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Volume 14
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Article

Novel Expressions for Certain Generalized Leonardo Polynomials and Their Associated Numbers

by
Waleed Mohamed Abd-Elhameed
1,*,
Omar Mazen Alqubori
2,
Abdulrahim A. Alluhaybi
3 and
Amr Kamel Amin
3
1
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
2
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia
3
Department of Mathematics, Adham University College, Umm Al-Qura University, Makkah 28653, Saudi Arabia
*
Author to whom correspondence should be addressed.
Axioms 2025, 14(4), 286; https://doi.org/10.3390/axioms14040286
Submission received: 22 February 2025 / Revised: 28 March 2025 / Accepted: 9 April 2025 / Published: 11 April 2025
(This article belongs to the Special Issue Research in Special Functions)
Versions Notes

Abstract

This article introduces new polynomials that extend the standard Leonardo numbers, generalizing Fibonacci and Lucas polynomials. A new power form representation is developed for these polynomials, which is crucial for deriving further formulas. This article also presents two connection formulas linking these generalized polynomials to the Fibonacci and Lucas polynomials, as well as several identities involving some generalized and specific Leonardo numbers. Additionally, new product formulas involving the generalized Leonardo polynomials with the Fibonacci and Lucas polynomials are provided, along with computations of definite integrals based on the derived formulas.
Keywords: recurrence relations; Leonardo numbers; Fibonacci and Lucas sequences; connection formulas; Zeilberger’s algorithm recurrence relations; Leonardo numbers; Fibonacci and Lucas sequences; connection formulas; Zeilberger’s algorithm

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MDPI and ACS Style

Abd-Elhameed, W.M.; Alqubori, O.M.; Alluhaybi, A.A.; Amin, A.K. Novel Expressions for Certain Generalized Leonardo Polynomials and Their Associated Numbers. Axioms 2025, 14, 286. https://doi.org/10.3390/axioms14040286

AMA Style

Abd-Elhameed WM, Alqubori OM, Alluhaybi AA, Amin AK. Novel Expressions for Certain Generalized Leonardo Polynomials and Their Associated Numbers. Axioms. 2025; 14(4):286. https://doi.org/10.3390/axioms14040286

Chicago/Turabian Style

Abd-Elhameed, Waleed Mohamed, Omar Mazen Alqubori, Abdulrahim A. Alluhaybi, and Amr Kamel Amin. 2025. "Novel Expressions for Certain Generalized Leonardo Polynomials and Their Associated Numbers" Axioms 14, no. 4: 286. https://doi.org/10.3390/axioms14040286

APA Style

Abd-Elhameed, W. M., Alqubori, O. M., Alluhaybi, A. A., & Amin, A. K. (2025). Novel Expressions for Certain Generalized Leonardo Polynomials and Their Associated Numbers. Axioms, 14(4), 286. https://doi.org/10.3390/axioms14040286

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