On ̄h-Jacobsthal and ̄h-Jacobsthal–Lucas sequences, and related quaternions

Giuseppina Anatriello; Giovanni Vincenzi

doi:10.2478/auom-2019-0030

.blurhash-client-img { display: none !important; }

On ̄h-Jacobsthal and ̄h-Jacobsthal–Lucas sequences, and related quaternions

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 27 (2019): Issue 3 (December 2019)

By: Giuseppina Anatriello and Giovanni Vincenzi

Open Access

|Dec 2019

Abstract

In this paper, inspired by recent articles of A. Szynal-Liana & I. Włoch and F. T. Aydin & S. Yüce (see [26] and [2]), we will introduce the ̄h-Jacobsthal quaternions and the ̄h-Jacobsthal–Lucas sequences and their associated quaternions. The new results that we have obtained extend most of those obtained in [26].

References

[1] G. Anatriello and G. Vincenzi, Tribonacci-like sequences and generalized Pascal’s triangles, Int. J. Math. Educ. Sci. Technol.45 (8) (2014) 1220–1232.10.1080/0020739X.2014.914283
Search in Google Scholar Back to article
[2] F. T. Aydin and S. Yüce, A New Approach to Jacobsthal Quaternions. Filomat31:18 (2017), 5567–557910.2298/FIL1718567T
Search in Google Scholar Back to article
[3] H. Campos, P. Catarino, A. P. Aires, P. Vasco and A. Borges, On some identities of k Jacobsthal–Lucas Numbers, Int. J. of Math. Analysis8 (10) (2014) 489–494.10.12988/ijma.2014.4249
Search in Google Scholar Back to article
[4] P. Catarino, On some identities for k-Fibonacci sequence, Int. J. Contemp. Math. Sci.9 (1) (2014) 37–42.10.12988/ijcms.2014.311120
Search in Google Scholar Back to article
[5] P. Catarino, The h(x)-Fibonacci Quaternion Polynomials: Some Combinatorial Properties, Adv. Appl. Cli ord Algebr., 26(1), 71-79.(2016).10.1007/s00006-015-0606-1
Search in Google Scholar Back to article
[6] P. Catarino and M. L. Morgado, On Generalized Jacobsthal and Jacobsthal–Lucas polynomials, An. Şt. Univ. Ovidius Constanţa, 24(3),(2016), 61–78.10.1515/auom-2016-0048
Search in Google Scholar Back to article
[7] G. Cerda-Morales, On a Generalization for Tribonacci Quaternions, Mediterr. J. Math.14 (6) (2017) 239.10.1007/s00009-017-1042-3
Search in Google Scholar Back to article
[8] C.B. Çimen and A. İpek A, On Pell quaternions and Pell-Lucas quaternions, Adv. Appl. Clifford Algebr.26 (1)(2016) 39–51.10.1007/s00006-015-0571-8
Search in Google Scholar Back to article
[9] C. B. Çimen and A. İpek, On Jacobsthal and Jacobsthal–Lucas Octonions, Mediterr. J. Math.14(2017), no. 2, Art. 37, 13 pp.10.1007/s00009-017-0873-2
Search in Google Scholar Back to article
[10] S. Falcòn andÀ. Plaza, On the Fibonacci k-numbers, Chaos Solitons & Fractals, (2007) 32(5) 1615–1624.10.1016/j.chaos.2006.09.022
Search in Google Scholar Back to article
[11] A. Fiorenza and G. Vincenzi, Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients, Chaos, Solitons & Fractals (2011) 44 (1):145–152.10.1016/j.chaos.2011.01.003
Search in Google Scholar Back to article
[12] A. Fiorenza and G. Vincenzi, From Fibonacci sequence to the golden ratio,J. Math., 2013. Art. ID 204674, 3.10.1155/2013/204674
Search in Google Scholar Back to article
[13] S. Halici, On complex Fibonacci quaternions, Adv. Appl. Clifford Algebras23 (2013) 105–112.10.1007/s00006-012-0337-5
Search in Google Scholar Back to article
[14] S. Halici, On Fibonacci quaternions, Adv. Appl. Clifford Algebras. 22 (2102) 321–327.10.1007/s00006-011-0317-1
