axioms14060449 doi: 10.3390/axioms14060449
[23]	T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York, 2001. https://doi.org/10.1002/9781118033067
[24]	J. P. M. Binet, Memoire sur lintegration des equations lineaires aux differences finies, dun ordre quelconque, a coefficients variables, C. R. Acad. Sci. Paris, 17 (1843), 559–567. Available from: https://mathshistory.st-andrews.ac.uk/Extras/Binet_Papers/.
[25]	T. Koshy, Fibonacci and Lucas Numbers With Applications, John Wiley & Sons, 2018. https://doi.org/10.1002/9781118742297
[26]	U. Bednarz, I. Wloch, M. Wolowiec-Musial, Total graph interpretation of the numbers of the Fibonacci type, J. Appl. Math., 2015 (2015), 837917. https://doi.org/10.1155/2015/837917 doi: 10.1155/2015/837917
[27]	A. Szynal-Liana, I. Wloch, Hypercomplex Numbers of the Fibonacci Type, Oficyna Wydawnicza Politechniki Rzeszowskiej, Rzeszów, 2019.
[28]	C. Kızılateş, W. Ramirez, Y. Quintana, D. Bedoya, M. F. Heredia-Moyano, Mixed-type Fibonacci-Mittag-Leffler and Lucas-Mittag-Leffler polynomials: Some properties, Comput. Appl. Math., 45 (2026), 32. https://doi.org/10.1007/s40314-025-03379-5 doi: 10.1007/s40314-025-03379-5
[29]	S. Halici, On Fibonacci quaternions, Adv. Appl. Clifford Algebras, 22 (2012), 321–327. https://doi.org/10.1007/s00006-011-0317-1 doi: 10.1007/s00006-011-0317-1
[30]	M. R. Iyer, A note on Fibonacci quaternions, Fibonacci Quart., 7 (1969), 225–229. https://doi.org/10.1080/00150517.1969.12431146 doi: 10.1080/00150517.1969.12431146
[31]	S. K. Nurkan, I. A. Güven, A note on bicomplex Fibonacci and Lucas numbers, Int. J. Pure Appl. Math., 120 (2018), 365–377. https://doi.org/10.12732/ijpam.v120i3.7 doi: 10.12732/ijpam.v120i3.7
[32]	A. Szynal-Liana, I. Wloch, The Fibonacci hybrid numbers, Util. Math., 110 (2019), 3–10. Available from: https://utilitasmathematica.com/index.php/Index/issue/view/59.
[33]	B. Kuloglu, E. Ozkan, M. Marin, Fibonacci and Lucas polynomials in $n$-gon, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat., 31 (2023), 127–140. https://doi.org/10.2478/auom-2023-0023 doi: 10.2478/auom-2023-0023
[34]	A. Shahid, H. L. Huang, M. M. Bhatti, M. Marin, Numerical computation of magnetized bioconvection nanofluid flow with temperature-dependent viscosity and Arrhenius kinetic, Math. Comput. Simul., 200 (2022), 377–392. https://doi.org/10.1016/j.matcom.2022.04.032 doi: 10.1016/j.matcom.2022.04.032
[35]	S. Askar, A. E. Abouelregal, M. Marin, A. Foul, Photo-thermoelasticity heat transfer modeling with fractional differential actuators for stimulated nano-semiconductor media, Symmetry, 15 (2023), 656. https://doi.org/10.3390/sym15030656 doi: 10.3390/sym15030656
[36]	A. E. Abouelregal, M. Marin, A. Öchsner, A modified spatiotemporal nonlocal thermoelasticity theory with higher-order phase delays for a viscoelastic micropolar medium exposed to short-pulse laser excitation, Contin. Mech. Thermodyn., 37 (2025), 15. https://doi.org/10.1007/s00161-024-01342-z doi: 10.1007/s00161-024-01342-z
[37]	W. M. Abd-Elhameed, Y. H. Youssri, N. El-Sissi, M. Sadek, New hypergeometric connection formulae between Fibonacci and Chebyshev polynomials, Ramanujan J., 42 (2017), 347–361. https://doi.org/10.1007/s11139-015-9712-x doi: 10.1007/s11139-015-9712-x
[38]	R. Frontczak, T. Goy, Chebyshev–Fibonacci polynomial relations using generating functions, Integers, 21 (2021), A100. https://doi.org/10.5281/zenodo.10839350 doi: 10.5281/zenodo.10839350
[39]	T. Kim, D. S. Kim, D. V. Dolgy, J. W. Park, Sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials, J. Inequal. Appl., 2018 (2018), 148. https://doi.org/10.1186/s13660-018-1744-5 doi: 10.1186/s13660-018-1744-5
[40]	T. Kim, D. S. Kim, On generalized Chebyshev polynomials, preprint, arXiv: 2607.29661, 2026. https://doi.org/10.48550/arXiv.2607.29661
[41]	G. Kizilaslan, T. Kim, The $k$-augmented Pascal matrix and its properties, J. Math., 2025 (2025), 6280240. https://doi.org/10.1155/jom/6280240 doi: 10.1155/jom/6280240
[42]	T. Kim, D. S. Kim, Degenerate Euler-Seidel matrix method and their applications, Math. Methods Appl. Sci., 49 (2026), 8209–8223. https://doi.org/10.1002/mma.70468 doi: 10.1002/mma.70468

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