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Novel Formulae of Certain Generalized Jacobi Polynomials

Author

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  • Waleed Mohamed Abd-Elhameed

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

Abstract

The main goal of this article is to investigate theoretically a kind of orthogonal polynomials, namely, generalized Jacobi polynomials ( G J P s ). These polynomials can be expressed as certain combinations of Legendre polynomials. Some basic formulas of these polynomials such as the power form representation and inversion formula of these polynomials are first introduced, and after that, some interesting formulas concerned with these polynomials are established. The formula of the derivatives of the moments of these polynomials is developed. As special cases of this formula, the moment and high-order derivative formulas of the G J P s are deduced. New expressions for the high-order derivatives of the G J P s , but in terms of different symmetric and non-symmetric polynomials, are also established. These expressions lead to some interesting connection formulas between the G J P s and some various polynomials.

Suggested Citation

  • Waleed Mohamed Abd-Elhameed, 2022. "Novel Formulae of Certain Generalized Jacobi Polynomials," Mathematics, MDPI, vol. 10(22), pages 1-25, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:22:p:4237-:d:971191
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    References listed on IDEAS

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    1. Harendra Singh & Rajesh K. Pandey & Hari Mohan Srivastava, 2019. "Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials," Mathematics, MDPI, vol. 7(3), pages 1-24, February.
    2. Waleed Mohamed Abd-Elhameed & Badah Mohamed Badah, 2021. "New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas," Mathematics, MDPI, vol. 9(13), pages 1-28, July.
    3. E. H. Doha & W. M. Abd-Elhameed, 2012. "Efficient Solutions of Multidimensional Sixth-Order Boundary Value Problems Using Symmetric Generalized Jacobi-Galerkin Method," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-19, May.
    4. 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.
    5. Ampol Duangpan & Ratinan Boonklurb & Tawikan Treeyaprasert, 2019. "Finite Integration Method with Shifted Chebyshev Polynomials for Solving Time-Fractional Burgers’ Equations," Mathematics, MDPI, vol. 7(12), pages 1-24, December.
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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.

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