The 3-parameter generalized quaternionic sequences with $\mathcal{DGC}$ Leonardo numbers via doubling
Çiğdem Zeynep YILMAZ, Gülsüm Yeliz SAÇLI
Received March 19, 2025. Published online May 18, 2026.
Abstract: The main theme of this paper is to investigate a generalized quaternionic sequence with dual-generalized complex ($\mathcal{DGC}$) Leonardo numbers via doubling depending on $3$-parameters, $\lambda_{i \in\{1,2,3\}}$, using Cayley-Dickson doubling procedure. Firstly, we briefly describe the $\mathcal{DGC}$ Fibonacci, $\mathcal{DGC}$ Lucas and $\mathcal{DGC}$ Leonardo sequences via doubling. After defining the $3$-parameter generalized quaternionic (3PGQic) sequence with $\mathcal{DGC}$ Leonardo numbers via doubling, we give the Binet's formula, the generating function, Catalan's, Cassini's and d'Ocagne's identities for this sequence. Finally, we discuss the special cases in which well-known quaternionic sequences with $\mathcal{DGC}$ Leonardo numbers are obtained.
References: [1] M. Akar, S. Yüce, S. Şahin: On the dual hyperbolic numbers and the complex hyperbolic numbers. J. Comput. Sci. Comput. Math. 8 (2018), 1-6. DOI 10.20967/jcscm.2018.01.001
[2] Y. Alp, E. G. Kocer: Hybrid Leonardo numbers. Chaos Solitons Fractals 150 (2021), Article ID 111128, 5 pages. DOI 10.1016/j.chaos.2021.111128 | MR 4278051 | Zbl 1498.11047
[3] Y. Alp, E. G. Koçer: Some properties of Leonardo numbers. Konuralp J. Math. 9 (2021), 183-189. MR 4252432
[4] G. Bilgici: Fibonacci 3-parameter generalized quaternions. European J. Sci. Technology 41 (2022), 357-361. DOI 10.31590/ejosat.1166686
[5] P. M. M. C. Catarino, A. Borges: On Leonardo numbers. Acta Math. Univ. Comen., New Ser. 89 (2020), 75-86. MR 4053337 | Zbl 1481.11016
[6] H. H. Cheng, S. Thompson: Dual polynomials and complex dual numbers for analysis of spatial mechanisms. ASME 1996 Design Engineering Technical Conferences and Computers in Engineering, Conference Volume 2B ASME, New York (1996), Article ID 96-DETC/MECH-1221, V02BT02A031, 12 pages. DOI 10.1115/96-DETC/MECH-1221
[7] H. H. Cheng, S. Thompson: Singularity analysis of spatial mechanisms using dual polynomials and complex dual numbers. J. Mech. Design 121 (1999), 200-205. DOI 10.1115/1.2829444
[8] J. Cockle: On systems of algebra involving more than one imaginary and on equations of the fifth degree. Phil. Mag. 35 (1849), 434-437. DOI 10.1080/14786444908646384
[9] A. Cohen, M. Shoham: Principle of transference: An extension to hyper-dual numbers. Mech. Mach. Theory 125 (2018), 101-110. DOI 10.1016/j.mechmachtheory.2017.12.007
[10] L. E. Dickson: On the theory of numbers and generalized quaternions. Am. J. Math. 46 (1924), 1-16. DOI 10.2307/2370658 | MR 1506514 | JFM 50.0094.02
[11] J. A. Fike, J. J. Alonso: Automatic differentiation through the use of hyper-dual numbers for second derivatives. Recent Advances in Algorithmic Differentiation Lecture Notes in Computational Science and Engineering 87. Springer, Berlin (2012), 163-173. DOI 10.1007/978-3-642-30023-3_15 | MR 3241221 | Zbl 1251.65026
[12] J. A. Fike, S. Jongsma, J. Alonso, E. Van Der Weide: Optimization with gradient and Hessian information calculated using hyper-dual numbers. 29th AIAA Applied Aerodynamics Conference AIAA, Reston (2011), Article ID AIAA 2011-3807, 19 pages. DOI 10.2514/6.2011-3807
[13] L. W. Griffiths: Generalized quaternion algebras and the theory of numbers. Am. J. Math. 50 (1928), 303-314. DOI 10.2307/2371761 | MR 1506671 | JFM 54.0164.01
[14] N. Gürses, G. Y. Şentürk, S. Yüce: A study on dual-generalized complex and hyperbolic-generalized complex numbers. Gazi Univ. J. Sci. 34 (2021), 180-194. DOI 10.35378/gujs.653906
[15] N. Gürses, G. Y. Şentürk, S. Yüce: A comprehensive survey of dual-generalized complex Fibonacci and Lucas numbers. Sigma J. Eng. Natural Sci. 40 (2022), 179-187. DOI 10.14744/sigma.2022.00014
[16] W. R. Hamilton: On quaternions; or on a new system of imaginaries in algebra. Phil. Mag. (3) 25 (1844), 10-13. DOI 10.1080/14786444408644923
[17] W. R. Hamilton: Lectures on Quaternions. Hodges and Smith, London (1853).
