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
References
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.
Wydział Matematyki i Fizyki Stosowanej, Politechnika Rzeszowska im. Ignacego Łukasiewicza Poland
https://orcid.org/0000-0001-5181-1725
Wydział Matematyki i Fizyki Stosowanej, Politechnika Rzeszowska im. Ignacego Łukasiewicza Poland
Wydział Matematyki i Fizyki Stosowanej, Politechnika Rzeszowska im. Ignacego Łukasiewicza Poland
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