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Year 2019, Volume 7, Issue 2, 492 - 501, 15.10.2019

Abstract

References

  • [1] Biss, D.K., Dugger, D., and Isaksen, D.C., Large annihilators in Cayley-Dickson algebras, Communication in Algebra, 2008.
  • [2] Bilgici, G., Tokes¸er, U¨ ,. U¨ nal, Z., Fibonacci and Lucas Sedenions, Journal of Integer Sequences, Article 17.1.8, 20, 1-11. 2017.
  • [3] Cariow, A., and Cariowa, G., Algorithm for Multiplying Two octonions, Radioelectronics and Communications Systems (Allerton Press, Inc. USA), vol.55, No 10, (2012), pp. 464–473, 2012.
  • [4] Cariow, A., Cariowa G., An Algorithm for Fast Multiplication of Sedenios, Information Proccessing Letters, Volume 113, Issue, 9, 324-331, 2013.
  • [5] Cariow, A., and Cariowa, G., An Algoritm for multiplication of trigintaduonions, Journal of Theoretical and Applied Computer Science, Vol. 8, No. 1,pp. 50-75, 2014.
  • [6] Catarino, P. The Modified Pell and Modified k-Pell Quaternions and Octonions. Advances in Applied Clifford Algebras 26, 577-590, 2016.
  • [7] Catarino, P., k-Pell, k-Pell–Lucas and modified k-Pell sedenions, Asian-European Journal of Mathematics, 2018.
  • [8] Cerda, G., Matrix Methods in Horadam Sequences, Bol. Mat. 19(2), 97-106, 2012.
  • [9] C¸ imen, C., ˙Ipek, A., On Jacobsthal and Jacobsthal-Lucas Sedenios and Several Identities Involving These Numbers, Mathematica Aeterna, Vol. 7, No.4,447-454, 2017.
  • [10] C¸ imen, C., ˙Ipek, A, On Jacobsthal and Jacobsthal-Lucas Octonions, Mediterr. J. Math., 14:37, 1-13, 2017.
  • [11] G¨ul, K., On k-Fibonacci and k-Lucas Trigintaduonions, International Journal of Contemporary Mathematical Sciences, Vol. 13, no. 1, 1 - 10, 2018.
  • [12] Halici, S., Karatas¸, A., On a Generalization for Fibonacci Quaternions. Chaos Solitons and Fractals 98, 178–182, 2017.
  • [13] Horadam, A.F., A Generalized Fibonacci Sequence, American Mathematical Monthly, Vol. 68, pp. 455-459, 1961.
  • [14] Horadam, A. F., Complex Fibonacci Numbers and Fibonacci quaternions, Amer. Math. Monthly 70, 289–291, 1963.
  • [15] Horadam, A. F., Basic Properties of a Certain Generalized Sequence of Numbers, The Fibonacci Quarterly 3.3, 161-176, 1965.
  • [16] Horadam, A. F., Special Properties of The Sequence wn(a;b; p;q), The Fibonacci Quarterly, Vol. 5, No. 5, pp. 424-434, 1967.
  • [17] Horadam, A. F., Generating functions for powers of a certain generalized sequence of numbers. Duke Math. J 32, 437-446, 1965.
  • [18] Imaeda, K., Imaeda, M., Sedenions: algebra and analysis, Applied Mathematics and Computation, 115, 77-88, 2000.
  • [19] ˙Ipek, A., and C¸ imen, C., On (p,q)-Fibonacci Octonions, Mathematica Aeterna, Vol. 6, No.6, 923-932, 2016.
  • [20] Karatas¸, A., and Halici, S., Horadam Octonions. An. S¸ t. Univ. Ovidus Constanta, Vol. 25(3), 97-108, 2017.
  • [21] Kec¸ilioglu O, Akkus¸, I., The Fibonacci Octonions, Adv. Appl. Clifford Algebr. 25, 151–158, 2015.
  • [22] Moreno, G., The zero divisors of the Cayley-Dickson algebras over the real numbers, Bol. Soc. Mat. Mexicana (3) 4 , 13-28,1998.
  • [23] Polatlı, E., A Generalization of Fibonacci and Lucas Quaternions, Advances in Applied Clifford Algebras, 26 (2), 719-730, 2016.
  • [24] Makarov, O.M., An algorithm for the multiplication of two quaternions, U.S.S.R. Comput. MathsMath. Phys. Vol. 17, pp. 221-222, 1978.
  • [25] Szynal-Liana, A. and I. Wloch I., The Pell quaternions and the Pell octonions. Advances in Applied Clifford Algebras 26.1, 435-440, 2016.
  • [26] Tasci, D., On k-Jacobsthal and k-Jacobsthal-Lucas Quaternions, Journal of Science and Arts, year 17, No. 3(40), pp. 469-476, 2017.

