IDEAS home Printed from https://ideas.repec.org/a/spr/indpam/v53y2022i4d10.1007_s13226-021-00214-5.html
   My bibliography  Save this article

New formulas including convolution, connection and radicals formulas of k-Fibonacci and k-Lucas polynomials

Author

Listed:
  • W. M. Abd-Elhameed

    (Cairo University)

  • N. A. Zeyada

    (Cairo University
    University of Jeddah)

Abstract

This paper is dedicated to deriving some new formulas of the two classes of polynomials, namely k-Fibonacci and k-Lucas polynomials. New connection formulas between these two classes are developed. We show that the connection coefficients are expressed explicitly in terms of hypergeometric functions of the type $$_2F_{1}(z)$$ 2 F 1 ( z ) , for certain z. Some linearization formulas of the two classes of polynomials are also given. New convolution identities involving the derivatives of the two classes of k-Fibonacci and k-Lucas polynomials are derived. As an important application of employing the two classes of polynomials, reduction formulas of some odd radicals are established.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:indpam:v:53:y:2022:i:4:d:10.1007_s13226-021-00214-5
    DOI: 10.1007/s13226-021-00214-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s13226-021-00214-5
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s13226-021-00214-5?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. Falcón, Sergio & Plaza, Ángel, 2007. "On the Fibonacci k-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 32(5), pages 1615-1624.
    2. W. M. Abd-Elhameed & N. A. Zeyada, 2018. "New identities involving generalized Fibonacci and generalized Lucas numbers," Indian Journal of Pure and Applied Mathematics, Springer, vol. 49(3), pages 527-537, September.
    3. Falcón, Sergio & Plaza, Ángel, 2009. "On k-Fibonacci sequences and polynomials and their derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1005-1019.
    4. Ye, Xiaoli & Zhang, Zhizheng, 2017. "A common generalization of convolved generalized Fibonacci and Lucas polynomials and its applications," Applied Mathematics and Computation, Elsevier, vol. 306(C), pages 31-37.
    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. Waleed Mohamed Abd-Elhameed & Amr Kamel Amin, 2023. "Novel Formulas of Schröder Polynomials and Their Related Numbers," Mathematics, MDPI, vol. 11(2), pages 1-23, January.

    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. 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.
    2. 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.
    3. Nalli, Ayse & Haukkanen, Pentti, 2009. "On generalized Fibonacci and Lucas polynomials," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3179-3186.
    4. 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.
    5. 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.
    6. 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.
    7. Natalia Bednarz, 2021. "On ( k , p )-Fibonacci Numbers," Mathematics, MDPI, vol. 9(7), pages 1-13, March.
    8. 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.
    9. Fatih Yılmaz & Aybüke Ertaş & Jiteng Jia, 2022. "On Harmonic Complex Balancing Numbers," Mathematics, MDPI, vol. 11(1), pages 1-15, December.
    10. Falcón, Sergio & Plaza, Ángel, 2009. "The metallic ratios as limits of complex valued transformations," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 1-13.
    11. Falcon, Sergio, 2016. "The k–Fibonacci difference sequences," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 153-157.
    12. Falcón, Sergio & Plaza, Ángel, 2009. "On k-Fibonacci sequences and polynomials and their derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1005-1019.
    13. Yang, Jizhen & Zhang, Zhizheng, 2018. "Some identities of the generalized Fibonacci and Lucas sequences," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 451-458.
    14. 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.
    15. Postavaru, Octavian, 2023. "An efficient numerical method based on Fibonacci polynomials to solve fractional differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 212(C), pages 406-422.
    16. Waleed Mohamed Abd-Elhameed & Andreas N. Philippou & Nasr Anwer Zeyada, 2022. "Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals," Mathematics, MDPI, vol. 10(13), pages 1-18, July.
    17. Esmaeili, Morteza & Esmaeili, Mostafa, 2009. "Polynomial Fibonacci–Hessenberg matrices," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2820-2827.
    18. Ye, Xiaoli & Zhang, Zhizheng, 2017. "A common generalization of convolved generalized Fibonacci and Lucas polynomials and its applications," Applied Mathematics and Computation, Elsevier, vol. 306(C), pages 31-37.
    19. Yuankui Ma & Wenpeng Zhang, 2018. "Some Identities Involving Fibonacci Polynomials and Fibonacci Numbers," Mathematics, MDPI, vol. 6(12), pages 1-8, December.
    20. 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.

    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:spr:indpam:v:53:y:2022:i:4:d:10.1007_s13226-021-00214-5. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.