Some Properties and Identities of Hyperbolic Generalized k-Horadam Quaternions and Octonions

Document Type : Original paper


1 Department of Mathematics, Central University of Jharkhand, India, 835205

2 Department of Mathematics, Zonguldak Bulent Ecevit University, 67100, Turkey


The aim of this paper is to introduce the hyperbolic generalized $k$-Horadam quaternions and octonions and investigate their algebraic properties. We present some properties and identities of these quaternions and octonions for generalized $k$-Horadam numbers. Moreover, we give some determinants related to the hyperbolic generalized $k$-Horadam quaternions and octonions. Finally, we evaluate its determinants through the Chebyshev polynomials of the second kind and give an illustrative example as well.


Main Subjects

[1] J. Baez, The octonions, Bull. Amer. Math. Soc. 39 (2002), no. 2, 145–205.
[2] D. Bród, A. Szynal-Liana, and I. Wloch, On some combinatorial properties of bihyperbolic numbers of the Fibonacci type, Math. Methods Appl. Sci. 44 (2021), no. 6, 4607–4615.
[3] A. Cariow and G. Cariowa, An unified approach for developing rationalized algorithms for hypercomplex number multiplication, Electric Review 91 (2015), no. 2, 36–39.
[4] A. Cariow, G. Cariowa, and J. Knapinski, Derivation of a low multiplicative complexity algorithm for multiplying hyperbolic octonions, arXiv:1502.06250, 2015.
[5] P. Catarino, A note on certain matrices with $h(x)$–Fibonacci quaternion polynomials, J. Difference Equ. Appl. 22 (2016), no. 2, 343–351.
[6] J.H. Conway and D.A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, AK Peters/CRC Press, 2003.
[7] S. Falcon, On the generating matrices of the $k$-Fibonacci numbers, Proyecciones (Antofagasta) 32 (2013), no. 4, 347–357.–09172013000400004
[8] R. Frontczak, A short remark on Horadam identities with binomial coefficients, Ann. Math. Inform., vol. 54, Eszterházy Károly Egyetem Líceum Kiadó, 2021, pp. 5–13.
[9] A.D. Godase, Hyperbolic $k$-Fibonacci and $k$-Lucas octonions, Notes Number Theory Discrete Math. 26 (2020), no. 3, 176–188.–188
[10] A.D. Godase, Hyperbolic $k$-Fibonacci and $k$-Lucas quaternions, The Mathematıcs Student 90 (2021), no. 1-2, 103–116.
[11] S. Halici and A. Karataş, On a generalization for Fibonacci quaternions, Chaos, Solitons & Fractals 98 (2017), 178–182.
[12] W.R. Hamilton, Elements of Quaternions, Longmans, Green, & Company, 1866.
[13] A.F. Horadam, A generalized Fibonacci sequence, Amer. Math. Monthly 68 (1961), no. 5, 455–459.
[14] A.F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly 70 (1963), no. 3, 289–291.
[15] T. Horzum and E.G. Kocer, On some properties of Horadam polynomials, Int. Math. Forum, vol. 4, 2009, pp. 1243–1252.
[16] O. Keçilioğlu and I. Akkus, The Fibonacci octonions, Advances in Applied Clifford Algebras 25 (2015), no. 1, 151–158.–014–0468–y
[17] C Kızılateş, On quaternions with incomplete Fibonacci and Lucas components, Util. Math. 110 (2019), 263–269.
[18] C. Kizilateş, A note on Horadam hybrinomials, Fundamental J. Math. Appl. 5 (2022), no. 1, 1–9.
[19] C. Kızılateş, P. Catarino, and N. Tuğlu, On the bicomplex generalized Tribonacci quaternions, Mathematics 7 (2019), no. 1, Article ID: 80.
[20] C. Kizilateş and T. Kone, On higher order Fibonacci quaternions, The J. Analysis 29 (2021), no. 4, 1071–1082.
 [21] C. Kızılateş and E. Polatlı, New families of Fibonacci and Lucas octonions with $q$-integer components, Indian J. Pure Appl. Math. 52 (2021), no. 1, 231–240.–021–00073–0
[22] M. Kumari, K. Prasad, and H. Mahato, On $k$-Mersenne and $k$-Mersenne-Lucas octonions, arXiv:2207.12243, 2022
[23] D.W. Niu, W.H. Li, and F. Qi, On signs of several Toeplitz–Hessenberg determinants whose elements contain central Delannoy numbers, Commun. Comb. Optim. 8 (2023), no. 4, 665–671.
[24] E. Özkan, M. Uysal, and A.D. Godase, Hyperbolic $k$-Jacobsthal and $k$-Jacobsthal-Lucas quaternions, Indian J. Pure Appl. Math. 53 (2022), no. 4, 956–967.–021–00202–9
[25] E. Polatlı, A generalization of Fibonacci and Lucas quaternions, Adv. Appl. Clifford Algebr. 26 (2016), no. 2, 719–730–015–0626–x
[26] K. Prasad and H. Mahato, Cryptography using generalized Fibonacci matrices with Affine-Hill cipher, J. Discrete Math. Sci. Cryptogr. 25 (2022), no. 8, 2341–2352.
[27] K. Prasad, H. Mahato, and M. Kumari, On the generalized $k$-Horadam-like sequences, Algebra, Analysis, and Associated Topics, Trends in Mathematics, Birkhäuser, Cham, 2022, pp. 11–26.
[28] G.Y. Şentürk, N. Gürses, and S. Yüce, Fundamental properties of extended Horadam numbers, Notes Number Theory Discret. Math. 27 (2021), no. 4, 219–235.–235
[29] T.D. Şentürk, . Bilgici, A. Daşdemir, and Z. Ünal, A study on Horadam hybrid numbers, Turkish J. Math. 44 (2020), no. 4, 1212–1221.–1908–77
[30] G. Udrea, A note on the sequence $(W_n)n≥e 0$ of A.F. Horadam, Portugaliae Math. 53 (1996), no. 2, 143–156.
[31] M. Uysal, M. Kumari, B. Kuloğlu, K. Prasad, and E. Özkan, On the hyperbolic $k$-Mersenne and $k$-Mersenne-Lucas octonions, Kragujevac J. Math. 49 (2025), no. 5, 765–779.
[32] Y. Yazlik and N. Taskara, A note on generalized $k$-Horadam sequence, Comput. Math. Appl. 63 (2012), no. 1, 36–41.
[33] F. Yılmaz and M. Özkan, On the generalized Gaussian fibonacci numbers and Horadam hybrid numbers: A unified approach, Axioms 11 (2022), no. 6, Article ID: 255.