Horadam $2^{k}$-Ions

Year 2019, Volume 7, Issue 2, 492 - 501, 15.10.2019

Abstract

In this paper, we generalize Fibonacci quaternion, octonion, sedenion, trigintaduonion, etc. and define Horadam $2^{k}$-ions and investigate their properties. Each Horadam (such as Fibonacci, Lucas, Pell) quaternions, octonions and sedenions are Horadam $2^{k}$-ions. We also present connection to some earlier works.

References

  • [1] Biss, D.K., Dugger, D., and Isaksen, D.C., Large annihilators in Cayley-Dickson algebras, Communication in Algebra, 2008.
  • [2] Bilgici, G., Tokes¸er, U¨ ,. U¨ nal, Z., Fibonacci and Lucas Sedenions, Journal of Integer Sequences, Article 17.1.8, 20, 1-11. 2017.
  • [3] Cariow, A., and Cariowa, G., Algorithm for Multiplying Two octonions, Radioelectronics and Communications Systems (Allerton Press, Inc. USA), vol.55, No 10, (2012), pp. 464–473, 2012.
  • [4] Cariow, A., Cariowa G., An Algorithm for Fast Multiplication of Sedenios, Information Proccessing Letters, Volume 113, Issue, 9, 324-331, 2013.
  • [5] Cariow, A., and Cariowa, G., An Algoritm for multiplication of trigintaduonions, Journal of Theoretical and Applied Computer Science, Vol. 8, No. 1,pp. 50-75, 2014.
  • [6] Catarino, P. The Modified Pell and Modified k-Pell Quaternions and Octonions. Advances in Applied Clifford Algebras 26, 577-590, 2016.
  • [7] Catarino, P., k-Pell, k-Pell–Lucas and modified k-Pell sedenions, Asian-European Journal of Mathematics, 2018.
  • [8] Cerda, G., Matrix Methods in Horadam Sequences, Bol. Mat. 19(2), 97-106, 2012.
  • [9] C¸ imen, C., ˙Ipek, A., On Jacobsthal and Jacobsthal-Lucas Sedenios and Several Identities Involving These Numbers, Mathematica Aeterna, Vol. 7, No.4,447-454, 2017.
  • [10] C¸ imen, C., ˙Ipek, A, On Jacobsthal and Jacobsthal-Lucas Octonions, Mediterr. J. Math., 14:37, 1-13, 2017.
  • [11] G¨ul, K., On k-Fibonacci and k-Lucas Trigintaduonions, International Journal of Contemporary Mathematical Sciences, Vol. 13, no. 1, 1 - 10, 2018.
  • [12] Halici, S., Karatas¸, A., On a Generalization for Fibonacci Quaternions. Chaos Solitons and Fractals 98, 178–182, 2017.
  • [13] Horadam, A.F., A Generalized Fibonacci Sequence, American Mathematical Monthly, Vol. 68, pp. 455-459, 1961.
  • [14] Horadam, A. F., Complex Fibonacci Numbers and Fibonacci quaternions, Amer. Math. Monthly 70, 289–291, 1963.
  • [15] Horadam, A. F., Basic Properties of a Certain Generalized Sequence of Numbers, The Fibonacci Quarterly 3.3, 161-176, 1965.
  • [16] Horadam, A. F., Special Properties of The Sequence wn(a;b; p;q), The Fibonacci Quarterly, Vol. 5, No. 5, pp. 424-434, 1967.
  • [17] Horadam, A. F., Generating functions for powers of a certain generalized sequence of numbers. Duke Math. J 32, 437-446, 1965.
  • [18] Imaeda, K., Imaeda, M., Sedenions: algebra and analysis, Applied Mathematics and Computation, 115, 77-88, 2000.
  • [19] ˙Ipek, A., and C¸ imen, C., On (p,q)-Fibonacci Octonions, Mathematica Aeterna, Vol. 6, No.6, 923-932, 2016.
  • [20] Karatas¸, A., and Halici, S., Horadam Octonions. An. S¸ t. Univ. Ovidus Constanta, Vol. 25(3), 97-108, 2017.
  • [21] Kec¸ilioglu O, Akkus¸, I., The Fibonacci Octonions, Adv. Appl. Clifford Algebr. 25, 151–158, 2015.
  • [22] Moreno, G., The zero divisors of the Cayley-Dickson algebras over the real numbers, Bol. Soc. Mat. Mexicana (3) 4 , 13-28,1998.
  • [23] Polatlı, E., A Generalization of Fibonacci and Lucas Quaternions, Advances in Applied Clifford Algebras, 26 (2), 719-730, 2016.
  • [24] Makarov, O.M., An algorithm for the multiplication of two quaternions, U.S.S.R. Comput. MathsMath. Phys. Vol. 17, pp. 221-222, 1978.
  • [25] Szynal-Liana, A. and I. Wloch I., The Pell quaternions and the Pell octonions. Advances in Applied Clifford Algebras 26.1, 435-440, 2016.
  • [26] Tasci, D., On k-Jacobsthal and k-Jacobsthal-Lucas Quaternions, Journal of Science and Arts, year 17, No. 3(40), pp. 469-476, 2017.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Melih GÖCEN
Zonguldak Bülent Ecevit Üniversitesi
0000-0001-8669-6122
Türkiye


