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On Convolved Fibonacci Polynomials

Author

Listed:
  • Waleed Mohamed Abd-Elhameed

    (Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt)

  • Omar Mazen Alqubori

    (Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia)

  • Anna Napoli

    (Department of Mathematics and Computer Science, University of Calabria, 87036 Rende, Italy)

Abstract

This work delves deeply into convolved Fibonacci polynomials (CFPs) that are considered generalizations of the standard Fibonacci polynomials. We present new formulas for these polynomials. An expression for the repeated integrals of the CFPs in terms of their original polynomials is given. A new approach is followed to obtain the higher-order derivatives of these polynomials from the repeated integrals formula. The inversion and moment formulas for these polynomials, which we find, are the keys to developing further formulas for these polynomials. The derivatives of the moments of the CFPs in terms of their original polynomials and different symmetric and non-symmetric polynomials are also derived. New product formulas of these polynomials with some polynomials, including the linearization formulas of these polynomials, are also deduced. Some closed forms for definite and weighted definite integrals involving the CFPs are found as consequences of some of the introduced formulas.

Suggested Citation

  • Waleed Mohamed Abd-Elhameed & Omar Mazen Alqubori & Anna Napoli, 2024. "On Convolved Fibonacci Polynomials," Mathematics, MDPI, vol. 13(1), pages 1-25, December.
    • Handle: RePEc:gam:jmathe:v:13:y:2024:i:1:p:22-:d:1553143
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    References listed on IDEAS

    as
    1. Zakariae Cheddour & Abdelhakim Chillali & Ali Mouhib, 2023. "Generalized Fibonacci Sequences for Elliptic Curve Cryptography," Mathematics, MDPI, vol. 11(22), pages 1-17, November.
    2. Clemente Cesarano & William Ramírez & Stiven Díaz & Adnan Shamaoon & Waseem Ahmad Khan, 2023. "On Apostol-Type Hermite Degenerated Polynomials," Mathematics, MDPI, vol. 11(8), pages 1-13, April.
    3. 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.
    4. Dae San Kim & Dmitry V. Dolgy & Dojin Kim & Taekyun Kim, 2019. "Representing by Orthogonal Polynomials for Sums of Finite Products of Fubini Polynomials," Mathematics, MDPI, vol. 7(4), pages 1-16, March.
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