Search in Google Scholar Back to article
[15] S. Halici and K Adnan, On a generalization for fibonacci quaternions, Chaos, Solitons & Fractals98 (2017) 178–182.10.1016/j.chaos.2017.03.037
Search in Google Scholar Back to article
[16] R. Herz-Fischler, Letter to the editor. Fibonacci Quart. (1986) 24(4) 382.10.1080/00150517.1986.12429752
Search in Google Scholar Back to article
[17] M. Frascari and L. Volpi Ghirardini, Contra divinam proportionem. In: Williams Kim, editor. Nexus II: Architecture and Mathematics. Fucecchio, Florence, Italy: Edizioni Dell’ Erba; (1998) 65–66.
Search in Google Scholar Back to article
[18] A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Am. Math. Month.70 (1963) 289–291.10.2307/2313129
Search in Google Scholar Back to article
[19] A. F. Horadam, Jacobsthal representation numbers, Fibonacci Quarterly34 (1998) 40–54.10.1080/00150517.1996.12429096
Search in Google Scholar Back to article
[20] A. F. Horadam, Quaternion recurrence relations, Ulam Quarterly (2) (1993) 23–33.
Search in Google Scholar Back to article
[21] A. İpek, On (p, q)-Fibonacci quaternions and their Binet’ s Formulas, generating functions and certain binomial sums. Adv. Appl. Cli ord Algebras,27(2) (2017), 1343–1351.10.1007/s00006-016-0704-8
Search in Google Scholar Back to article
[22] M. R. Iyer, A note on Fibonacci quaternions, Fibonacci Quaterly3 (1969), 225–229.10.1080/00150517.1969.12431146
Search in Google Scholar Back to article
[23] G. S Krishnaswami and S. Sachdev, Algebra and Geometry of Hamilton’s Quaternions, Resonance, (2016) 529–544.10.1007/s12045-016-0358-9
Search in Google Scholar Back to article
[24] S. K. Nurkan and İA. Güen, Dual Fibonacci quaternions, Adv. Appl. Clifford Algebras25(2) (2015), 403–414.10.1007/s00006-014-0488-7
Search in Google Scholar Back to article
[25] A. Szynal-Liana and I. Włoch, The Pell quaternions and the Pell octonions, Adv. Appl. Clifford Algebras.26 (2016), 435–440.10.1007/s00006-015-0570-9
Search in Google Scholar Back to article
[26] A. Szynal-Liana and I. Włoch, A Note on Jacobsthal Quaternions, Adv. Appl. Clifford Algebras. 26 (2016), 441–447.10.1007/s00006-015-0622-1
Search in Google Scholar Back to article
[27] D. Taşci, On k-Jacobsthal and k-Jacobsthal–Lucas Quaternions, Journal of Science and Arts. 3 (40), (2017) 469-476.
Search in Google Scholar Back to article
[28] G. Vincenzi and S. Siani, Fibonacci-like sequences and generalized Pascal’s triangles, Int. J. Math. Educ. in Sci. Technol.45 (4) (2014) 609–614.10.1080/0020739X.2013.851806
Search in Google Scholar Back to article
[29] J. P. Ward, Quaternions and Cayley Numbers: Algebra and Applications, Kluwer Academic Publishers, London, 199710.1007/978-94-011-5768-1
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/auom-2019-0030 | Journal eISSN: 1844-0835 | Journal ISSN: 1224-1784

Journal RSS Feed

Language: English

Page range: 5 - 23

Submitted on: Dec 8, 2018

Accepted on: Jan 29, 2019

Published on: Dec 21, 2019

Published by: Ovidius University of Constanta

In partnership with: Paradigm Publishing Services

Publication frequency: 3 issues per year

Keywords:

Jacobsthal numbers,

Jacobsthal–Lucas numbers,

recurrence relations,

Related subjects:

© 2019 Giuseppina Anatriello, Giovanni Vincenzi, published by Ovidius University of Constanta
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 27 (2019): Issue 3 (December 2019)