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Kısıtlamasız k – Fibonacci ve k – Lucas genelleştirilmiş kuaterniyonları

Year 2021, , 179 - 188, 15.01.2021
https://doi.org/10.17714/gumusfenbil.726136

Abstract

Literatürde Fibonacci ve Lucas sayılarının birçok genelleştirilmesi bulunmaktadır Bu genelleştirmelere bir örnek olarak k – Fibonacci ve k – Lucas sayıları verilebilir. Bu çalışmada kısıtlamasız k – Fibonacci ve k – Lucas genelleştirilmiş kuaterniyonları tanımlanmıştır. Kısıtlamasız kelimesinden, kuaterniyonların sıralı tabanındaki versörlerin katsayılarının keyfi k – Fibonacci ve k – Lucas sayısı olarak atanabilmesi kastedilmektedir. Bu doğrultuda, kısıtlamasız k – Fibonacci ve k – Lucas genelleştirilmiş kuaterniyonlarının üreteç fonksiyonları ve Binet formülleri elde edildikten sonra, bilinen bazı özdeşliklerin genelleştirmeleri verilmiştir.

References

  • Akyigit, M., Kosal, H.H. ve Tosun, M. (2013). Split Fibonacci quaternions. Advances in Applied Clifford Algebras, 23(3), 535-545, https://doi.org/10.1007/s00006-013-0401-9
  • Akyigit, M., Kosal, H.H. ve Tosun, M. (2014). Fibonacci generalized quaternions. Advances in Applied Clifford Algebras, 24(3), 631-641, https://doi.org/10.1007/s00006-014-0458-0
  • Bilgici, G., Tokeser, U. ve Unal, Z. (2017). k-Fibonacci and k-Lucas generalized quaternions. Konuralp Journal of Mathematics, 5(2), 102 – 113.
  • Catarino, P. (2016). The modified Pell and the modified k-Pell quaternions and octonions. Advances in Applied Clifford Algebras, 26(2), 577-590, https://doi.org.10.1007/s00006-015-0611-4
  • Çimen, C.B. ve İpek, A. (2016). On Pell quaternions and Pell-Lucas quaternions. Advances in Applied Clifford Algebras, 26(1), 39-51, https://doi.org/10.1007/s00006-015-0571-8
  • Falcon, S. ve Plaza, A. (2007). The k-Fibonacci sequence and the Pascal 2-triangle. Chaos Solitons Fractals, 33(1), 38-49, https://doi.org/10.1016/j.chaos.2006.10.022
  • Falcon, S. (2011). On the k-Lucas numbers. International Journal of Contemporary Mathematical Sciences, 21, 1039-1050.
  • Halıcı, S. (2012). On Fibonacci quaternions. Advances in Applied Clifford Algebras, 22(2), 321-327, https://doi.org/10.1007/s00006-011-0317-1
  • Hamilton, W.R. (1853). Lectures on quaternions. Dublin: Hodges and Smith.
  • Horadam, A. F. (1963). Complex Fibonacci numbers and Fibonacci quaternions. The American Mathematical Monthly, 70, 289-291.
  • Iyer, M. R. (1969). Some result on Fibonacci quaternions. Fibonacci Quarterly, 7(2), 201-210.
  • Koshy, T. (2001). Fibonacci and Lucas numbers with applications. Canada: A Wiley-Interscience Publication.
  • Koshy, T. (2014). Pell and Pell-Lucas numbers with applications. New York: Springer-Verlag.
  • Polatlı, E. ve Kesim, S. (2015). A Note on Catalan's identity for the k- Fibonacci quaternions. Journal of Integer Sequence, 18, 1-4.
  • Ramirez, J. L. (2015). Some combinatorial properties of the k-Fibonacci and the k-Lucas quaternions. Analele Stiintifice ale Universitatii Ovidius Constanta Seria Matematica, 23(2), 201-212, https://doi.org/10.1515/auom-2015-0037
  • Tokeser, U., Unal, Z. ve Bilgici, G. (2017). Split Pell and split Pell-Lucas quaternions. Advances in Applied Clifford Algebras, 27(2), 1881-1893, https://doi.org/10.1007/s00006-016-0747-x

Kısıtlamasız k – Fibonacci ve k – Lucas genelleştirilmiş kuaterniyonları

Year 2021, , 179 - 188, 15.01.2021
https://doi.org/10.17714/gumusfenbil.726136

Abstract

There are many generalizations of Fibonacci and Lucas numbers. One of them is k – Fibonacci and k – Lucas numbers. In this study, we introduce unrestricted k – Fibonacci and k – Lucas generalized quaternions. The word “unrestricted” means that we can determine the coefficients of the versors of the basis of the quaternions arbitrarily. In this manner, we give generating functions and Binet formulas for the unrestricted k – Fibonacci and k – Lucas generalized quaternions and obtain generalizations of some well – known identities.

