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$k$-FIBONACCI AND $k$-LUCAS GENERALIZED QUATERNIONS

Year 2017, Volume: 5 Issue: 2, 102 - 113, 15.10.2017

Abstract

We investigate the properties of $k-$Fibonacci and $k-$Lucas quaternions over the generalized quaternion algebra. After presenting generating functions and Binet's formulas for these types of quaternions, we calculate several well-known identities such as Catalan's, Cassini's and d'Ocagne's identities for $k-$Fibonacci and $k-$Lucas generalized quaternions.

References

  • [1] Akyigit, M., Kosal, H.H. and Tosun, M., Split Fibonacci Quaternions, Adv. Appl. Clifford Algebr. Vol:23 (2014), 535-545.
  • [2] Akyigit, M., Kosal, H.H. and Tosun, M., Fibonacci Generalized Quaternions, Adv. Appl. Clifford Algebr. Vol:24 (2014), 631-641.
  • [3] Cimen, C.B. and Ipek, A., On Pell Quaternions and Pell-Lucas Quaternions, Adv. Appl. Clifford Algebr., Vol:26 (2016), 39-51.
  • [4] Catarino, P., The Modi ed Pell and the Modi ed k-Pell Quaternions and Octonions, Adv. Appl. Clifford Algebr., Vol:26 (2016), 577-590.
  • [5] Catarino, P. and Vasco, P., On Dual k􀀀Pell Quaternions and Octonions, Mediterr. J. Math., Vol:14 (2017), 75-87.
  • [6] Falcon, S. and Plaza, A., The k-Fibonacci Sequence and the Pascal 2-Triangle, Chaos Solitons Fractals, Vol:33, No.1 (2007), 38-49.
  • [7] Falcon, S., On the k-Lucas Numbers, Int. J. Contemp. Math. Sci. Vol:21 (2011), 1039-1050.
  • [8] Halici, S., On Fibonacci Quaternions, Adv. Appl. Clifford Algebr. Vol:22 (2012), 321-327.
  • [9] Harman, C.J., Complex Fibonacci Numbers, Fibonacci Quart. Vol:19, No.1 (1981), 82-86. [10] Horadam, A. F. , Complex Fibonacci Numbers and Fibonacci Quaternions, Amer. Math. Monthly No:70 (1963), 289-291.
  • [11] Iyer, M. R., Some Result on Fibonacci Quaternions, Fibonacci Quart. Vol:7, No.2 (1969), 201-210.
  • [12] Koshy, T., Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience Publication, Canada, 2001.
  • [13] Koshy, T., Pell and Pell-Lucas Numbers with Applications, Springer-Verlag New York, USA, 2014.
  • [14] Polatli,E., Kizilates, C. and Kesim, S., On Split k􀀀Fibonacci and k-Lucas Quaternions, Adv. Appl. Clifford Algebr. Vol:26 (2016), 353-362.
  • [15] Polatli, E. and Kesim, S., A Note on Catalan's Identity for the k􀀀 Fibonacci Quaternions, J. Integer Seq. Vol:18 (Article 15.8.2 ) (2015), 1-4.
  • [16] Ramirez, J. L., Some Combinatorial Properties of the k-Fibonacci and the k-Lucas Quaternions, An. St. Univ. Ovidius Constanta Ser. Mat. Vol:23 No.2 (2015), 201-212.
  • [17] Tokeser, U., Unal, Z. and Bilgici, G., Split Pell and Split Pell-Lucas Quaternions, Adv. Appl. Clifford Algebr. Vol: 27 (2017), 1881-1893.
  • [18] Szynal-Liana, A. and Wloch, I. The Pell Quaternions and the Pell Octonions, Adv. Appl. Clifford Algebr. Vol:26 (2016), 435-440.
Year 2017, Volume: 5 Issue: 2, 102 - 113, 15.10.2017

