On Generalized Commutative Jacobsthal Quaternions
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Authors: D. BRóD, A. SZYNAL-LIANA AND I. WłOCH
DOI: 10.46793/KgJMat2605.787B
Abstract:
In this paper, we introduce and study generalized commutative Jacobsthal quaternions and their one-parameter generalization. We present some fundamental properties of them, among others the Binet formula, Catalan, Cassini, d’Ocagne and Vajda identities. Moreover, we give the generating functions and summation formulas for these numbers.
Keywords:
Jacobsthal numbers, quaternions, generalized quaternions, Binet formula, Catalan identity, Cassini identity.
References:
[1] F. T. Aydın and S. Yüce, A new approach to Jacobsthal quaternions, Filomat 31(18) (2017), 5567–5579. http://dx.doi.org/10.2298/FIL1718567T
[2] D. Bród, On a new Jacobsthal-type sequence, Ars Combin. 150 (2020), 21–29.
[3] G. Cerda-Morales, Identities for third order Jacobsthal quaternions, Adv. Appl. Clifford Algebr. 27 (2017), 1043–1053. https://doi.org/10.1007/s00006-016-0654-1
[4] A. Dasdemir, On The Jacobsthal numbers by matrix methods, SDU Journal of Science (E-Journal) 71 (2012), 69–76.
[5] A. Dasdemir, The representation, generalized Binet formula and sums of the generalized Jacobsthal p-sequence, Hittite Journal of Science and Engineering 3(2) (2016), 99–104. https://doi.org/10.17350/HJSE19030000038
[6] F. Falcon, On the k-Jacobsthal numbers, American Review of Mathematics and Statistics 2(1) (2014), 67–77.
[7] A. Godase, Hyperbolic k-Fibonacci and k-Lucas quaternions, The Mathematics Student 90(1-2) (2021) 103–116.
[8] A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly 70 (1963), 289–291. https://doi.org/10.2307/2313129
[9] A. F. Horadam, Jacobsthal representation numbers, Fibonacci Quart. 34 (1996), 40–54.
[10] M.R. Iyer, Some results on Fibonacci quaternions, Fibonacci Quart. 7(1) (1969), 201–210.
[11] E. Özkan and M. Taştan, A new families of Gauss k-Jacobsthal numbers and Gauss k-Jacobsthal-Lucas numbers and their polynomials, Journal of Science and Arts 4(53) (2020), 893–908. http://dx.doi.org/10.46939/J.Sci.Arts-20.4-a10
[12] E. Özkan and M. Uysal, On quaternions with higher order Jacobsthal numbers components, Gazi University Journal of Science 36(1) (2023), 336–347. https://doi.org/10.35378/gujs.1002454
[13] E. Özkan, M. Uysal and A. D. Godase, Hyperbolic k-Jacobsthal and k-Jacobsthal-Lucas quaternions, Indian J. Pure Appl. Math. 53 (2022), 956–967. https://doi.org/10.1007/s13226-021-00202-9
[14] D. A. Pinotsis, Segre quaternions, spectral analysis and a four-dimensional Laplace equation, in: M. Ruzhansky and J. Wirth (Eds.), Progress in Analysis and its Applications, World Scientific, Singapore, 2010, 240–246.
[15] A. Szynal-Liana and I. Włoch, A note on Jacobsthal quaternions, Adv. Appl. Clifford Algebr. 26 (2016), 441–447. https://doi.org/10.1007/s00006-015-0622-1
[16] A. Szynal-Liana and I. Włoch, Generalized commutative quaternions of the Fibonacci type, Bol. Soc. Mat. Mex. 28(1) (2022). https://doi.org/10.1007/s40590-021-00386-4
[17] A. Szynal-Liana and I. Włoch, Hypercomplex Numbers of the Fibonacci Type, Oficyna Wydawnicza Politechniki Rzeszowskiej, Rzeszów, 2019, 25–46.
[18] M. Uysal and E. Özkan, Higher-order Jacobsthal-Lucas quaternions, Axioms (2022), Article ID 671, 11 pages. https://doi.org/10.3390/axioms11120671
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