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Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2021

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Volume 2021 |Article ID 9967855 | https://doi.org/10.1155/2021/9967855

Muajebah Hidan, Mohamed Akel, Salah Mahmoud Boulaaras, Mohamed Abdalla, "On Behavior Laplace Integral Operators with Generalized Bessel Matrix Polynomials and Related Functions", Journal of Function Spaces, vol. 2021, Article ID 9967855, 10 pages, 2021. https://doi.org/10.1155/2021/9967855

On Behavior Laplace Integral Operators with Generalized Bessel Matrix Polynomials and Related Functions

Academic Editor: Xinguang Zhang
Received10 Mar 2021
Revised12 Apr 2021
Accepted20 Apr 2021
Published30 Apr 2021

Abstract

Recently, the applications and importance of integral transforms (or operators) with special functions and polynomials have received more attention in various fields like fractional analysis, survival analysis, physics, statistics, and engendering. In this article, we aim to introduce a number of Laplace and inverse Laplace integral transforms of functions involving the generalized and reverse generalized Bessel matrix polynomials. In addition, the current outcomes are yielded to more outcomes in the modern theory of integral transforms.

1. Introduction

Recently, the integral transforms (or operators) have been extensively used tools in solving certain boundary value problems and certain integral equations. They are also useful in evaluating infinite integrals involving special functions or in solving differential equations of mathematical physics (see, e.g., [16] and the references cited therein). Laplace transform is a type of the integral transforms that is the most popular and widely used in several branches of astronomy, engineering, applied statistics, probability distributions, and applied mathematics (see, for instance, [713]).

A number of studies on the generalizations of Laplace transform associated with special polynomials have been contributed by Ortigueira and Machado [14], Jarad and Abdeljawad [15], Ganie and Jain [16], and Saifa et al. [17].

In 1949, Krall and Frink [18] introduced and discussed several properties of the generalized Bessel polynomials (GBPs), which are given by

These polynomials, which seem to have been considered first by Bochner [19], are also mentioned in Romanovsky [20] and Krall [21].

Recently, these polynomials have been investigated in diverse ways and turned out to be applicable in a number of research fields (see, to exemplify, [2225]).

Additionally, various extensions of the classical orthogonal polynomials to matrix setting were investigated. The matrix generalization of the generalized Bessel polynomials , , for parameters (square) matrices and , was also introduced in diverse ways ([26]; see also [27]). Various studies of the generalized Bessel matrix polynomials have been presented and discussed (see [27, 28]).

Recently, many works established Laplace integral transforms of special functions like Gauss’s and Kummer’s functions [29], generalized hypergeometric functions [30, 31], Aleph-Functions [32], and Bessel functions [33]. Whereas, some formulas corresponding to integral transforms of orthogonal matrix polynomials are little known and traceless in the literature. This motivates us to discuss Laplace integral transforms for functions involving generalized Bessel matrix polynomials. In particular, we obtain a number of useful Laplace and inverse Laplace type integrals of the generalized Bessel matrix polynomials together with ceratin elementary matrix functions, exponential function, logarithmic function, generalized hypergeometric matrix functions, and Bessel functions and products of generalized Bessel matrix polynomials. We also discuss some interesting and special cases of our main results.

2. Preliminaries

Here, we state some basic definitions and preliminaries which will be used in the article (see, for details, [3436]).

Here and in the following sections, and denote the sets of complex numbers and positive integers, respectively, and We denote by the space of complex matrices endowed with classical norm defined by

This norm satisfies the inequality where and are in

Definition 1. For any matrix in , the spectrum is the set of all eigenvalues of for which we denote where refers to the spectral abscissa of and for which . A matrix is said to be positive stable if and only if .

Definition 2 (see [35, 36]). If and , then the matrix exponential is given to be where is the identity matrix in

Definition 3 (see [37]). Let be a positive stable matrix in with is invertible for all integers , the Gamma matrix function and the Digamma matrix function are defined, respectively, as follows: where and are reciprocal and derivative of the Gamma matrix function.

Note that the scalar Gamma and Digamma functions are easily found when in (5) and (6), respectively (see, e.g., [38, Section 1.1])).

Definition 4 (see [?]). For all in we assume and the Pochhammer symbol (the shifted factorial) is defined by

Lemma 5 (see [34]). Let be a matrix in such that and Then, exists, and we have

Definition 6 (see [39]). Let and be finite positive integers, the generalized hypergeometric matrix function is given by where and are commutative matrices in with are invertible for all integers and . In [39], Abdalla discussed regions of convergence of (2.6).

