Advertisement

Advances in Applied Clifford Algebras

, Volume 26, Issue 2, pp 719–730 | Cite as

A Generalization of Fibonacci and Lucas Quaternions

  • Emrah Polatlı
Article

Abstract

In this paper, we give a generalization of the Fibonacci and Lucas quaternions. We obtain the Binet formulas, generating functions, and some certain identities for these quaternions which include generalizations of some results of Halici.

Keywords

Generalized Fibonacci quaternions Generalized Lucas quaternions Extended Binet formulas 

Mathematics Subject Classification

Primary 11B39 Secondary 11B37 11R52 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akyiğit M., Kösal H.H., Tosun M.: Split Fibonacci quaternions. Adv. Appl. Clifford Algebras 23, 535–545 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Akyiğit M., Kösal H.H., Tosun M.: Fibonacci generalized quaternions. Adv. Appl. Clifford Algebras 24, 631–641 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bilgici G.: Two generalizations of Lucas sequence. Appl. Math. Comput. 245, 526–538 (2014)MathSciNetMATHGoogle Scholar
  4. 4.
    Catarino P.: A note on h(x)-Fibonacci quaternion polynomials. Chaos Solitons Fractals 77, 1–5 (2015)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Edson M., Yayenie O.: A new generalization of Fibonacci sequence and extended Binet’s formula. INTEGERS 9, 639–654 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Edson M., Lewis S., Yayenie O.: The k-periodic Fibonacci sequence and an extended Binet’s formula. INTEGERS 11, 739–751 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gürlebeck K., Sprößig W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Chichester (1997)MATHGoogle Scholar
  8. 8.
    Halici S.: On Fibonacci quaternions. Adv. Appl. Clifford Algebras 22, 321–327 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Horadam A.F.: Complex Fibonacci numbers and Fibonacci quaternions. Am. Math. Mon. 70, 289–291 (1963)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Horadam A.F.: Quaternion recurrence relations. Ulam Q. 2, 22–33 (1993)MathSciNetMATHGoogle Scholar
  11. 11.
    Iakin A.L.: Generalized quaternions of higher order. Fibonacci Q. 15, 343–346 (1977)MathSciNetMATHGoogle Scholar
  12. 12.
    Iakin A.L.: Generalized quaternions with quaternion components. Fibonacci Q. 15, 350–352 (1977)MathSciNetMATHGoogle Scholar
  13. 13.
    Iakin A.L.: Extended Binet forms for generalized quaternions of higher order. Fibonacci Q. 19, 410–413 (1981)MathSciNetMATHGoogle Scholar
  14. 14.
    Irmak N., Alp M.: Some identities for generalized Fibonacci and Lucas sequences. Hacet. J. Math. 42, 331–338 (2013)MathSciNetMATHGoogle Scholar
  15. 15.
    Iyer M.R.: Some results on Fibonacci quaternions. Fibonacci Q. 7, 201–210 (1969)MathSciNetMATHGoogle Scholar
  16. 16.
    Iyer M.R.: A note on Fibonacci quaternions. Fibonacci Q. 7, 225–229 (1969)MathSciNetMATHGoogle Scholar
  17. 17.
    Koshy, T.: Fibonacci and Lucas Numbers with Applications. A Wiley-Interscience Publication (2001)Google Scholar
  18. 18.
    Polatli, E., Kesim, S.: On quaternions with generalized Fibonacci and Lucas number components. Adv. Differ. Equ. 2015, 1–8 (2015)Google Scholar
  19. 19.
    Ramírez J.L.: Some combinatorial properties of the k-Fibonacci and the k-Lucas quaternions. An. St. Univ. Ovidius Constanta 23, 201–212 (2015)MathSciNetGoogle Scholar
  20. 20.
    Sahin M.: The Gelin-Cesàro identity in some conditional sequences. Hacet. J. Math. 40, 855–861 (2011)MathSciNetMATHGoogle Scholar
  21. 21.
    Sahin M.: The generating function of a family of the sequences in terms of the continuant. Appl. Math. Comput. 217, 5416–5420 (2011)MathSciNetMATHGoogle Scholar
  22. 22.
    Swamy M.N.S.: On generalized Fibonacci quaternions. Fibonacci Q. 11, 547–549 (1973)MathSciNetMATHGoogle Scholar
  23. 23.
    Vajda S.: Fibonacci and Lucas Numbers, and the Golden Section. Ellis Horwood, Chichester (1989)MATHGoogle Scholar
  24. 24.
    Ward J.P.: Quaternions and Cayley Numbers. Kluwer, Dordrecht (1997)CrossRefMATHGoogle Scholar
  25. 25.
    Yayenie O.: A note on generalized Fibonacci sequences. Appl. Math. Comput. 217, 5603–5611 (2011)MathSciNetMATHGoogle Scholar
  26. 26.
    Yayenie O.: New identities for generalized Fibonacci sequences and new generalization of Lucas sequences. Southeast Asian Bull. Math. 36, 739–752 (2012)MathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and ArtsBulent Ecevit UniversityZonguldakTurkey

Personalised recommendations