Abstract
In this article, we investigate the algebra of quaternions and octonions with components as the higher order Mersenne numbers. We examine the inter-relations of these newly introduced sequences with the Mersenne numbers and their quaternions and octonions counterparts. Moreover, we obtain the Binet type formula, recurrence relation, Catalan’s identity, d’Ocagne’s identity, Vajda’s identity, Honsberger’s identity, generating functions, summations formulas, etc. of them.
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References
Catarino, P., Campos, H., Vasco, P.: On the Mersenne sequence. Ann. Math. Inform. 46, 37–53 (2016)
Çimen, C.B., Ipek, A.: On Jacobsthal and Jacobsthal–Lucas octonions. Mediterran. J. Math. 14, 1–13 (2017)
Dasdemir, A., Bilgici, G.: Gaussian Mersenne numbers and Mersenne quaternions. Not. Num. Theory Disc. Math. 25(3), 87–96 (2019)
Hamilton, W.R.: Elements of Quaternions. Green & Company, Longmans (1866)
Horadam, A.F.: Complex Fibonacci numbers and Fibonacci quaternions. Am. Math. Mon. 70(3), 289–291 (1963)
Karataş, A., Halici, S.: Horadam octonions. Analele ştiinţifice ale Universităţii" Ovidius" Constanţa. Seria Matematică 25(3), 97–106 (2017)
Keçilioğlu, O., Akkus, I.: The Fibonacci octonions. Adv. Appl. Clifford Algebras 25, 151–158 (2015)
Kızılateş, C., Kone, T.: On higher order Fibonacci hyper complex numbers. Chaos Solit. Fract. 148, 111044 (2021)
Kızılateş, C., Kone, T.: On higher order Fibonacci quaternions. J. Anal. 29(4), 1071–1082 (2021). https://doi.org/10.1007/s41478-020-00295-1
Kumari, M., Prasad, K., Mahato, H.: On the \(k\)-Mersenne and k-Mersenne–Lucas octonions. TWMS J. App. Eng. Math. 14 (4) (2024) (to appear)
Kumari, M., Prasad, K., Frontczak, R.: On the k-Fibonacci and k-Lucas spinors. Not. Num. Theory Discr. Math. 29(2), 322–335 (2023). https://doi.org/10.7546/nntdm.2023.29.2.322-335
Kumari, M., Prasad, K., Tanti, J.: On the generalization of Mersenne and Gaussian Mersenne polynomials. J. Anal. 32, 931–947 (2024). https://doi.org/10.1007/s41478-023-00666-4
Özimamoğlu, H.: On hyper complex numbers with higher order Pell numbers components. J. Anal. 31, 2443–2457 (2023)
Özkan, E., Uysal, M.: On quaternions with higher order Jacobsthal numbers components. Gazi Univ. J. Sci. 36(1), 336–347 (2023)
Polatli, E., Kizilates, C., Kesim, S.: On split k-Fibonacci and k-Lucas quaternions. Adv. Appl. Clifford Algebras 26, 353–362 (2016)
Prasad, K., Kumari, M., Mohanta, R., Mahato, H.: The sequence of higher order Mersenne numbers and associated binomial transforms (2023). arXiv preprint arXiv:2307.08073.
Prasad, K., Mahato, H., Kumari, M., Mohanty, R.: On the generalized Leonardo Pisano octonions. Natl. Acad. Sci. Lett., 1–7 (2023). https://doi.org/10.1007/s40009-023-01291-2
Randić, M., Morales, D.A., Araujo, O.: Higher-order Fibonacci numbers. J. Math. Chem. 20, 79–94 (1996)
Soykan, Y.: A study on generalized Mersenne numbers. J. Progress. Res. Math. 18(3), 90–108 (2021)
Soykan, Y.: On generalized p-Mersenne numbers. Earthline J. Math. Sci. 8(1), 83–120 (2022)
Tian, Y.: Matrix representations of octonions and their applications. Adv. Appl. Clifford Algebras 10(1), 61–90 (2000)
Ünal, Z., Tokeşer, Ü., Bilgici, G.: Some properties of dual Fibonacci and dual Lucas octonions. Adv. Appl. Clifford Algebras 27, 1907–1916 (2017)
Uysal, M., Özkan, E.: Higher-order Jacobsthal–Lucas quaternions. Axioms 11(12), 671 (2022)
Uysal, M., Kumari, M., Kuloglu, B., Prasad, K., Özkan, E.: On the hyperbolic k-Mersenne and k-Mersenne–Lucas octonions. Kragujevac J. Math. 49(05), 765–779 (2025). https://doi.org/10.46793/KgJMat2504.615P
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Kumari, M., Prasad, K. & Mohanta, R. Algebra of quaternions and octonions involving higher order Mersenne numbers. Proc.Indian Natl. Sci. Acad. (2024). https://doi.org/10.1007/s43538-024-00297-8
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DOI: https://doi.org/10.1007/s43538-024-00297-8