Note that for in (10), we have the Binomial type matrix function [39] as follows:

Also, for in (10), we get the hypergeometric matrix function (cf. [40]).

Further, the substitution in (10) leads to the classical generalized hypergeometric functions [38, Section 1.5], see also, [41].

Definition 7 (see [26]). Let and be commuting matrices in such that is an invertible matrix. For any natural number , the generalized Bessel matrix polynomial is defined as

In addition, the reverse generalized Bessel matrix polynomial is given by (see [27])

Obviously, the generalized Bessel matrix polynomial when is easily found to be the scalar generalized Bessel polynomials (1.1).

Definition 8. Let be a function of specified for . Then, the Laplace transform of is defined by provided that the improper integral exists, is the kernel of the transformation and the function is called the inverse Laplace transform of (see [1, Chapter 3]; see also [7]).

The following Lemma, which may be easily derivable from (14), will be desired in the sequel.

Lemma 9. Let be a positive stable and invertible matrix in and . Then, we have where is the incomplete Gamma matrix function [42].

3. Laplace Type Integrals of Functions Involving and

In this section, we investigate several Laplace-type transforms of functions involving generalized and reverse generalized Bessel matrix polynomials asserted in the following theorems:

Theorem 10. Let , , , and . Also, let and be matrices in such that and are invertible for all For the function we have

Proof. From the expansion series of the in (12) and upon using (15) in Lemma 9, we obtain Thus, we get the required result (19).

Theorem 11. Let , , , and . Also, let and be matrices in such that are invertible for all and satisfies the spectral condition (7). Further, let Then,

Proof. Starting from Definition 7, and applying the relation (15), it follows that Thus, the result (22) is established.

Theorem 12. Let , , , and . Also, let and be matrices in such that are invertible for all and satisfies the spectral condition (7). If Then,

Proof. For convenience, let the left-hand side of (25) be denoted by and by invoking the series expression of (12) to , we obtain therefore, (25) as desired.

Theorem 13. Let , , , and . Also, let and be matrices in such that and are invertible for all For the function we have where is the incomplete Gamma matrix function defined in [42].

Proof. To prove (28), we consider According to (16) in Lemma 9, we get This completes the proof of Theorem 13.

Theorem 14. Let , , , and . Also let and be matrices in such that are invertible for all and satisfies the spectral condition (7). Further, let Then,

Proof. To prove (32), we require the relation (15) and Definition 7, thus we arrive at This completes the proof of Theorem 14.

Theorem 15. Let , , , and . Also, let and be matrices in such that are invertible for all and satisfies the spectral condition (7). Further, let Then,

Proof. Applying the following formula (see [39]) We thus find that Making use of (15), we observe that Thus, after a simplification, we get the required result (35).

Theorem 16. Let , , , and . Also, let and be matrices in such that and are invertible for all For the function then, we have where is the Digamma matrix function defined in (6).

Proof. The proof of this Theorem is quite straight forward as Upon using (2,2), we have Hence, We thus arrive at Therefore, we get We thus have From the above equations, we get the required result as follows:

Theorem 17. Let , , , and . Also, let and be matrices in such that and are invertible for all Further, let Then, where is the generalized hypergeometric type matrix functions defined in (10) such that if and if .

Proof. Using Definitions (10) and (12) and upon using (15), we obtain Thus, after a simplification, we obtain the result (49) in Theorem 3.11.

Theorem 18. Let , , , and . Also, let be matrix in such that and are invertible for all For the function Then, we have where is the Bessel function of the first kind of order defined by (see, e.g., [38, 41, 43])

Proof. According to (12) and (53) and upon sing (15), it follows that This completes the proof of Theorem 18.

4. Inverse Laplace Type Integrals of Functions Involving

Here, we obtain the following inverse Laplace type transforms of generalized Bessel matrix polynomials with products of some functions in the following theorem:

Theorem 19. Let , , , and . Also, let be matrix in such that If Then,

Proof. It is sufficient to find Laplace transform of This finalizes the proof of Theorem 19.

Theorem 20. Let , , , and . Also, let be matrix in such that Further, let Then,

Proof. By invoking to (15) and (53), we consider Putting we obtain