References

  • Akyigit, M., Kosal, H.H. ve Tosun, M. (2013). Split Fibonacci quaternions. Advances in Applied Clifford Algebras, 23(3), 535-545, https://doi.org/10.1007/s00006-013-0401-9
  • Akyigit, M., Kosal, H.H. ve Tosun, M. (2014). Fibonacci generalized quaternions. Advances in Applied Clifford Algebras, 24(3), 631-641, https://doi.org/10.1007/s00006-014-0458-0
  • Bilgici, G., Tokeser, U. ve Unal, Z. (2017). k-Fibonacci and k-Lucas generalized quaternions. Konuralp Journal of Mathematics, 5(2), 102 – 113.
  • Catarino, P. (2016). The modified Pell and the modified k-Pell quaternions and octonions. Advances in Applied Clifford Algebras, 26(2), 577-590, https://doi.org.10.1007/s00006-015-0611-4
  • Çimen, C.B. ve İpek, A. (2016). On Pell quaternions and Pell-Lucas quaternions. Advances in Applied Clifford Algebras, 26(1), 39-51, https://doi.org/10.1007/s00006-015-0571-8
  • Falcon, S. ve Plaza, A. (2007). The k-Fibonacci sequence and the Pascal 2-triangle. Chaos Solitons Fractals, 33(1), 38-49, https://doi.org/10.1016/j.chaos.2006.10.022
  • Falcon, S. (2011). On the k-Lucas numbers. International Journal of Contemporary Mathematical Sciences, 21, 1039-1050.
  • Halıcı, S. (2012). On Fibonacci quaternions. Advances in Applied Clifford Algebras, 22(2), 321-327, https://doi.org/10.1007/s00006-011-0317-1
  • Hamilton, W.R. (1853). Lectures on quaternions. Dublin: Hodges and Smith.
  • Horadam, A. F. (1963). Complex Fibonacci numbers and Fibonacci quaternions. The American Mathematical Monthly, 70, 289-291.
  • Iyer, M. R. (1969). Some result on Fibonacci quaternions. Fibonacci Quarterly, 7(2), 201-210.
  • Koshy, T. (2001). Fibonacci and Lucas numbers with applications. Canada: A Wiley-Interscience Publication.
  • Koshy, T. (2014). Pell and Pell-Lucas numbers with applications. New York: Springer-Verlag.
  • Polatlı, E. ve Kesim, S. (2015). A Note on Catalan's identity for the k- Fibonacci quaternions. Journal of Integer Sequence, 18, 1-4.
  • Ramirez, J. L. (2015). Some combinatorial properties of the k-Fibonacci and the k-Lucas quaternions. Analele Stiintifice ale Universitatii Ovidius Constanta Seria Matematica, 23(2), 201-212, https://doi.org/10.1515/auom-2015-0037
  • Tokeser, U., Unal, Z. ve Bilgici, G. (2017). Split Pell and split Pell-Lucas quaternions. Advances in Applied Clifford Algebras, 27(2), 1881-1893, https://doi.org/10.1007/s00006-016-0747-x
There are 16 citations in total.

Details

Primary Language Turkish
Journal Section Articles
Authors

Göksal Bilgici 0000-0001-9964-5578

Publication Date January 15, 2021
Submission Date April 23, 2020
Acceptance Date December 23, 2020
Published in Issue Year 2021

Cite

APA Bilgici, G. (2021). Kısıtlamasız k – Fibonacci ve k – Lucas genelleştirilmiş kuaterniyonları. Gümüşhane Üniversitesi Fen Bilimleri Dergisi, 11(1), 179-188. https://doi.org/10.17714/gumusfenbil.726136