On a new two-parameter generalization of dual-hyperbolic Jacobsthal numbers

Dorota Bród, Anetta Szynal-Liana, Iwona Włoch

Abstract


In this paper we introduce two-parameter generalization of dualhyperbolic Jacobsthal numbers: dual-hyperbolic (s,p)-Jacobsthal numbers. We present some properties of them, among others the Binet formula, Catalan, Cassini, d’Ocagne identities. Moreover, we give the generating function, matrix generator and summation formula for these numbers.

Keywords


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

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References


Akar, M., Yuce S., Sahin, S., On the dual hyperbolic numbers and the complex hyperbolic numbers, Journal of Computer Science & Computational Mathematics 8 (1) (2018) DOI: 10.20967/jcscm.2018.01.001.

Bród, D., On a two-parameter generalization of Jacobsthal numbers and its graph interpretation, Ann. Univ. Mariae Curie-Skłodowska Sect. A 72 (2) (2018), 21–28.

Bród, D., On split r-Jacobsthal quaternions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 74 (1) (2020), 1–14.

Bród, D., Szynal-Liana, A., Włoch, I., Two generalizations of dual-hyperbolic balancing numbers, Symmetry 12 (11) (2020), 1866.

Cihan, A., Azak, A. Z., Gungor, M. A., Tosun, M., A study on dual hyperbolic Fibonacci and Lucas numbers, An. Stiint¸. Univ. “Ovidius” Constant¸a Ser. Mat. 27 (1) (2019), 35–48.

Clifford, W. K., Preliminary sketch of biquaternions, Proc. Lond. Math. Soc. 4 (1873), 381–395.

Cockle, J., On certain functions resembling quaternions, and on a new imaginary in algebra, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 33 (1848), 435–439.

Cockle, J., On a new imaginary in algebra, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 34 (1849), 37–47.

Cockle, J., On the symbols of algebra, and on the theory of tesarines, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 34 (1849), 406–410.

Cockle, J., On impossible equations, on impossible quantities, and on tesarines, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 37 (1850), 281–283.

Dasdemir, A., The representation, generalized Binet formula and sums of the generalized Jacobsthal p-sequence, Hittite Journal of Science and Engineering 3 (2) (2016), 99–104.

Falcon, S., On the k-Jacobsthal numbers, American Review of Mathematics and Statistics 2 (1) (2014), 67–77.

Horadam, A. F., Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly 70 (1963), 289–291.

Kizilates, C., Kone, T., On quaternions with incomplete Fibonacci and Lucas numbers components, Util. Math. 110 (2019), 263–269.

Kizilates, C., Kone, T., On higher order Fibonacci quaternions, J. Anal. (2021), DOI: 10.1007/s41478-020-00295-1.

Kizilates, C., Kone, T., On higher order Fibonacci hyper complex numbers, Chaos Solitons Fractals 148 (2021), 111044.

Polatli, E., Kizilates, C., Kesim S., On split k-Fibonacci and k-Lucas quaternions, Adv. Appl. Clifford Algebr. 26 (2016), 353–362.

Rochon, D., Shapiro, M., On algebraic properties of bicomplex and hyperbolic numbers, Analele Universitatii Oradea, Fascicola Matematica 11 (2004), 71–110.

Soykan, V., Tasdemir, E., Okumus, I., On dual hyperbolic numbers with generalized Jacobsthal numbers components, preprint.

Szynal-Liana, A., Włoch, I., A note on Jacobsthal quaternions, Adv. Appl. Clifford Algebr. 26 (2016), 441–447.

Szynal-Liana, A., Włoch, I., The Pell quaternions and the Pell octonions, Adv. Appl. Clifford Algebr. 26 (2016), 435–440.

Szynal-Liana, A., Włoch, I., Hypercomplex numbers of the Fibonacci type, Oficyna Wydawnicza Politechniki Rzeszowskiej, Rzeszów, 2019.

Uygun, S., The (s,t)-Jacobsthal and (s,t)-Jacobsthal Lucas sequences, Appl. Math. Sci. 9 (70) (2015), 3467–3476.




DOI: http://dx.doi.org/10.17951/a.2021.75.1.1-14
Date of publication: 2021-07-24 12:06:58
Date of submission: 2021-07-21 19:06:45


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