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Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals

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  • Waleed Mohamed Abd-Elhameed

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

  • Andreas N. Philippou

    (Department of Mathematics, University of Patras, 26504 Patras, Greece)

  • Nasr Anwer Zeyada

    (Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
    Department of Mathematics, College of Science, University of Jeddah, Jeddah 23218, Saudi Arabia)

Abstract

The goal of this study is to develop some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. Hypergeometric functions of the kind 2 F 1 ( z ) are included in all connection coefficients for a specific z . Several new connection formulae between some famous polynomials, such as Fibonacci, Lucas, Pell, Fermat, Pell–Lucas, and Fermat–Lucas polynomials, are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some new formulae linking some celebrated numbers are given and also some newly closed formulae of certain definite weighted integrals are deduced. Based on using the two generalized classes of Fibonacci and Lucas polynomials, some new reduction formulae of certain odd and even radicals are developed.

Suggested Citation

  • Waleed Mohamed Abd-Elhameed & Andreas N. Philippou & Nasr Anwer Zeyada, 2022. "Novel Results for Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals," Mathematics, MDPI, vol. 10(13), pages 1-18, July.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:13:p:2342-:d:855534
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    References listed on IDEAS

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    1. Ye, Xiaoli & Zhang, Zhizheng, 2017. "A common generalization of convolved generalized Fibonacci and Lucas polynomials and its applications," Applied Mathematics and Computation, Elsevier, vol. 306(C), pages 31-37.
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    Cited by:

    1. Waleed Mohamed Abd-Elhameed & Amr Kamel Amin, 2023. "Novel Formulas of Schröder Polynomials and Their Related Numbers," Mathematics, MDPI, vol. 11(2), pages 1-23, January.
    2. Waleed Mohamed Abd-Elhameed, 2022. "Novel Formulae of Certain Generalized Jacobi Polynomials," Mathematics, MDPI, vol. 10(22), pages 1-25, November.
    3. Mei Bai & Wenchang Chu & Dongwei Guo, 2022. "Reciprocal Formulae among Pell and Lucas Polynomials," Mathematics, MDPI, vol. 10(15), pages 1-11, July.
    4. Dongwei Guo & Wenchang Chu, 2022. "Sums of Pell/Lucas Polynomials and Fibonacci/Lucas Numbers," Mathematics, MDPI, vol. 10(15), pages 1-10, July.

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