One-Parameter Generalization of Dual-Hyperbolic Jacobsthal Numbers

Bród, Dorota; Szynal-Liana, Anetta; Włoch, Iwona

doi:10.2478/amsil-2023-0005

Abstract

In this paper, we introduce one-parameter generalization of dual-hyperbolic 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.

References

M. Akar, S. Yüce, and S. fiahin, On the dual hyperbolic numbers and the complex hyperbolic numbers, Journal of Computer Science & Computational Mathematics 8 (2018), no. 1, 1–7.
Search in Google Scholar Back to article
D. Bród, On a new Jacobsthal-type sequence, Ars Combin. 150 (2020), 21–29.
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
W.K. Clifford, Preliminary sketch of biquaternions, Proc. Lond. Math. Soc. 4 (1871/73), 381–395.
Search in Google Scholar Back to article
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
Open DOI Search in Google Scholar Back to article
J. Cockle, On a new imaginary in algebra, Lond. Edinb. Dubl. Phil. Mag. 34 (1849), no. 227, 37–47. DOI: 10.1080/14786444908646169
Open DOI Search in Google Scholar Back to article
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
Open DOI Search in Google Scholar Back to article
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
Open DOI Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
S. Falcon, On the k-Jacobsthal numbers, American Review of Mathematics and Statistics 2 (2014), no. 1, 67–77.
Search in Google Scholar Back to article
A.F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly 70 (1963), no. 3, 289–291.
Search in Google Scholar Back to article
D. Rochon and M. Shapiro, On algebraic properties of bicomplex and hyperbolic numbers, An. Univ. Oradea Fasc. Mat. 11 (2004), 71–110.
Search in Google Scholar Back to article
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
Open DOI Search in Google Scholar Back to article
A. Szynal-Liana and I. Włoch, A note on Jacobsthal quaternions, Adv. Appl. Clifford Algebr. 26 (2016), no. 1, 441–447.
Search in Google Scholar Back to article
A. Szynal-Liana and I. Włoch, The Pell quaternions and the Pell octonions, Adv. Appl. Clifford Algebr. 26 (2016), no. 1, 435–440.
Search in Google Scholar Back to article
A. Szynal-Liana and I. Włoch, Hypercomplex Numbers of the Fibonacci Type, Oficyna Wydawnicza Politechniki Rzeszowskiej, Rzeszów, 2019.
Search in Google Scholar Back to article

One-Parameter Generalization of Dual-Hyperbolic Jacobsthal Numbers

Abstract

Paradigm

My account