One-parameter generalization of dual-hyperbolic Jacobsthal numbers



Abstract

In this paper, we introduce one-parameter generalization of dualhyperbolic Jacobsthal numbers – dual-hyperbolic r-Jacobsthal numbers. We present some properties of them, among others the Binet formula, Catalan, Cassini, and d’Ocagne identities. Moreover, we give the generating function and summation formula for these numbers. The presented results are a generalization of the results for the dual-hyperbolic Jacobsthal numbers.


Keywords

Jacobsthal numbers; dual-hyperbolic numbers; dual-hyperbolic Jacobsthal numbers; Binet formula; Catalan identity; Cassini identity

M. Akar, S. Yüce, and S. Şahin, On the dual hyperbolic numbers and the complex hyperbolic numbers, Journal of Computer Science & Computational Mathematics 8 (2018), no. 1, 1–7.

D. Bród, On a new Jacobsthal-type sequence, Ars Combin. 150 (2020), 21–29.

A. Cihan, A.Z. Azak, M.A. Güngör, and M. Tosun, A study on dual hyperbolic Fibonacci and Lucas numbers, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 27 (2019), no. 1, 35–48.

W.K. Clifford, Preliminary sketch of biquaternions, Proc. Lond. Math. Soc. 4 (1871/73), 381–395.

J. Cockle, On certain functions resembling quaternions, and on a new imaginary in algebra, Lond. Edinb. Dubl. Phil. Mag. 33 (1848), no. 224, 435–439. DOI: 10.1080/14786444808646139

J. Cockle, On a new imaginary in algebra, Lond. Edinb. Dubl. Phil. Mag. 34 (1849), no. 227, 37–47. DOI: 10.1080/14786444908646169

J. Cockle, On the symbols of algebra, and on the theory of tesarines, Lond. Edinb. Dubl. Phil. Mag. 34 (1849), no. 231, 406–410. DOI: 10.1080/14786444908646257

J. Cockle, On impossible equations, on impossible quantities, and on tessarines, Lond. Edinb. Dubl. Phil. Mag. 37 (1850), no. 250, 281–283. DOI: 10.1080/14786445008646598

A. Daşdemir, The representation, generalized Binet formula and sums of the generalized Jacobsthal p-sequence, Hittite J. Sci. Eng. 3 (2016), no. 2, 99–104.

S. Falcon, On the k-Jacobsthal numbers, American Review of Mathematics and Statistics 2 (2014), no. 1, 67–77.

A.F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly 70 (1963), no. 3, 289–291.

D. Rochon and M. Shapiro, On algebraic properties of bicomplex and hyperbolic numbers, An. Univ. Oradea Fasc. Mat. 11 (2004), 71–110.

Y. Soykan, E. Taşdemir, and İ. Okumuş, On dual hyperbolic numbers with generalized Jacobsthal numbers components, Indian J. Pure Appl. Math. (2022). DOI: 10.1007/s13226-022-00301-1

A. Szynal-Liana and I. Włoch, A note on Jacobsthal quaternions, Adv. Appl. Clifford Algebr. 26 (2016), no. 1, 441–447.

A. Szynal-Liana and I. Włoch, The Pell quaternions and the Pell octonions, Adv. Appl. Clifford Algebr. 26 (2016), no. 1, 435–440.

A. Szynal-Liana and I. Włoch, Hypercomplex Numbers of the Fibonacci Type, Oficyna Wydawnicza Politechniki Rzeszowskiej, Rzeszów, 2019.

Download

Published : 2023-04-02


BródD., Szynal-LianaA., & WłochI. (2023). One-parameter generalization of dual-hyperbolic Jacobsthal numbers. Annales Mathematicae Silesianae, 37(2), 224-239. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/15439

Dorota Bród  dorotab@prz.edu.pl
Wydział Matematyki i Fizyki Stosowanej, Politechnika Rzeszowska im. Ignacego Łukasiewicza  Poland
https://orcid.org/0000-0001-5181-1725
Anetta Szynal-Liana 
Wydział Matematyki i Fizyki Stosowanej, Politechnika Rzeszowska im. Ignacego Łukasiewicza  Poland
Iwona Włoch 
Wydział Matematyki i Fizyki Stosowanej, Politechnika Rzeszowska im. Ignacego Łukasiewicza  Poland



Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.

  1. License
    This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license.
  2. Author’s Warranties
    The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s.
  3. User Rights
    Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor.
  4. Co-Authorship
    If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.