Abstract

References

  • [1] Akyigit, M., Kosal, H.H. and Tosun, M., Split Fibonacci Quaternions, Adv. Appl. Clifford Algebr. Vol:23 (2014), 535-545.
  • [2] Akyigit, M., Kosal, H.H. and Tosun, M., Fibonacci Generalized Quaternions, Adv. Appl. Clifford Algebr. Vol:24 (2014), 631-641.
  • [3] Cimen, C.B. and Ipek, A., On Pell Quaternions and Pell-Lucas Quaternions, Adv. Appl. Clifford Algebr., Vol:26 (2016), 39-51.
  • [4] Catarino, P., The Modi ed Pell and the Modi ed k-Pell Quaternions and Octonions, Adv. Appl. Clifford Algebr., Vol:26 (2016), 577-590.
  • [5] Catarino, P. and Vasco, P., On Dual k􀀀Pell Quaternions and Octonions, Mediterr. J. Math., Vol:14 (2017), 75-87.
  • [6] Falcon, S. and Plaza, A., The k-Fibonacci Sequence and the Pascal 2-Triangle, Chaos Solitons Fractals, Vol:33, No.1 (2007), 38-49.
  • [7] Falcon, S., On the k-Lucas Numbers, Int. J. Contemp. Math. Sci. Vol:21 (2011), 1039-1050.
  • [8] Halici, S., On Fibonacci Quaternions, Adv. Appl. Clifford Algebr. Vol:22 (2012), 321-327.
  • [9] Harman, C.J., Complex Fibonacci Numbers, Fibonacci Quart. Vol:19, No.1 (1981), 82-86. [10] Horadam, A. F. , Complex Fibonacci Numbers and Fibonacci Quaternions, Amer. Math. Monthly No:70 (1963), 289-291.
  • [11] Iyer, M. R., Some Result on Fibonacci Quaternions, Fibonacci Quart. Vol:7, No.2 (1969), 201-210.
  • [12] Koshy, T., Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience Publication, Canada, 2001.
  • [13] Koshy, T., Pell and Pell-Lucas Numbers with Applications, Springer-Verlag New York, USA, 2014.
  • [14] Polatli,E., Kizilates, C. and Kesim, S., On Split k􀀀Fibonacci and k-Lucas Quaternions, Adv. Appl. Clifford Algebr. Vol:26 (2016), 353-362.
  • [15] Polatli, E. and Kesim, S., A Note on Catalan's Identity for the k􀀀 Fibonacci Quaternions, J. Integer Seq. Vol:18 (Article 15.8.2 ) (2015), 1-4.
  • [16] Ramirez, J. L., Some Combinatorial Properties of the k-Fibonacci and the k-Lucas Quaternions, An. St. Univ. Ovidius Constanta Ser. Mat. Vol:23 No.2 (2015), 201-212.
  • [17] Tokeser, U., Unal, Z. and Bilgici, G., Split Pell and Split Pell-Lucas Quaternions, Adv. Appl. Clifford Algebr. Vol: 27 (2017), 1881-1893.
  • [18] Szynal-Liana, A. and Wloch, I. The Pell Quaternions and the Pell Octonions, Adv. Appl. Clifford Algebr. Vol:26 (2016), 435-440.
There are 17 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

GÖKSAL Bilgici

ÜMİT Tokaşer This is me

ZAFER Ünal

Publication Date October 15, 2017
Submission Date October 15, 2017
Acceptance Date June 7, 2017
Published in Issue Year 2017 Volume: 5 Issue: 2

Cite

APA Bilgici, G., Tokaşer, Ü., & Ünal, Z. (2017). $k$-FIBONACCI AND $k$-LUCAS GENERALIZED QUATERNIONS. Konuralp Journal of Mathematics, 5(2), 102-113.
AMA Bilgici G, Tokaşer Ü, Ünal Z. $k$-FIBONACCI AND $k$-LUCAS GENERALIZED QUATERNIONS. Konuralp J. Math. October 2017;5(2):102-113.
Chicago Bilgici, GÖKSAL, ÜMİT Tokaşer, and ZAFER Ünal. “$k$-FIBONACCI AND $k$-LUCAS GENERALIZED QUATERNIONS”. Konuralp Journal of Mathematics 5, no. 2 (October 2017): 102-13.
EndNote Bilgici G, Tokaşer Ü, Ünal Z (October 1, 2017) $k$-FIBONACCI AND $k$-LUCAS GENERALIZED QUATERNIONS. Konuralp Journal of Mathematics 5 2 102–113.
IEEE G. Bilgici, Ü. Tokaşer, and Z. Ünal, “$k$-FIBONACCI AND $k$-LUCAS GENERALIZED QUATERNIONS”, Konuralp J. Math., vol. 5, no. 2, pp. 102–113, 2017.
ISNAD Bilgici, GÖKSAL et al. “$k$-FIBONACCI AND $k$-LUCAS GENERALIZED QUATERNIONS”. Konuralp Journal of Mathematics 5/2 (October 2017), 102-113.
JAMA Bilgici G, Tokaşer Ü, Ünal Z. $k$-FIBONACCI AND $k$-LUCAS GENERALIZED QUATERNIONS. Konuralp J. Math. 2017;5:102–113.
MLA Bilgici, GÖKSAL et al. “$k$-FIBONACCI AND $k$-LUCAS GENERALIZED QUATERNIONS”. Konuralp Journal of Mathematics, vol. 5, no. 2, 2017, pp. 102-13.
Vancouver Bilgici G, Tokaşer Ü, Ünal Z. $k$-FIBONACCI AND $k$-LUCAS GENERALIZED QUATERNIONS. Konuralp J. Math. 2017;5(2):102-13.
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