Skip to main content
SpringerLink
Log in

Investigation of Dual-Complex Fibonacci, Dual-Complex Lucas Numbers and Their Properties

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

In this study, we define the dual complex Fibonacci and Lucas numbers. We give the generating functions and Binet formulas for these numbers. Moreover, the well-known properties e.g. Cassini and Catalan identities have been obtained for these numbers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akyiğit, M., Kösal, H.H., Tosun, M.: Split fibonacci quaternions. Adv. Appl. Clifford Algebras 23, 535–545 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Akyiğit, M., Kösal, H.H., Tosun, M.: Fibonacci generalized quaternions. Adv. Appl. Clifford Algebras 24, 631–641 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ata, E., Yaylı, Y.: Dual unitary matrices and unit dual quaternions. Diff. Geom. Dyn. Syst. 10, 1–12 (2008)

    MATH  MathSciNet  Google Scholar 

  4. Clifford, W.K.: Preliminary sketch of biquaternions. Proc. Lond. Math. Soc. 4(64), 381–395 (1873)

    MATH  MathSciNet  Google Scholar 

  5. Dunlap, R.A.: The golden ratio and fibonacci numbers. World Scientific Publishing, Singapore (1997)

    Book  MATH  Google Scholar 

  6. Halıcı, S.: On fibonacci quaternions. Adv. Appl. Clifford Algebras 22, 321–327 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Halıcı, S.: On complex fibonacci quaternions. Adv. Appl. Clifford Algebras 23, 105–112 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Horadam, A.F.: Complex fibonacci numbers and fibonacci quaternions. Am. Math. Mon. 70, 289–291 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  9. Iyer, M.R.: Some results on fibonacci quaternions. Fibonacci Quart. 7(2), 201–210 (1969)

    MATH  MathSciNet  Google Scholar 

  10. Knuth, D.: Negafibonacci numbers and hyperbolic plane. Annual meeting of the Math. Association of America, San Jose, California (2013)

  11. Koshy, T.: Fibonacci and lucas numbers with applications. Wiley, New York (2001)

    Book  MATH  Google Scholar 

  12. Kotelnikov, A. P.: Screw calculus and some of its applications to geometry and mechanics. Magister Dissertation, Kazan. Gos. Univ., Kazan (1895) (Russian)

  13. Majernik, V.: Multicompenent number systems. Acta Phys. Pol. A 90, 491–498 (1996)

    Article  MathSciNet  Google Scholar 

  14. Messelmi F.: Dual-complex numbers and their holomorphic functions. https://hal.archives-ouvertes.fr/hal-01114178 (2015). Accessed 9 May 2015

  15. Nurkan, S.K., Güven, İ.A.: Dual fibonacci quaternions. Adv. Appl. Clifford Algebras 25, 403–414 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  16. Nurkan, S.K., Güven, İ.A.: A Note on Bicomplex Fibonacci and Lucas Numbers. arXiv:1508.03972v1, (2015)

  17. Siegler, L.E.: Fibonaccis Liber Abaci: A Translation into Modern English of the Book of Calculation. Sources and Studies in the History of Mathematics and Physical Sciences, Springer, Berlin, Germany (2002)

  18. Study E.: Geometrie der Dynamen. Verlag Teubner, Leipzig (1903)

  19. Swamy, M.N.: On Generalized Fibonacci Quaternions. Fibonacci Quart. 5, 547–550 (1973)

    MATH  MathSciNet  Google Scholar 

  20. Vajda, S.: Fibonacci and Lucas Numbers and the Golden Section. Ellis Horwood Books Math. Appl. Ellis Horwood Ltd./Halsted Press, Chichester, England (1989)

    Google Scholar 

  21. Verner, E., Hoggatt, Jr.: Fibonacci and Lucas Numbers. The Fibonacci Association, Santa Clara, CA (1969)

  22. Weisstein, E.W.: Fibonacci Number. Mathworld (online mathematics reference work)

  23. Yüce, S., Aydın, F.T.: A New Aspect of Dual Fibonacci Quaternions. Adv. Appl. Clifford Algebras 26, 873–884 (2016)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank to the referees for their helpful suggestions and comments.

Author information

Authors and Affiliations

  1. Faculty of Arts and Sciences, Department of Mathematics, Sakarya University, 54187, Hendek, Sakarya, Turkey

    Mehmet Ali Güngör

  2. Faculty of Education, Mathematics and Science Education Department, Sakarya University, 54300, Hendek, Sakarya, Turkey

    Ayşe Zeynep Azak

Authors
  1. Mehmet Ali Güngör
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ayşe Zeynep Azak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ayşe Zeynep Azak.

Additional information

Communicated by Wolfgang Sprössig

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Güngör, M.A., Azak, A.Z. Investigation of Dual-Complex Fibonacci, Dual-Complex Lucas Numbers and Their Properties. Adv. Appl. Clifford Algebras 27, 3083–3096 (2017). https://doi.org/10.1007/s00006-017-0813-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00006-017-0813-z

Mathematics Subject Classification

Keywords

Search

Navigation