$k$-FIBONACCI AND $k$-LUCAS GENERALIZED QUATERNIONS
Year 2017,
Volume: 5 Issue: 2, 102 - 113, 15.10.2017
GÖKSAL Bilgici
,
ÜMİT Tokaşer
ZAFER Ünal
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
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Year 2017,
Volume: 5 Issue: 2, 102 - 113, 15.10.2017
GÖKSAL Bilgici
,
ÜMİT Tokaşer
ZAFER Ünal
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 Modied Pell and the Modied k-Pell Quaternions and Octonions, Adv. Appl. Clifford Algebr., Vol:26 (2016), 577-590.
- [5] Catarino, P. and Vasco, P., On Dual kPell 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 kFibonacci 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.