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Extending generalized Fibonacci sequences and their binet-type formula

Advances in Difference Equations volume 2006, Article number: 023849 (2006) Cite this article

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.

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References

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Section de Mathématique, LEGT - F. Arago, Académie de Reims, 1, rue F. Arago, Reims, 51100, France

    Mustapha Rachidi

  2. Faculty of Mathematics, Kyushu University, Hakozaki, Fukuoka, 812-8581, Japan

    Osamu Saeki

Authors
  1. Mustapha Rachidi
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  2. Osamu Saeki
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Correspondence to Mustapha Rachidi.

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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.

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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

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