Yüksel SOYKAN
Zonguldak Bülent Ecevit Üniversitesi
0000-0002-1895-211X
Türkiye

Publication Date October 15, 2019
Application Date July 2, 2019
Acceptance Date October 30, 2019
Published in Issue Year 2019, Volume 7, Issue 2

Cite

Bibtex @research article { konuralpjournalmath585452, journal = {Konuralp Journal of Mathematics (KJM)}, issn = {}, eissn = {2147-625X}, address = {}, publisher = {Mehmet Zeki SARIKAYA}, year = {2019}, volume = {7}, pages = {492 - 501}, doi = {}, title = {Horadam \$2\^\{k\}\$-Ions}, key = {cite}, author = {Göcen, Melih and Soykan, Yüksel} }
APA Göcen, M. & Soykan, Y. (2019). Horadam $2^{k}$-Ions . Konuralp Journal of Mathematics (KJM) , 7 (2) , 492-501 . Retrieved from https://dergipark.org.tr/en/pub/konuralpjournalmath/issue/31493/585452
MLA Göcen, M. , Soykan, Y. "Horadam $2^{k}$-Ions" . Konuralp Journal of Mathematics (KJM) 7 (2019 ): 492-501 <https://dergipark.org.tr/en/pub/konuralpjournalmath/issue/31493/585452>
Chicago Göcen, M. , Soykan, Y. "Horadam $2^{k}$-Ions". Konuralp Journal of Mathematics (KJM) 7 (2019 ): 492-501
RIS TY - JOUR T1 - Horadam $2^{k}$-Ions AU - Melih Göcen , Yüksel Soykan Y1 - 2019 PY - 2019 N1 - DO - T2 - Konuralp Journal of Mathematics (KJM) JF - Journal JO - JOR SP - 492 EP - 501 VL - 7 IS - 2 SN - -2147-625X M3 - UR - Y2 - 2019 ER -
EndNote %0 Konuralp Journal of Mathematics (KJM) Horadam $2^{k}$-Ions %A Melih Göcen , Yüksel Soykan %T Horadam $2^{k}$-Ions %D 2019 %J Konuralp Journal of Mathematics (KJM) %P -2147-625X %V 7 %N 2 %R %U
ISNAD Göcen, Melih , Soykan, Yüksel . "Horadam $2^{k}$-Ions". Konuralp Journal of Mathematics (KJM) 7 / 2 (October 2019): 492-501 .
AMA Göcen M. , Soykan Y. Horadam $2^{k}$-Ions. Konuralp J. Math.. 2019; 7(2): 492-501.
Vancouver Göcen M. , Soykan Y. Horadam $2^{k}$-Ions. Konuralp Journal of Mathematics (KJM). 2019; 7(2): 492-501.
IEEE M. Göcen and Y. Soykan , "Horadam $2^{k}$-Ions", Konuralp Journal of Mathematics (KJM), vol. 7, no. 2, pp. 492-501, Oct. 2019
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