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New Results for Certain Jacobsthal-Type Polynomials

Author

Listed:
  • Waleed Mohamed Abd-Elhameed

    (Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
    Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia)

  • Omar Mazen Alqubori

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

  • Amr Kamel Amin

    (Department of Mathematics, Adham University College, Umm AL-Qura University, Makkah 28653, Saudi Arabia
    Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef 62514, Egypt)

Abstract

This paper investigates a class of Jacobsthal-type polynomials (JTPs) that involves one parameter. We present several new formulas for these polynomials, including expressions for their derivatives, moments, and linearization formulas. The key idea behind the derivation of these formulas is based on developing a new connection formula that expresses the shifted Chebyshev polynomials of the third kind in terms of the JTPs. This connection formula is used to deduce a new inversion formula of the JTPs. Therefore, by utilizing the power form representation of these polynomials and their corresponding inversion formula, we can derive additional expressions for them. Additionally, we compute some definite integrals based on some formulas of these polynomials.

Suggested Citation

  • Waleed Mohamed Abd-Elhameed & Omar Mazen Alqubori & Amr Kamel Amin, 2025. "New Results for Certain Jacobsthal-Type Polynomials," Mathematics, MDPI, vol. 13(5), pages 1-27, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:5:p:715-:d:1597541
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    References listed on IDEAS

    as
    1. Tereszkiewicz, Agnieszka & Wawreniuk, Izabela, 2015. "Generalized Jacobsthal polynomials and special points for them," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 806-814.
    2. Nalli, Ayse & Haukkanen, Pentti, 2009. "On generalized Fibonacci and Lucas polynomials," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3179-3186.
    3. Waleed Mohamed Abd-Elhameed & Omar Mazen Alqubori & Anna Napoli, 2024. "On Convolved Fibonacci Polynomials," Mathematics, MDPI, vol. 13(1), pages 1-25, December.
    Full references (including those not matched with items on IDEAS)

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