On generalized commutative Leonardo quaternions and their generalization

Articles in Press

Document Type : Original paper

Authors

1 Department of Mathematics, University of Bitlis Eren, Bitlis, Turkey

2 Department of Discrete Mathematics, Rzeszow University of Technology, Rzeszów, Poland

Abstract

In this paper, we give some properties of the generalized commutative Leonardo quaternions, among others the Binet formula, generating function, and the general bilinear index-reduction formula which imply d'Ocagne, Vajda, Halton, Catalan, and Cassini identities. We also give the matrix representations and some sum formulas of the generalized commutative Leonardo quaternions. Moreover, we present a one-parameter generalization of the generalized commutative Leonardo quaternions and their properties. 

Keywords

Main Subjects

References

[1] U. Bednarz and M. Wołowiec-Musiał, Generalized Fibonacci–Leonardo numbers, Journal of Difference Equations and Applications 30 (2024), no. 1, 111–121. https://doi.org/10.1080/10236198.2023.2265509
[2] P.D. Beites and P. Catarino, On the Leonardo quaternions sequence, Hacet. J. Math. Stat. 53 (2024), no. 4, 1001–1023.
[3] D. Bród and A. Szynal-Liana, Generalized commutative Jacobsthal quaternions and some matrices, Ex. Countex. 3 (2023), 100102. https://doi.org/10.1016/j.exco.2023.100102
[4] D. Bród and A. Szynal-Liana, On some combinatorial properties of generalized commutative Pell and Pell-Lucas quaternions, Kragujevac J. Math. 49 (2025), no. 6, 889–897. https://doi.org/10.46793/KgJMat2506.889B
[5] D. Bród, A. Szynal-Liana, and I. Włoch, On generalized commutative Jacobsthal quaternions, Kragujevac J. Math. 50 (2026), no. 5, 787–799.
[6] P.M.M.C. Catarino and A. Borges, On Leonardo numbers, Acta Math. Univ. Comenian. 89 (2019), no. 1, 75–86.
[7] E.W. Dijkstra, Fibonacci Numbers and Leonardo Numbers, EWD797, 1981.
[8] T.A. Ell, N.L. Bihan, and S.J. Sangwine, Quaternion Fourier Transforms for Signal and Image Processing, John Wiley & Sons, 2014.
[9] H. Gökbaş, Gaussian quaternion involving Leonardo numbers, Sarajevo J. Math. 19 (2023), no. 2, 207–214. https://doi.org/10.5644/SJM.19.02.06
[10] H. Gökbaş, A new family of number sequences: Leonardo-Alwyn numbers, Armen. J. Math. 15 (2023), no. 6, 1–13.
https://doi.org/10.52737/18291163–2023.15.6-1-13
[11] W.R. Hamilton, Lectures on Quaternions, Hodges and Smith, 1853.
[12] K.K. Ian and Y.H. Xia, Linear quaternion differential equations: basic theory and fundamental results, Stud. Appl. Math. 141 (2018), no. 1, 3–45. https://doi.org/10.1111/sapm.12211
[13] Z. İşbilir, M. Akyiğit, and M. Tosun, Pauli–Leonardo quaternions, Notes Number Theory Discrete Math. 29 (2023), no. 1, 1–16. https://doi.org/10.7546/nntdm.2023.29.1.1-16.
[14] A. Karataş, On complex Leonardo numbers, Notes Number Theory Discrete Math. 28 (2022), no. 3, 458–465. https://doi.org/10.7546/nntdm.2022.28.3.458-465
[15] A. Karataş, Dual Leonardo numbers, AIMS Math. 8 (2023), no. 12, 30527–30536. https://doi.org/10.3934/math.20231560
[16] K. Kuhapatanakul and J. Chobsorn, On the generalized Leonardo numbers, Integers 22 (2022), #A48.
[17] M. Kumari, K. Prasad, H. Mahato, and P.M.M.C. Catarino, On the generalized Leonardo quaternions and associated spinors, Kragujevac J. Math. 50 (2026), no. 3, 425–438.
[18] K. Prasad and M. Kumari, The Leonardo polynomials and their algebraic properties, Proc. Indian Natl. Sci. 91 (2025), no. 2, 662–671. https://doi.org/10.1007/s43538-024-00348-0
[19] K. Prasad, M. Kumari, and C. Kızılateş, Some properties and identities of hyperbolic generalized $k$-Horadam quaternions and octonions, Commun. Comb. Optim. 9 (2024), no. 4, 707–724. https://doi.org/10.22049/cco.2023.28272.1495
[20] K. Prasad, R. Mohanty, M. Kumari, and H. Mahato, Some new families of generalized $k$-Leonardo and Gaussian Leonardo numbers, Commun. Comb. Optim. 9 (2024), no. 3, 539–553. https://doi.org/10.22049/cco.2023.28236.1485
[21] Y. Soykan, Generalized Horadam-Leonardo numbers and polynomials, Asian J. Adv. Res. 17 (2023), no. 8, 128–169.
[22] A. Szynal-Liana and I. Włoch, Generalized commutative quaternions of the Fibonacci type, Bol. Soc. Mat. Mex. 28 (2022), no. 1, 1. https://doi.org/10.1007/s40590-021-00386-4
[23] M. Turan, S.Ö. Karakuş, and S.K. Nurkan, A new perspective on bicomplex numbers with Leonardo number components, Commun. Fac. Sci. Univ. 72 (2023), no. 2, 340–351. https://doi.org/10.31801/cfsuasmas.1181930
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 08 January 2026

Files

Share

How to cite

Statistics