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

Authors: Muajebah Hidan; Mohamed Akel; Salah Mahmoud Boulaaras; Mohamed Abdalla
Journal: Journal of Function Spaces
 (2021)

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## 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.


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## Body

## 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., [1–6] 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, [7–13]).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
(1)Ynα,β;ξ=∑s=0nnsn+α−1sξβs.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, [22–25]).Additionally, various extensions of the classical orthogonal polynomials to matrix setting were investigated. The matrix generalization of the generalized Bessel polynomialsBnθ,ϕz, z∈ℂ, for parameters (square) matrices θ and ϕ, 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, [34–36]).Here and in the following sections,C and N denote the sets of complex numbers and positive integers, respectively, and N0=N∪0. We denote by Mrℂ the space of r×r complex matrices endowed with classical norm defined by
(2)∥θ∥=supy≠0∥θy∥∥y∥=sup∥θy∥,∥y∥=1.This norm satisfies the inequality∥θϕ∥≤∥θ∥∥ϕ∥, where θ and ϕ are in Mrℂ.Definition 1.
For any matrixθ in Mrℂ, the spectrum σθ is the set of all eigenvalues of θ for which we denote
(3)αθ=maxRη:η∈σθandβθ=minRη:η∈σθ,where αθ refers to the spectral abscissa of θ and for which βθ=−α−θ. A matrix θ∈Mrℂ is said to be positive stable if and only if βθ>0.Definition 2 (see [35, 36]).
Ifθ∈Mrℂ, and w∈C, then the matrix exponential eθw is given to be
(4)eθw=I+θw+⋯+θnn!wn+⋯,where I is the identity matrix in Mrℂ.Definition 3 (see [37]).
Letθ be a positive stable matrix in Mrℂ with θ+nI is invertible for all integers n∈N0, the Gamma matrix function Γθ and the Digamma matrix function ψθ are defined, respectively, as follows:
(5)Γθ=∫0∞e−uuθ−Idu;uθ−I=expθ−Ilnu.(6)ψθ=Γ−1θΓ′θ,where Γ−1θ and Γ′θ are reciprocal and derivative of the Gamma matrix function.Note that the scalar Gamma and Digamma functions are easily found whenr=1 in (5) and (6), respectively (see, e.g., [38, Section 1.1])).Definition 4 (see [?]).
For allθ in Mrℂ, we assume
(7)θ+kIisinvertibleforallk∈ℕ0,and the Pochhammer symbol (the shifted factorial) is defined by
(8)θr=θθ+I⋯θ+r−1I=Γ−1θΓθ+rI,r∈ℕ,I,r=0.Lemma 5 (see [34]).
Letθ be a matrix in Mrℂ such that ∥θ∥<1 and ∥I∥=1. Then, I+θ−1 exists, and we have
(9)I+θ−1=I−θ+θ2−θ3+θ4−θ5+⋯.Definition 6 (see [39]).
Letm and n be finite positive integers, the generalized hypergeometric matrix function is given by
(10)mFnθ;ϕ;z=∑k=0∞∏i=1mθik∏j=1nφjk−1zkk!,where θi,1≤i≤m and ϕj,1≤j≤n are commutative matrices in Mrℂ with ϕj+kI are invertible for all integers k∈N0 and 1≤i≤m. In [39], Abdalla discussed regions of convergence of (2.6).Note that form=1,n=0 in (10), we have the Binomial type matrix function 1F0θ;−;z [39] as follows:
(11)F10θ;−;z=1−z−θ=I+θz+θθ+Iz22!+⋯+θnznn!+⋯,z<1.Also, form=2,n=1 in (10), we get the hypergeometric matrix function 2F1 (cf. [40]).Further, the substitutionr=1 in (10) leads to the classical generalized hypergeometric functions [38, Section 1.5], see also, [41].Definition 7 (see [26]).
Letθ and ϕ be commuting matrices in Mrℂ such that ϕ is an invertible matrix. For any natural number n≥0, the nth generalized Bessel matrix polynomial Bnθ,θz is defined as
(12)Bnθ,ϕz=∑r=0nnrθ+n−1Irzϕ−1r=∑r=0n−1rr!−nIrθ+n−1Irzϕ−1r=2F0−nI,θ+n−1I;−;−zϕ−1.In addition, thenth reverse generalized Bessel matrix polynomial Θnθ,ϕz is given by (see [27])
(13)Θnθ,ϕz=znBnθ,ϕz−1=−1nΓ−1−θ−2n−2IΓ−θ+n−2I×1F1−nI;−θ−2n−2I;ϕz.Θnθ,ϕz=znBnθ,ϕz−1=−1nΓ−1−θ−2n−2IΓ−θ+n−2I×1F1−nI;−θ−2n−2I;ϕz.Obviously, thenth generalized Bessel matrix polynomial Bnθ,ϕz when r=1 is easily found to be the scalar generalized Bessel polynomials (1.1).Definition 8.
Letgτ be a function of τ specified for τ>0. Then, the Laplace transform of gτ is defined by
(14)Gλ=Lgτ:λ=∫0∞e−λτgτdτ,Rλ>0,provided that the improper integral exists, e−λu is the kernel of the transformation and the function gτ is called the inverse Laplace transform of Gλ (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.
Letθ be a positive stable and invertible matrix in Mrℂ and Rλ>0. Then, we have
(15)Lτθ:λ=∫0∞e−λττθdτ=λ−θ+IΓθ+I,(16)Lτθτ+1−1:λ=Γθ+IeλΓ−θ,λ,where Γθ,λ is the incomplete Gamma matrix function [42].
(17)Lgτeθτ:λ=GλI−θ,L−1λ−θ:τ=τθ−IΓ−1θ.