[18] W. R. Hamilton: Elements of Quaternions. Chelsea Publishing, New York (1969). DOI 10.1017/CBO9780511707162 | MR 0237284
[19] A. A. Harkin, J. B. Harkin: Geometry of generalized complex numbers. Math. Mag. 77 (2004), 118-129. DOI 10.1080/0025570X.2004.11953236 | MR 1573734 | Zbl 1176.30070
[20] M. Jafari, Y. Yaylı: Generalized quaternions and their algebratic properties. Commun. Fac. Sci. Univ. Ank., Sér. A1, Math. Stat. 64 (2015), 15-27. DOI 10.1501/Commua1_0000000724 | MR 3453638 | Zbl 1354.15022
[21] I. L. Kantor, A. S. Solodovnikov: Hypercomplex Numbers: An Elementary Introduction to Algebras. Springer, New York (1989). MR 0996029 | Zbl 0669.17001
[22] T. Koshy: Fibonacci and Lucas Numbers with Applications. Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts. Wiley, New York (2001). DOI 10.1002/9781118033067 | MR 1855020 | Zbl 0984.11010
[23] Y. Kulaç, M. Tosun: Some equations on $p$-complex Fibonacci numbers. 6th International Eurasian Conference On Mathematical Sciences And Applications AIP, New York (2018), Article ID 020024, 6 pages. DOI 10.1063/1.5020473 | Zbl 1469.11023
[24] V. Majerník: Multicomponent number systems. Acta Phys. Polon. A 90 (1996), 491-498. DOI 10.12693/APhysPolA.90.491 | MR 1426884
[25] F. Messelmi: Dual-complex numbers and their holomorphic function. Available at https://zenodo.org/records/22961 (2013), 11 pages. DOI 10.5281/zenodo.22961
[26] H. Mortazaasl, M. Jafari: A study on semi-quaternions algebra in semi-Euclidean 4-space. Math. Sci. Appl. E-Notes 1 (2013), 20-27. Zbl 1353.11106
[27] H. Pottmann, J. Wallner: Computational Line Geometry. Mathematics and Visualization. Springer, Berlin (2001). DOI 10.1007/978-3-642-04018-4 | MR 1849803 | Zbl 1006.51015
[28] B. Rosenfeld: Geometry of Lie Groups. Mathematics and its Applications (Dordrecht) 393. Kluwer, Dordrecht (1997). DOI 10.1007/978-1-4757-5325-7 | MR 1443207 | Zbl 0867.53002
[29] G. Y. Saçlı: On 3-parameter generalized quaternions with the Leonardo $p$-sequence. J. Sci. Arts 24 (2024), 21-30. DOI 10.46939/J.Sci.Arts-24.1-a03
[30] D. Savin, D. Flaut, C. Ciobanu: Some properties of the symbol algebras. Carpathian J. Math. 25 (2009), 239-245. MR 2731200 | Zbl 1249.17007
[31] G. Y. Şentürk: A brief study on dual-generalized complex Leonardo numbers. 5th International Conference on Mathematical Advances and Applications, Istanbul, Turkey, 11-14 May 2022.
[32] G. Y. Şentürk, N. Gürses, S. Yüce: Fundamental properties of extended Horadam numbers. Notes Number Theory Disc. Math. 27 (2021), 219-235. DOI 10.7546/nntdm.2021.27.4.219-235
[33] T. D. Şentürk, Z. Ünal: 3-parameter generalized quaternions. Comput. Methods Funct. Theory 22 (2022), 575-608. DOI 10.1007/s40315-022-00451-7 | MR 4473942 | Zbl 1496.15018
[34] Ç. Z. Yılmaz, G. Y. Saçlı: Investigating the dual quaternion extension of the $DGC$ Leonardo sequence. Honam Math. J. 46 (2024), 677-696. DOI 10.5831/hmj.2024.46.4.677 | MR 4818798 | Zbl 1563.11070
[35] Ç. Z. Yılmaz, G. Y. Saçlı: On some identities for the $DGC$ Leonardo sequence. Notes Number Theory Disc. Math. 30 (2024), 253-270. DOI 10.7546/nntdm.2024.30.2.253-270
Affiliations:Çiğdem Zeynep Yılmaz, Gülsüm Yeliz Saçlı (corresponding author), Yildiz Technical University, Faculty of Arts and Sciences, Department of Mathematics, 34220, Istanbul, Türkiye, e-mail: cigdem.yilmaz@yildiz.edu.tr, yeliz.sacli@yildiz.edu.tr
