Special matrix functions: characteristics, achievements and future directions

Abdalla, Mohamed

doi:10.1080/03081087.2018.1497585

Cited by 49 publications

(60 citation statements)

References 53 publications

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“…For an arbitrary matrix A ∈ C N×N , satisfy condition (8) then the n-th Laguerre matrix polynomials L A n (z) is defined by (see [8])…”

Section: Preliminariesmentioning

confidence: 99%

“…Suppose that A is a matrix in C N×N satisfying (8) and u ∈ C. Then the Shively's matrix polynomials of two variables has the following summation formulas…”

Section: Summation Formulas and Operational Representationmentioning

confidence: 99%

“…Recently, the matrix versions of the classical families orthogonal polynomials such as Jacobi, Hermite, Chebyshev, Legendre, Gegenbauer, Bessel and Humbert polynomials of one variables and some other polynomials were introduced by many authors for matrices in C N×N and various properties satisfied by them were given from the scalar case. Rather than giving an exhaustive list of references, we refer the reader to the article [8]. Theory of generalized and multivariable orthogonal matrix polynomials has provided new means of analysis to deal with the majority of problems in mathematical physics which find broad practical applications.…”

Section: Introductionmentioning

confidence: 99%

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Two Variables Shivley’s Matrix Polynomials

Bakhet

Hidan

et al. 2019

Symmetry

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“…For an arbitrary matrix A ∈ C N×N , satisfy condition (8) then the n-th Laguerre matrix polynomials L A n (z) is defined by (see [8])…”

Section: Preliminariesmentioning

confidence: 99%

“…Suppose that A is a matrix in C N×N satisfying (8) and u ∈ C. Then the Shively's matrix polynomials of two variables has the following summation formulas…”

Section: Summation Formulas and Operational Representationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Two Variables Shivley’s Matrix Polynomials

Bakhet

Hidan

et al. 2019

Symmetry

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“…In the past few years, matrix functions and polynomials defined in terms of power matrix series [18][19][20][21] attracted the attention to new and exciting problems, specifically that involve the generalized hypergeometric matrix functions and future directions. It is worth mentioning the recent work of Shang [22] who uncovered some of the information of topological properties hidden in the smaller eigenvalues of A in matrix functions of the form f (A) = ∑ ∞ k=0 c k A k .…”

Section: Introductionmentioning

confidence: 99%

“…Recently, many authors (see, e.g., [18][19][20][21]) proposed the extension of the classical polynomials to the matrix framework. The operational matrix of fractional derivatives has been studied for various types of orthogonal polynomials, including Chebyshev polynomials [27].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2020

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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.

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On Fourier–Bessel matrix transforms and applications

Abdalla

Boulaaras

Akel

2021

Math Methods in App Sciences

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The Fourier–Bessel transform is an integral transform and is also known as the Hankel transform. This transform is a very important tool in solving many problems in mathematical sciences, physics, and engineering. Very recently, Abdalla (AIMS Mathematics 6: [2021], 6122–6139) introduced certain Hankel integral transforms associated with functions involving generalized Bessel matrix polynomials and various applications. Motivated by this work, we introduce the Fourier–Bessel matrix transform (FBMT) containing Bessel matrix function of the first kind as a kernel. The corresponding inversion formula and several illustrative examples of this transform have been presented. A relation between the Laplace transform and the FBMT has been obtained. Furthermore, a convolution of the FBMT is constructed with some properties. In addition, some applications are proposed in the present research. Finally, we outlined the significant links for the preceding outcomes of some particular cases with our results.

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Special matrix functions: characteristics, achievements and future directions

Cited by 49 publications

References 53 publications

Two Variables Shivley’s Matrix Polynomials

Two Variables Shivley’s Matrix Polynomials

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

On Fourier–Bessel matrix transforms and applications

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