Skip to main content
SpringerLink
Log in

On r-circulant matrices with generalized bi-periodic Fibonacci numbers

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, we calculate the Frobenius norm, and give upper and lower bounds for the spectral norm of r-circulant matrices whose entries are defined in terms of generalized bi-periodic Fibonacci numbers. We also provide explicit formulas for the computation of eigenvalues and determinants of these matrices.

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. Alptekin, E.G., Mansour, T., Tuglu, N.: Norms of circulant and semicirculant matrices with Horadam’s numbers. ARS Combinatoria 85, 353–359 (2007)

  2. Bahsi, M.: On the norms of circulant matrices with the generalized Fibonacci and Lucas numbers. TWMS J. Pure Appl. Math. 6(1), 84–92 (2015)

    MathSciNet  MATH  Google Scholar 

  3. Belbachir, H., Belkhir, A.: On some generalizations of Horadam’s numbers. Filomat 32(14), 5037–5052 (2018)

  4. Bilgici, G.: Two generalizations of Lucas sequence. Appl. Math. Comput. 245, 526–538 (2014)

    Article  MathSciNet  Google Scholar 

  5. Chandoul, A.: On the norms of \(r\)-circulant matrices with generalized Fibonacci numbers. J. Algebra Comb. Discrete Appl. 4, 13–21 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Cline, R.E., Plemmons, R.J., Worm, G.: Generalized inverses of certain Toeplitz matrices. Linear Algebr. Appl. 8, 25–33 (1974)

    Article  MathSciNet  Google Scholar 

  7. Edson, M., Yayenie, O.: A new generalization of Fibonacci sequences and extended Binet’s Formula. Integers 9(A48), 639–654 (2009)

  8. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

  9. Köme, C., Yazlik, Y.: On the spectral norms of \(r\)-circulant matrices with the biperiodic Fibonacci and Lucas numbers. J. Inequal. Appl. 2017, 192 (2017)

    Article  MathSciNet  Google Scholar 

  10. Lind, D.A.: A Fibonacci circulant. Fibonacci Quart. 8(5), 449–455 (1970)

    MathSciNet  MATH  Google Scholar 

  11. Merikoski, J.K., Haukkanen, P., Mattila, M., Tossavainen, T.: On the spectral and Frobenius norm of a generalized Fibonacci \(r\)-circulant matrix. Spec. Matrices 6, 23–36 (2018)

    Article  MathSciNet  Google Scholar 

  12. Nalli, A., Şen, M.: On the norms of circulant matrices with generalized Fibonacci numbers. Selçuk J. Appl. Math. 11(1), 107–116 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Panario, D., Sahin, M., Wang, Q.: A family of Fibonacci-like conditional sequences. Integers 13, A78 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Shen, S., Cen, J.: On the bounds for the norms of \(r\)-circulant matrices with the Fibonacci and Lucas numbers. Appl. Math. Comput. 216, 2891–2897 (2010)

    Article  MathSciNet  Google Scholar 

  15. Solak, S.: On the norms of circulant matrices with the Fibonacci and Lucas numbers. Appl. Math. Comput. 160(1), 125–132 (2005)

    Article  MathSciNet  Google Scholar 

  16. Solak, S.: Erratum to “On the norms of circulant matrices with the Fibonacci and Lucas numbers” [Appl. Math. Comput. 160(1) 125–132 (2005)]. Appl. Math. Comput. 190(2), 1855–1856 (2007)

  17. Stone, B.J.: Best possible ratios of certain matrix norms. Numer. Math. 4, 114–116 (1962)

    Article  MathSciNet  Google Scholar 

  18. Tan, E.: General sum formula for the bi-periodic Fibonacci and Lucas numbers. Integers 17, A42 (2017)

    MathSciNet  MATH  Google Scholar 

  19. Tan, E.: Some properties of bi-periodic Horadam sequences. Notes Number Theory Discrete Math. 23(4), 56–65 (2017)

    MATH  Google Scholar 

  20. Tan, E., Leung, H.-H.: Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences. Adv. Differ. Equ. 2020, 26 (2020)

    Article  MathSciNet  Google Scholar 

  21. Yayenie, O.: A note on generalized Fibonacci sequence. Appl. Math. Comput. 217, 5603–5611 (2011)

    Article  MathSciNet  Google Scholar 

  22. Yayenie, O.: New identities for generalized Fibonacci sequences and new generalization of Lucas sequences. Southeast Asian Bull. Math. 36, 739–752 (2012)

    MathSciNet  MATH  Google Scholar 

  23. Yazlik, Y., Taskara, N.: On the norms of an \(r\)-circulant matrix with the generalized \(k\)-Horadam numbers. J. Inequal. Appl. 2013, 394 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We are very grateful to anonymous referees for various comments that have led to a number of improvements in the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Amasya University, 05000, Amasya, Turkey

    Mehmet Daǧlı

  2. Department of Mathematics, Ankara University, 06100, Ankara, Turkey

    Elif Tan & Oktay Ölmez

Authors
  1. Mehmet Daǧlı
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elif Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Oktay Ölmez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mehmet Daǧlı.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was supported by The Scientific Research Coordination Unit of Amasya University. Project Number: FMB-BAP 20-0474.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Daǧlı, M., Tan, E. & Ölmez, O. On r-circulant matrices with generalized bi-periodic Fibonacci numbers. J. Appl. Math. Comput. 68, 2003–2014 (2022). https://doi.org/10.1007/s12190-021-01610-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-021-01610-0

Keywords

Mathematics Subject Classification

Search

Navigation