Skip to main content

Extending generalized Fibonacci sequences and their binet-type formula

Abstract

We study the extension problem of a given sequence defined by a finite order recurrence to a sequence defined by an infinite order recurrence with periodic coefficient sequence. We also study infinite order recurrence relations in a strong sense and give a complete answer to the extension problem. We also obtain a Binet-type formula, answering several open questions about these sequences and their characteristic power series.

[12345678910]

References

  1. 1.

    Bernoussi B, Motta W, Rachidi M, Saeki O: Approximation of -generalized Fibonacci sequences and their asymptotic Binet formula. The Fibonacci Quarterly 2001,39(2):168–180.

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Bernoussi B, Motta W, Rachidi M, Saeki O: On periodic -generalized Fibonacci sequences. The Fibonacci Quarterly 2004,42(4):361–367.

    MathSciNet  Google Scholar 

  3. 3.

    Dubeau F: On r -generalized Fibonacci numbers. The Fibonacci Quarterly 1989,27(3):221–229.

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Dubeau F, Motta W, Rachidi M, Saeki O: On weighted r -generalized Fibonacci sequences. The Fibonacci Quarterly 1997,35(2):102–110.

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Levesque C: On m -th order linear recurrences. The Fibonacci Quarterly 1985,23(4):290–293.

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Miles EP Jr.: Generalized Fibonacci numbers and associated matrices. The American Mathematical Monthly 1960,67(8):745–752. 10.2307/2308649

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Motta W, Rachidi M, Saeki O: On -generalized Fibonacci sequences. The Fibonacci Quarterly 1999,37(3):223–232.

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Motta W, Rachidi M, Saeki O: Convergent -generalized Fibonacci sequences. The Fibonacci Quarterly 2000,38(4):326–333.

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Mouline M, Rachidi M: Suites de Fibonacci généralisées et chaînes de Markov. Real Academia de Ciencias Exactas, Físicas y Naturales de Madrid. Revista 1995,89(1–2):61–77.

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Mouline M, Rachidi M: -Generalized Fibonacci sequences and Markov chains. The Fibonacci Quarterly 2000,38(4):364–371.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Mustapha Rachidi.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Rachidi, M., Saeki, O. Extending generalized Fibonacci sequences and their binet-type formula. Adv Differ Equ 2006, 023849 (2006). https://doi.org/10.1155/ADE/2006/23849

Download citation

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation