3-Parameter Generalized Quaternions and Dual Quaternions with Repunit Number Components
By: Işıl Arda Kösal and Murat Tosun
References
- G. Bilgici, Fibonacci 3-parameter generalized quaternions, Avrupa Bilim ve Teknoloji Dergisi, 41 (2022), 357–361.
- G. Bilgici, Jacobsthal 3-parameter generalized quaternions, Advanced and Contemporary Studied in Natural Science and Mathematics, Chapter 4, 46–53, 2024.
- G. Cerda-Morales, On a generalization for Tribonacci quaternions, Mediterr. J. Math. 14 (2017), Article Number: 239, 12 pages.
- G. Cerda-Morales, Identities for third order Jacobsthal quaternions Adv. Appl. Clifford Algebr. 27 (2017), 1043–1053.
- G. Cerda-Morales, On third-order ¯h-Jacobsthal and third-order ¯h-Jacobsthal-Lucas sequences, and related quaternions, Preprints (2020), 13 pages.
- G. Cerda-Morales, Third-order Jacobsthal generalized quaternions, J. Geom. Symmetry Phys. 50 (2018), 11–27.
- G. Cerda-Morales, Dual third-order Jacobsthal quaternions, Proyecciones (Antofagasta) 37(4) (2018), 731–747.
- R. Chaker, A. Boua, Some results on generalized quaternions algebra with generalized Fibonacci quaternions, An. S¸tiint¸ Univ. Al. I. Cuza Iaşi. Mat. (N.S.) LXIX(2) (2023), 233–246.
- J. Cockle, On systems of algebra involving more than one imaginary; and on equations of the fifth degree, Philos. Mag. 35(238) (1849), 434–437.
- E. A. Costa, P. M. Catarino, P. J. Vasco, F. R. Alves, A Brief Study on the k-Dimensional Repunit Sequence. Rn, 10(1) (2025), 9.
- E. A. Costa, P. M. Catarino, D. C. Santos, A study of the symmetry of the tricomplex repunit sequence with repunit sequence. Symmetry, 17(1) (2024), 28.
- E. A. Costa, D. C. dos Santos, P. M. M. C. Catarino, Some Tridiagonal Matrix of the Repunit Sequence. Latin American Journal of Mathematics, 4(1) (2025), 1–12.
- E. A. Costa, F. S. Carvalho, On repunit polynomials sequence. Brazilian Electronic Journal of Mathematics. 5(2024), 1–15.
- E. A. Costa, G. A. Costa, P. M. M. C. Catarino, One-Zero numbers: A new Horadam-type sequence, Intermaths. 5(2)(2024), 80-92.
- E. A. Costa, D. C. Santos, P. M. M. C. Catarino, E. V. P. Spreafico, On Gaussian and Quaternion Repunit Numbers. Revista de Matemática da UFOP, 2 (2024).
- E. A. Costa, D. C. dos Santos, F. S. Monteiro, V. M. A. de Souza, On the Repunit sequence at negative indices. Revista de Matemática da UFOP, 1 (2024).
- L. E. Dickson, On the theory of numbers and generalized quaternions, Amer. J. Math. 46 (1924), 1–16.
- O. Dişkaya, H. Menken, On the (s, t)-Padovan and (s, t)-Perrin quaternions, J. Adv. Math. 12 (2019), 186–192.
- O. Dişkaya, H. Menken, On the split (s, t)-Padovan and (s, t)-Perrin quaternions, Int. J. Appl. Math. Inform. 13 (2019), 25–28.
- Z. Ercan and S. Yüce, On properties of the dual quaternions, European Journal of Pure and Applied Mathematics, 4(2) (2011), 142–146.
- C. Flaut, V. Shpakivskyi, On generalized Fibonacci quaternions and Fibonacci-Narayana quaternions, Adv. Appl. Clifford Algebr. 23(3) (2013), 673–688.
- L. W. Griffiths, Generalized quaternion algebras and the theory of numbers, Amer. J. Math. 50 (1928), 303–314.
- S.S. Gupta, Repunit Numbers. In: Exploring the Beauty of Fascinating Numbers. Springer Praxis Books(). Springer, Singapore (2025).
- H. Günay, Quaternions and Applications of Some Generalized Third Order Sequences, Selçuk University, The Graduate School of Natural and Applied Science of Selçuk University, 2019.
- H. Günay, N. Taşkara, Some properties of Padovan quaternion, Asian-Eur. J. Math. 12(06) (2019), 2040017.
- N. Gürses, Z. ˙Işbilir, On the combined Jacobsthal-Padovan generalized quaternions, Adv. Studies: Euro-Tbilisi Math. J. 15(2) (2022), 55–70.
- W. R. Hamilton, On quaternions; or on a new system of imaginaries in algebra, Philos. Mag. 25(3) (1844), 489–495.
- W. R. Hamilton, Lectures on Quaternions, Hodges and Smith, Dublin, 1853.
- W. R. Hamilton, Elements of Quaternions, Chelsea Pub. Com. New York, 1969.
- Z. İşbilir, N. Gürses, Padovan and Perrin generalized quaternions, Math. Methods Appl. Sci. 45(18) (2022), 12060–12076.
