RETRACTED ARTICLE: A Determinantal Expression for the Fibonacci Polynomials in Terms of a Tridiagonal Determinant

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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.

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  • 13 October 2020

    This article has been retracted. Please see the retraction notice for more detail: <ExternalRef><RefSource>https://doi.org/10.1007/s41980-020-00472-9</RefSource><RefTarget Address="10.1007/s41980-020-00472-9" TargetType="DOI"/></ExternalRef>.

References

  1. 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

    Google Scholar 

  2. 2.

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

    Google Scholar 

  3. 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

    MathSciNet  Article  MATH  Google Scholar 

  4. 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

    MathSciNet  Article  MATH  Google Scholar 

  5. 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

    MathSciNet  Article  MATH  Google Scholar 

  6. 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

    MathSciNet  Article  MATH  Google Scholar 

  7. 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

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

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

  9. 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

    MathSciNet  Article  MATH  Google Scholar 

  10. 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

    MathSciNet  Article  MATH  Google Scholar 

  11. 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

    MathSciNet  Article  MATH  Google Scholar 

  12. 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

    MathSciNet  Article  MATH  Google Scholar 

  13. 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

    MathSciNet  Article  MATH  Google Scholar 

  14. 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. 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

    MathSciNet  Article  MATH  Google Scholar 

  16. 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. 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

    MathSciNet  Article  MATH  Google Scholar 

  18. 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)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

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

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Correspondence to Bai-Ni Guo.

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This article has been retracted. Please see the retraction notice for more detail:https://doi.org/10.1007/s41980-020-00472-9

Communicated by Amir Akbary.

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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

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Keywords

  • Determinantal expression
  • Fibonacci number
  • Fibonacci polynomial
  • Tridiagonal determinant
  • Hessenberg determinant

Mathematics Subject Classification

  • 11B39
  • 11B83
  • 11C20
  • 11Y55
  • 15A15
  • 65F40