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AIMS Mathematics

2025, Volume 10, Issue 2: 2930-2957. doi: 10.3934/math.2025136
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Research article

New expressions for certain polynomials combining Fibonacci and Lucas polynomials

  • 1.
    Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt; Email: waleed@cu.edu.eg
  • 2.
    Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabia; Email: Ommohamad3@uj.edu.sa
  • Received: 22 November 2024 Revised: 31 December 2024 Accepted: 05 February 2025 Published: 17 February 2025

  • MSC : 11B39, 11B83, 33C45

  • We establish a new sequence of polynomials that combines the Fibonacci and Lucas polynomials. We will refer to these polynomials as merged Fibonacci-Lucas polynomials (MFLPs). We will show that we can represent these polynomials by combining two certain Fibonacci polynomials. This formula will be essential for determining the power form representation of these polynomials. This representation and its inversion formula for these polynomials are crucial to derive new formulas about the MFLPs. New derivative expressions for these polynomials are given as combinations of several symmetric and non-symmetric polynomials. We also provide the inverse formulas for these formulas. Some new product formulas involving the MFLPs have also been derived. We also provide some definite integral formulas that apply to the derived formulas.

    Citation: Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori. New expressions for certain polynomials combining Fibonacci and Lucas polynomials[J]. AIMS Mathematics, 2025, 10(2): 2930-2957. doi: 10.3934/math.2025136

    Related Papers:

  • We establish a new sequence of polynomials that combines the Fibonacci and Lucas polynomials. We will refer to these polynomials as merged Fibonacci-Lucas polynomials (MFLPs). We will show that we can represent these polynomials by combining two certain Fibonacci polynomials. This formula will be essential for determining the power form representation of these polynomials. This representation and its inversion formula for these polynomials are crucial to derive new formulas about the MFLPs. New derivative expressions for these polynomials are given as combinations of several symmetric and non-symmetric polynomials. We also provide the inverse formulas for these formulas. Some new product formulas involving the MFLPs have also been derived. We also provide some definite integral formulas that apply to the derived formulas.



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