Fractional order of Legendre-type matrix polynomials

M Zayed,M Abdalla,M Hidan,M Abul-Ez

doi:10.1186/s13662-020-02975-5

Abstract

Recently, special functions of fractional order calculus have had many applications in various areas of mathematical analysis, physics, probability theory, optimization theory, graph theory, control systems, earth sciences, and engineering. Very recently, Zayed et al. (Mathematics 8:136, 2020) introduced the shifted Legendre-type matrix polynomials of arbitrary fractional orders and their various applications utilizing Rodrigues matrix formulas. In this line of research, we use the fractional order of Rodrigues formula to provide further investigation on such Legendre polynomials from a different point of view. Some properties, such as hypergeometric representations, continuation properties, recurrence relations, and differential equations, are derived. Moreover, Laplace’s first integral form and orthogonality are obtained.

Highlights

The recent advances in fractional order calculus (FOC) are dominated by its multidisciplinary applications
The “special functions of fractional order calculus” (SF of FOC) as generalized fractional calculus operators of some classical special functions were found by Kiryakova [8, 9] and Agarwal [2]
Analogous to the classical case, it is noticed that Rodrigues matrix formula is a useful approach to define a sequence of orthogonal matrix polynomials

Summary

Introduction

The recent advances in fractional order calculus (FOC) are dominated by its multidisciplinary applications. Zayed et al Advances in Difference Equations (2020) 2020:506 lishing many interesting properties of the matrix polynomials These generalized matrix formulas allow to define new classes of special matrix functions and matrix polynomials and to include fractional order differentiation. 3, we define the Legendre-type matrix polynomials of arbitrary fractional orders. 4. Laplace’s first integral form and the orthogonality of Legendre-type matrix polynomials of arbitrary fractional orders are proved in Sect. Definition 2.6 ([1, 17]) Let D be a positive stable matrix in CN×N and μ ∈ C such that Re(μ) > 0, the Riemann–Liouville fractional integral of order μ is defined as follows: Iμ ζ D = 1 ζ (ζ – z)μ–1ζ D dz, D ∈ CN×N ,. For α ∈ (–1, 0), the Legendre-type matrix functions of order α are defined by

Recurrence relations and matrix differential equations

Laplace’s first integral form

Orthogonality property

Conclusion