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On the Fractional Order Rodrigues Formula for the Shifted Legendre-Type Matrix Polynomials

Author

Listed:
  • Mohra Zayed

    (Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia
    These authors contributed equally to this work.)

  • Mahmoud Abul-Ez

    (Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt
    These authors contributed equally to this work.)

  • Mohamed Abdalla

    (Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia
    Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt
    These authors contributed equally to this work.)

  • Nasser Saad

    (School of Mathematical and Computational Sciences, University of Prince Edward Island, Charlottetown, PE C1A 4P3, Canada
    These authors contributed equally to this work.)

Abstract

The generalization of Rodrigues’ formula for orthogonal matrix polynomials has attracted the attention of many researchers. This generalization provides new integral and differential representations in addition to new mathematical results that are useful in theoretical and numerical computations. Using a recently studied operational matrix for shifted Legendre polynomials with the variable coefficients fractional differential equations, the present work introduces the shifted Legendre-type matrix polynomials of arbitrary (fractional) orders utilizing some Rodrigues matrix formulas. Many interesting mathematical properties of these matrix polynomials are investigated and reported in this paper, including recurrence relations, differential properties, hypergeometric function representation, and integral representation. Furthermore, the orthogonality property of these polynomials is examined in some particular cases. The developed results provide a matrix framework that generalizes and enhances the corresponding scalar version and introduces some new properties with proposed applications. Some of these applications are explored in the present work.

Suggested Citation

  • Mohra Zayed & Mahmoud Abul-Ez & Mohamed Abdalla & Nasser Saad, 2020. "On the Fractional Order Rodrigues Formula for the Shifted Legendre-Type Matrix Polynomials," Mathematics, MDPI, vol. 8(1), pages 1-23, January.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:136-:d:310169
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    References listed on IDEAS

    as
    1. J. A. Tenreiro Machado & Manuel F. Silva & Ramiro S. Barbosa & Isabel S. Jesus & Cecília M. Reis & Maria G. Marcos & Alexandra F. Galhano, 2010. "Some Applications of Fractional Calculus in Engineering," Mathematical Problems in Engineering, Hindawi, vol. 2010, pages 1-34, November.
    2. Edmundo Capelas de Oliveira & José António Tenreiro Machado, 2014. "A Review of Definitions for Fractional Derivatives and Integral," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-6, June.
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