On Quaternion Gaussian Bronze Fibonacci Numbers

Catarino, Paula; Ricardo, Sandra

doi:10.2478/amsil-2022-0013

Abstract

In the present work, a new sequence of quaternions related to the Gaussian Bronze numbers is defined and studied. Binet’s formula, generating function and certain properties and identities are provided. Tridiagonal matrices are considered to determine the general term of this sequence.

References

[1] M. Akyiğit, H.H. Kösal, and M. Tosun, Split Fibonacci quaternions, Adv. Appl. Clifford Algebr. 23 2013, no. 3, 535–545.10.1007/s00006-013-0401-9
Search in Google Scholar Back to article
[2] M. Asci and E. Gurel, Gaussian Jacobsthal and Gaussian Jacobsthal Lucas numbers, Ars Combin. 111 2013, 53–63.
Search in Google Scholar Back to article
[3] G. Berzsenyi, Gaussian Fibonacci numbers, Fibonacci Quart. 15 1977, no. 3, 233–236.10.1080/00150517.1977.12430445
Search in Google Scholar Back to article
[4] G. Bilgici and P. Catarino, Unrestricted Pell and Pell–Lucas quaternions, Int. J. Math. Syst. Sci. 1 2018, no. 3, Article ID: 816, 4 pp.10.24294/ijmss.v1i3.816
Search in Google Scholar Back to article
[5] P. Catarino, The modified Pell and the modified k-Pell quaternions and octonions, Adv. Appl. Clifford Algebr. 26 2016, no. 2, 577–590.10.1007/s00006-015-0611-4
Search in Google Scholar Back to article
[6] P. Catarino, Bicomplex k-Pell quaternions, Comput. Methods Funct. Theory 19 (2019), no. 1, 65–76.10.1007/s40315-018-0251-5
Search in Google Scholar Back to article
[7] P. Catarino, On k-Pell hybrid numbers, J. Discrete Math. Sci. Cryptogr. 22 (2019), no. 1, 83–89.10.1080/09720529.2019.1569822
Search in Google Scholar Back to article
[8] P. Catarino and A. Borges, A note on incomplete Leonardo numbers, Integers 20 (2020), Paper No. A43, 7 pp.
Search in Google Scholar Back to article
[9] P. Catarino and A. Borges, On Leonardo numbers, Acta Math. Univ. Comenian. (N.S.) 89 (2020), no. 1, 75–86.
Search in Google Scholar Back to article
[10] P. Catarino and H. Campos, A note on Gaussian modified Pell numbers, J. Inf. Optim. Sci. 39 (2018), no. 6, 1363–1371.10.1080/02522667.2018.1471267
Search in Google Scholar Back to article
[11] P. Catarino and P. Vasco, The generalized order-m (k-Pell) numbers, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 66 (2020), no. 1, 55–65.
Search in Google Scholar Back to article
[12] Y. Choo, A generalized quaternion with generalized Fibonacci number components, Appl. Math. Sci. 14 (2020), no. 1, 31–38.10.12988/ams.2020.912160
Search in Google Scholar Back to article
[13] S. Falcon, On the generating matrices of the k-Fibonacci numbers, Proyecciones 32 (2013), no. 4, 347–357.10.4067/S0716-09172013000400004
Search in Google Scholar Back to article
[14] S. Halici, On quaternion-Gaussian Lucas numbers, Math. Methods Appl. Sci. 44 (2021), no. 9, 7601–7606.10.1002/mma.6410
Search in Google Scholar Back to article
[15] S. Halici and G. Cerda-Morales, On quaternion-Gaussian Fibonacci numbers and their properties, An. fitiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 29 (2021), no. 1, 71–82.10.2478/auom-2021-0005
Search in Google Scholar Back to article
[16] S. Halici and A. Karataş, On a generalization for Fibonacci quaternions, Chaos Solitons Fractals 98 (2017), 178–182.10.1016/j.chaos.2017.03.037
