Skip to main content
SpringerLink
Log in

RETRACTED ARTICLE: A Determinantal Expression for the Fibonacci Polynomials in Terms of a Tridiagonal Determinant

  • Original Paper
  • Published:
Bulletin of the Iranian Mathematical Society Aims and scope Submit manuscript

This article was retracted on 13 October 2020

This article has been updated

Abstract

In the paper, after concisely reviewing and surveying some known results, the authors find a determinantal expression for the Fibonacci polynomials and, consequently, for the Fibonacci numbers, in terms of a tridiagonal determinant.

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

Change history

References

  1. Bourbaki, N.: Elements of Mathematics: Functions of a Real Variable: Elementary Theory, Translated from the 1976 French Original by Philip Spain. Elements of Mathematics (Berlin). Springer, Berlin (2004). https://doi.org/10.1007/978-3-642-59315-4

    Book  Google Scholar 

  2. Dunlap, R.A.: The Golden Ratio and Fibonacci Numbers. World Scientific Publishing Co., Inc., River Edge (1997). https://doi.org/10.1142/9789812386304

    Book  MATH  Google Scholar 

  3. Guo, B.-N., Mező, I., Qi, F.: An explicit formula for the Bernoulli polynomials in terms of the \(r\)-Stirling numbers of the second kind. Rocky Mt. J. Math. 46(6), 1919–1923 (2016). https://doi.org/10.1216/RMJ-2016-46-6-1919

    Article  MathSciNet  MATH  Google Scholar 

  4. Guo, B.-N., Qi, F.: Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind. J. Comput. Appl. Math. 272, 251–257 (2014). https://doi.org/10.1016/j.cam.2014.05.018

    Article  MathSciNet  MATH  Google Scholar 

  5. Guo, B.-N., Qi, F.: Generalization of Bernoulli polynomials. Int. J. Math. Ed. Sci. Technol. 33(3), 428–431 (2002). https://doi.org/10.1080/002073902760047913

    Article  MathSciNet  MATH  Google Scholar 

  6. Guo, B.-N., Qi, F.: Some identities and an explicit formula for Bernoulli and Stirling numbers. J. Comput. Appl. Math. 255, 568–579 (2014). https://doi.org/10.1016/j.cam.2013.06.020

    Article  MathSciNet  MATH  Google Scholar 

  7. Higgins, V., Johnson, C.: Inverse spectral problems for collections of leading principal submatrices of tridiagonal matrices. Linear Algebra Appl. 489, 104–122 (2016). https://doi.org/10.1016/j.laa.2015.10.004

    Article  MathSciNet  MATH  Google Scholar 

  8. Janjić, M.: Determinants and recurrence sequences. J. Integer Seq. 15(3) (2012). (article 12.3.5, 21 pages)

  9. Kittappa, R.K.: A representation of the solution of the \(n\)th order linear difference equation with variable coefficients. Linear Algebra Appl. 193, 211–222 (1993). https://doi.org/10.1016/0024-3795(93)90278-V

    Article  MathSciNet  MATH  Google Scholar 

  10. Martin, R.S., Wilkinson, J.H.: Handbook series linear algebra: similarity reduction of a general matrix to Hessenberg form. Numer. Math. 12(5), 349–368 (1968). https://doi.org/10.1007/BF02161358

    Article  MathSciNet  MATH  Google Scholar 

  11. Qi, F.: A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers. J. Comput. Appl. Math. 351, 1–5 (2019). https://doi.org/10.1016/j.cam.2018.10.049

    Article  MathSciNet  MATH  Google Scholar 

  12. Qi, F.: Derivatives of tangent function and tangent numbers. Appl. Math. Comput. 268, 844–858 (2015). https://doi.org/10.1016/j.amc.2015.06.123

    Article  MathSciNet  MATH  Google Scholar 

  13. Qi, F.: Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind. Filomat 28(2), 319–327 (2014). https://doi.org/10.2298/FIL1402319O

    Article  MathSciNet  MATH  Google Scholar 

  14. Qi, F.: Notes on a double inequality for ratios of any two neighbouring non-zero Bernoulli numbers. Turk. J. Anal. Number Theory 6(5), 129–131 (2018). https://doi.org/10.12691/tjant-6-5-1

    Article  Google Scholar 

  15. Qi, F., Chapman, R.J.: Two closed forms for the Bernoulli polynomials. J. Number Theory 159, 89–100 (2016). https://doi.org/10.1016/j.jnt.2015.07.021

    Article  MathSciNet  MATH  Google Scholar 

  16. Qi, F., Guo, B.-N.: Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials. Mediterr. J. Math. 14(3) (2017). https://doi.org/10.1007/s00009-017-0939-1. (article 140, 14 pages)

  17. Qi, F., Zhang, X.-J.: An integral representation, some inequalities, and complete monotonicity of the Bernoulli numbers of the second kind. Bull. Korean Math. Soc. 52(3), 987–998 (2015). https://doi.org/10.4134/BKMS.2015.52.3.987

    Article  MathSciNet  MATH  Google Scholar 

  18. Wei, C.-F., Qi, F.: Several closed expressions for the Euler numbers. J. Inequal. Appl. 2015, 219 (2015). https://doi.org/10.1186/s13660-015-0738-9. (8 pages)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors express many thanks to anonymous referees for their careful corrections to and valuable comments on the original version of this paper.

Author information

Authors and Affiliations

  1. College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, 028043, Inner Mongolia, China

    Feng Qi

  2. School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, 454010, Henan, China

    Feng Qi & Bai-Ni Guo

  3. Teda No. 2 Primary School, No. 98, The Fifth Street, Economic and Technological Development Zone, Tianjin, 300457, China

    Jing-Lin Wang

  4. School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin, 300387, China

    Feng Qi & Jing-Lin Wang

Authors
  1. Feng Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jing-Lin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bai-Ni Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bai-Ni Guo.

Additional information

Communicated by Amir Akbary.

Publisher's Note

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

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qi, F., Wang, JL. & Guo, BN. RETRACTED ARTICLE: A Determinantal Expression for the Fibonacci Polynomials in Terms of a Tridiagonal Determinant. Bull. Iran. Math. Soc. 45, 1821–1829 (2019). https://doi.org/10.1007/s41980-019-00232-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41980-019-00232-4

Keywords

Mathematics Subject Classification

Search

Navigation