- Z. İşbilir, N. Gürses, Pell-Padovan generalized quaternions, Notes Number Theory Discrete Math. 27(1) (2021), 171–187.
- Z. İşbilir, N. Gürses, A study on the association of the 3-primes and reverse 3-primes generalized quaternions, 1st International Symposium on Recent Advances in Fundamental and Applied Sciences (ISFAS-2021), 2021.
- Z. İşbilir, N. Gürses, Horadam 3-parameter generalized quaternions, Honam Math. J. 46(3) (2023), 407–427.
- Z. İşbilir and N. Gürses, Padovan, Perrin and Pell-Padovan dual quaternions, Turkish Journal of Mathematics and Computer Science 15(1) (2023), 125–144.
- Z. İşbilir and N. Gürses, (2023). Examination of generalized Tribonacci dual quaternions. Acta et Commentationes Universitatis Tartuensis de Mathematica, 27(2), 235-255.
- Z. İşbilir, N. Gürses, M. Tosun, On the 3-parameter generalized quaternions with generalized tribonacci numbers components, Filomat, 39(9) (2025), 3003–3027.
- M. Jafari, Genelleştirilmiş Hamilton Operatörleri ve Lie Grupları , AnkaraÜniversitesi, Fen Bilimleri Enstitüsü, (2012).
- M. Jafari, Y. Yaylı, Generalized quaternions and rotation in 3-space E3αβ, TWMS J. Pure Appl. Math. 6(2) (2015), 224–232.
- M. Jafari, Y. Yaylı, Generalized quaternions and their algebraic properties, Commun. Fac. Sci. Ank. Series A1 64(1) (2015), 15–27.
- C. Kızılateş, P. Catarino, and N. Tuğlu, On the bicomplex generalized Tribonacci quaternions, Mathematics 7 (2019), 80.
- V. Majernik, Quaternion formulation of the Galilean space-time transformation, Acta Physica Slovaca. 56(1) (2006), 9–14.
- A. B. Mamagani, M. Jafari, On properties of generalized quaternion algebra, J. Nov. Appl. Sci. 2(12) (2013), 683–689.
- H. Mortazaasl, M. Jafari, A study of semi-quaternions algebra in semi-Euclidean 4-space, Mathematical Science and Applications E-Notes 2(1) (2013), 20–27.
- S. K. Nurkan, ˙I. Arslan Güven, Dual Fibonacci quaternions, Advances in Applied Clifford Algebras. 25(2) (2015), 403–414.
- H. Pottmann, J. Wallner, Computational Line Geometry, Springer-Verlag Berlin Heidelberg, New York, 2001.
- B. Rosenfeld, Geometry of Lie Groups, Kluwer Academic Publishers, 1997.
- D. Savin, C. Flaut, and C. Ciobanu, Some properties of the symbol algebras, Carpathian J. Math. 25(2) (2009), 239–245.
- N. J. A. Sloane, The online encyclopedia of integer sequences, (1964) http://oeis.org/.
- D. C. Santos, E. A. Costa, A note on Repunit number sequence. Inter-maths, 5(1) (2024), 54-66.
- D. C. Santos, E. A. Costa, Um passeio pela sequência repunidade. CQDRevista Eletrônica Paulista de Matemática, p. 241–254, 2023.
- D. C. Santos, E. A. Costa, P. M. Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence. Axioms, 14(3) (2025), 203.
- T. D. S¸entürk, 3-parameter generalized quaternions, Institue of Science, Kastamonu University, PhD Thesis, 2020.
- T. D. S¸entürk, Z.Ünal, 3-parameter generalized quaternions, Comput. Methods Funct. Theory. 22(3) (2022), 575–608.
- T. D. S¸entürk, G. Bilgici, A. Daşdemir, Z.Ünal, A study on Horadam hybrid numbers, Turk. J. Math. 44(4) (2020), Article 10, 1212–1221.
- D. Taşcı, Padovan and Pell-Padovan quaternions, J. Sci. Arts 42(1) (2018), 125–132.
- T. Unger, N. Markin, Quadratic forms and space-time block codes from generalized quaternion and biquaternion algebras, IEEE Trans. Inform. Theory. 57(9) (2011), 6148–6156.
- Z.Ünal, Dual Fibonacci ve Lucas 3-parametreli genellestirilmis kuaterniyonları. In International Conference on Scientific and Academic Research (Vol. 1, pp. 34-36), (2023).
- S. Yates, Repunits and Repetends. [S.l.]: Inc., Boynton Beach, Florida, 1982.15
- S. Yüce and F. Torunbalcı Aydın, Generalized dual Fibonacci quaternions, Applied Mathematics E-Notes. 16(309) (2016), 276–289.
- S. Yüce and F. Torunbalcı Aydın, A new aspect of dual Fibonacci quaternions, Adv. Appl. Clifford Algebras. 26 (2016), 873–884.
Language: English
Page range: 111 - 138
Submitted on: Aug 29, 2025
Accepted on: Dec 30, 2025
Published on: Jul 11, 2026
Published by: Ovidius University of Constanta
In partnership with: Paradigm Publishing Services
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© 2026 Işıl Arda Kösal, Murat Tosun, published by Ovidius University of Constanta
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