Search in Google Scholar Back to article
[17] S. Halici and S. Öz, On some Gaussian Pell and Pell–Lucas numbers, Ordu Univ. J. Sci. Tech. 6 (2016), no. 1, 8–18.
Search in Google Scholar Back to article
[18] W.R. Hamilton, Elements of Quaternions, Longmans, Green, & Co., London, 1866.
Search in Google Scholar Back to article
[19] A.F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly 70 (1963), 289–291.10.2307/2313129
Search in Google Scholar Back to article
[20] L. Huang and W. So, Quadratic formulas for quaternions, Appl. Math. Lett. 15 (2002), no. 5, 533–540.10.1016/S0893-9659(02)80003-9
Search in Google Scholar Back to article
[21] J.H. Jordan, Gaussian Fibonacci and Lucas numbers, Fibonacci Quart. 3 (1965), 315–318.10.1080/00150517.1965.12431411
Search in Google Scholar Back to article
[22] M.Y. Kartal, Gaussian Bronze Fibonacci numbers, EJONS Int. J. Math. Eng. Nat. Sci. 4 (2020), no. 13, 19–25.10.38063/ejons.175
Search in Google Scholar Back to article
[23] E. Kılıç and D. Tasci, On the generalized Fibonacci and Pell sequences by Hessenberg matrices, Ars Combin. 94 (2010), 161–174.
Search in Google Scholar Back to article
[24] E. Kılıç, D. Tasci, and P. Haukkanen, On the generalized Lucas sequences by Hessenberg matrices, Ars Combin. 95 (2010), 383–395.
Search in Google Scholar Back to article
[25] C. Kızılateş, A new generalization of Fibonacci hybrid and Lucas hybrid numbers, Chaos Solitons Fractals 130 (2020), 109449, 5 pp.10.1016/j.chaos.2019.109449
Search in Google Scholar Back to article
[26] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York, 2001.10.1002/9781118033067
Search in Google Scholar Back to article
[27] C. Levesque, On m-th order linear recurrences, Fibonacci Quart. 23 (1985), no. 4, 290–293.10.1080/00150517.1985.12429796
Search in Google Scholar Back to article
[28] E. Polatli, C. Kizilates, and S. Kesim, On split k-Fibonacci and k-Lucas quaternions, Adv. Appl. Clifford Algebr. 26 (2016), no. 1, 353–362.10.1007/s00006-015-0591-4
Search in Google Scholar Back to article
[29] N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, The OEIS Foundation Inc., http://oeis.org/.
Search in Google Scholar Back to article
[30] J.L. Synge, Quaternions, Lorentz Transformations, and the Conway-Dirac-Eddington Matrices, Communications of the Dublin Institute for Advanced Studies, Series A, 21, Dublin Inst. for Adv. Stud., Dublin, 1972.
Search in Google Scholar Back to article
[31] A. Szynal-Liana and I. Włoch, A note on Jacobsthal quaternions, Adv. Appl. Clifford Algebr. 26 (2016), no. 1, 441–447.10.1007/s00006-015-0622-1
Search in Google Scholar Back to article
[32] D. Tasci, On k-Jacobsthal and k-Jacobsthal–Lucas quaternions, J. Sci. Arts 40 (2017), no. 3, 469–476.
Search in Google Scholar Back to article
[33] Ü. Tokeşer, Z. Ünal, and G. Bilgici, Split Pell and Pell–Lucas quaternions, Adv. Appl. Clifford Algebr. 27 2017, no. 2, 1881–1893.10.1007/s00006-016-0747-x
Search in Google Scholar Back to article
[34] T. Yağmur, Split Jacobsthal and Jacobsthal–Lucas quaternions, Commun. Math. Appl. 10 (2019), no. 3, 429–438.10.26713/cma.v10i3.902
Search in Google Scholar Back to article

On Quaternion Gaussian Bronze Fibonacci Numbers

Abstract

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