IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v44y2011i1p145-152.html
   My bibliography  Save this article

Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients

Author

Listed:
  • Fiorenza, Alberto
  • Vincenzi, Giovanni

Abstract

For complex linear homogeneous recursive sequences with constant coefficients we find a necessary and sufficient condition for the existence of the limit of the ratio of consecutive terms. The result can be applied even if the characteristic polynomial has not necessarily roots with modulus pairwise distinct, as in the celebrated Poincaré’s theorem. In case of existence, we characterize the limit as a particular root of the characteristic polynomial, which depends on the initial conditions and that is not necessarily the unique root with maximum modulus and multiplicity. The result extends to a quite general context the way used to find the Golden mean as limit of ratio of consecutive terms of the classical Fibonacci sequence.

Suggested Citation

  • Fiorenza, Alberto & Vincenzi, Giovanni, 2011. "Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 44(1), pages 145-152.
    • Handle: RePEc:eee:chsofr:v:44:y:2011:i:1:p:145-152
    DOI: 10.1016/j.chaos.2011.01.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077911000087
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2011.01.003?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Andrew J. Fleming, 2002. "Plant mathematics and Fibonacci's flowers," Nature, Nature, vol. 418(6899), pages 723-723, August.
    2. Amar J. S. Klar, 2002. "Plant mathematics: Fibonacci's flowers," Nature, Nature, vol. 417(6889), pages 595-595, June.
    3. Kocer, E. Gokcen & Tuglu, Naim & Stakhov, Alexey, 2009. "On the m-extension of the Fibonacci and Lucas p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1890-1906.
    4. Büyükkılıç, F. & Demirhan, D., 2009. "Cumulative growth with fibonacci approach, golden section and physics," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 24-32.
    5. Falcón, Sergio & Plaza, Ángel, 2007. "On the Fibonacci k-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 32(5), pages 1615-1624.
    6. Falcón, Sergio & Plaza, Ángel, 2009. "On k-Fibonacci sequences and polynomials and their derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1005-1019.
    7. Akbulak, Mehmet & Bozkurt, Durmuş, 2009. "On the order-m generalized Fibonacci k-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1347-1355.
    8. Stakhov, Alexey & Rozin, Boris, 2006. "Theory of Binet formulas for Fibonacci and Lucas p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1162-1177.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Alberto Fiorenza & Giovanni Vincenzi, 2013. "From Fibonacci Sequence to the Golden Ratio," Journal of Mathematics, Hindawi, vol. 2013, pages 1-3, March.
    2. Renato Fiorenza, 2022. "Existence of the Limit of Ratios of Consecutive Terms for a Class of Linear Recurrences," Mathematics, MDPI, vol. 10(12), pages 1-8, June.
    3. Ivana Matoušová & Pavel Trojovský, 2020. "On Coding by (2, q )-Distance Fibonacci Numbers," Mathematics, MDPI, vol. 8(11), pages 1-24, November.
    4. Florek, Wojciech, 2018. "A class of generalized Tribonacci sequences applied to counting problems," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 809-821.
    5. Anatriello, Giuseppina & Németh, László & Vincenzi, Giovanni, 2022. "Generalized Pascal’s triangles and associated k-Padovan-like sequences," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 192(C), pages 278-290.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. E. Gokcen Kocer & Huriye Alsan, 2022. "Generalized Hybrid Fibonacci and Lucas p-numbers," Indian Journal of Pure and Applied Mathematics, Springer, vol. 53(4), pages 948-955, December.
    2. Nalli, Ayse & Haukkanen, Pentti, 2009. "On generalized Fibonacci and Lucas polynomials," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3179-3186.
    3. Chung-Chuan Chen & Lin-Ling Huang, 2021. "Some New Identities for the Generalized Fibonacci Polynomials by the Q(x) Matrix," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 13(2), pages 1-21, April.
    4. Florek, Wojciech, 2018. "A class of generalized Tribonacci sequences applied to counting problems," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 809-821.
    5. Falcón, Sergio & Plaza, Ángel, 2009. "On k-Fibonacci sequences and polynomials and their derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1005-1019.
    6. Esmaeili, Morteza & Esmaeili, Mostafa, 2009. "Polynomial Fibonacci–Hessenberg matrices," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2820-2827.
    7. Kocer, E. Gokcen & Tuglu, Naim & Stakhov, Alexey, 2009. "On the m-extension of the Fibonacci and Lucas p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1890-1906.
    8. Flavio Pressacco & Giacomo Plazzotta & Laura Ziani, 2014. "K-Fibonacci sequences and minimal winning quota in Parsimonious game," Working Papers hal-00950090, HAL.
    9. Kilic, E. & Stakhov, A.P., 2009. "On the Fibonacci and Lucas p-numbers, their sums, families of bipartite graphs and permanents of certain matrices," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2210-2221.
    10. W. M. Abd-Elhameed & N. A. Zeyada, 2022. "New formulas including convolution, connection and radicals formulas of k-Fibonacci and k-Lucas polynomials," Indian Journal of Pure and Applied Mathematics, Springer, vol. 53(4), pages 1006-1016, December.
    11. A. D. Godase, 2019. "Properties of k-Fibonacci and k-Lucas octonions," Indian Journal of Pure and Applied Mathematics, Springer, vol. 50(4), pages 979-998, December.
    12. Esmaeili, M. & Gulliver, T.A. & Kakhbod, A., 2009. "The Golden mean, Fibonacci matrices and partial weakly super-increasing sources," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 435-440.
    13. Flaut, Cristina & Shpakivskyi, Vitalii & Vlad, Elena, 2017. "Some remarks regarding h(x) – Fibonacci polynomials in an arbitrary algebra," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 32-35.
    14. Ilija Tanackov & Ivan Pavkov & Željko Stević, 2020. "The New New-Nacci Method for Calculating the Roots of a Univariate Polynomial and Solution of Quintic Equation in Radicals," Mathematics, MDPI, vol. 8(5), pages 1-18, May.
    15. Flaut, Cristina & Savin, Diana, 2019. "Some remarks regarding l-elements defined in algebras obtained by the Cayley–Dickson process," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 112-116.
    16. Hatir, E. & Noiri, T., 2009. "On δ–β-continuous functions," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 205-211.
    17. Natalia Bednarz, 2021. "On ( k , p )-Fibonacci Numbers," Mathematics, MDPI, vol. 9(7), pages 1-13, March.
    18. Yasemin Taşyurdu, 2019. "Generalized (p,q)-Fibonacci-Like Sequences and Their Properties," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 11(6), pages 1-43, December.
    19. Fatih Yılmaz & Aybüke Ertaş & Jiteng Jia, 2022. "On Harmonic Complex Balancing Numbers," Mathematics, MDPI, vol. 11(1), pages 1-15, December.
    20. Falcón, Sergio & Plaza, Ángel, 2009. "The metallic ratios as limits of complex valued transformations," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 1-13.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:44:y:2011:i:1:p:145-152. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.