New Results for Certain Jacobsthal-Type Polynomials

Waleed Mohamed Abd-Elhameed (), Omar Mazen Alqubori and Amr Kamel Amin
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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
Amr Kamel Amin: Department of Mathematics, Adham University College, Umm AL-Qura University, Makkah 28653, Saudi Arabia

Mathematics, 2025, vol. 13, issue 5, 1-27

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.

Keywords: Jacobsthal polynomials; orthogonal polynomials; analytic and inversion formulas; linearization and connection coefficients; moment formulas; definite integrals (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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