## 3. Laplace Type Integrals of Functions InvolvingBnθ,ϕz and Θnθ.ϕz

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.
Letz,λ∈ℂ, Rλ>0, n∈ℕ0, and r∈ℕ. Also, let θ,ϕ and A be matrices in Mrℂ such that βA>0 and ϕ+kI are invertible for all k∈ℕ0. For the function
(18)g1z=zA−IBnθ,ϕz,we have
(19)G1λ=Lg1z:λ=λ−AΓA3F0−nI,θ+n−1I,A−;−λϕ−1.Proof.
From the expansion series of theBnθ,ϕz in (12) and upon using (15) in Lemma 9, we obtain
(20)G1λ=∑s=0n−nIsθ+n−1Is−ϕ−1ss!LzA+s−1I=∑s=0n−nIsθ+n−1Is−ϕ−1ss!λ−A+sIΓA+sI=λ−AΓA∑s=0n−nIsθ+n−1IsAs−λϕ−1ss!.
Thus, we get the required result (19).Theorem 11.
Letz,λ∈ℂ, Rλ>0, n∈ℕ0, and r∈ℕ. Also, let θ,ϕ and A be matrices in Mrℂ such that βA>0,ϕ+kI are invertible for all k∈ℕ0 and I−A satisfies the spectral condition (7). Further, let
(21)g2z=zA−n+1IΘnθ,ϕ;z.
Then,(22)G2λ=Lg2z:λ=λ−AΓA2F1−nI,θ+n−1II−A;λθ−1.Proof.
Starting from Definition7, and applying the relation (15), it follows that
(23)G2λ=∑s=0n−nIsθ+n−1Is−ϕ−1ss!LzA−s+1I=∑s=0n−nIsθ+n−1Is−ϕ−1ss!λ−A−sIΓA−sI=λ−AΓA∑s=0n−nIsθ+n−1IsI−As−1λϕ−1ss!.
Thus, the result (22) is established.Theorem 12.
Letz,μ,λ∈ℂ, Rλ−μ>0, n∈ℕ0, and r∈ℕ. Also, let θ,ϕ and A be matrices in Mrℂ such that βA>0,ϕ+kI are invertible for all k∈ℕ0 and I−A satisfies the spectral condition (7). If
(24)g3z=zA−IeμzBnθ,ϕz−1.
Then,(25)G3λ=Lg3z:λ=λ−μ−AΓA2F1−nI,θ+n−1II−A;λ−μϕ−1.Proof.
For convenience, let the left-hand side of (25) be denoted by S and by invoking the series expression of (12) to S, we obtain
(26)S=∑k=0n−nIsθ+n−1Ik−ϕ−1ss!∫0∞zA−s−1Ie−−μ+λzdz=∑s=0n−nIsθ+n−1Is−ϕ−1ss!−μ+λ−A−sIΓA−sI=ΓAλ−μ−A∑s=0n−nIsθ+n−1IsI−As−1λ−μϕ−1ss!,therefore, (25) as desired.Theorem 13.
Letz,w,λ∈ℂ, Rλ>0, n∈ℕ0, and r∈ℕ. Also, let θ,ϕ and A be matrices in Mrℂ such that βA>0 and ϕ+kI are invertible for all k∈ℕ0. For the function
(27)g4z=zA−Iz+w−1Bnθ,ϕz,we have
(28)G4λ=Lg4z:λ=wA−IΓAeλw×∑k=0n−nIkθ+n−1IkAkk!ΓI−A−kI;λw−wϕ−1k,where ΓA,z is the incomplete Gamma matrix function defined in [42].Proof.
To prove (28), we consider
(29)G4λ=∫0∞zA−Iz+w−1Bnθ,ϕze−λzdz=∑k=0n−nIkθ+n−1Ikk!−ϕ−1k×∫0∞zA+k−1Iz+w−1e−λzdz.
According to (16) in Lemma 9, we get
(30)G4λ=∑k=0n−nIkθ+n−1Ikk!ΓA+kI×wA+k−1IeλwΓ1−kI−A,wλ−ϕ−1k=ΓAwA−Iewλ∑k=0n−nIkθ+n−1IkAkk!×Γ1−kI−A,wλ−wϕ−1k.
This completes the proof of Theorem13.Theorem 14.
Letz,λ,ν∈ℂ, Rλ>0, Rν>0,n,m∈ℕ0, and r∈ℕ. Also let θ,ϕ and A be matrices in Mrℂ such that βA>0,ϕ+kI are invertible for all k∈ℕ0,1+nI−A and 2−nI−A−θ satisfies the spectral condition (7). Further, let
(31)g5z=zA−IBnθ,λIz−1BmνI,ϕz−1.
Then,(32)G5λ=g5z:λ=λ−AΓAΓI−AΓ2I−A−θ×Γ−11+nI−AΓ−12−nI−A−θ×3F2−mI,ν+n−1I,2I−A−θ1+nI−A,2−nI−A−θ;λϕ−1.Proof.
To prove (32), we require the relation (15) and Definition 7, thus we arrive at
(33)G5λ=∑s=0n∑j=0m−nIsθ+n−1Is−λ−1ss!×−mIjνI+m−1Ij−ϕ−1jj!LzA−s+j+1I=∑s=0n∑j=0m−nIsϕ+n−1Is−λ−1ss!×−mIjνI+m−1Ij−ϕ−1rr!ΓA−s+jIλ−A−s+jI=λ−AΓA∑j=0m−mIjνI+m−1Ij−ϕ−1jj!I−Aj−1×∑s=0n−nIsϕ+n−1Iss!1−jI−As−1=λ−AΓAΓI−AΓ2I−A−θΓ−1I−θ+nIΓ−12I−A−θ−nI×∑j=0m−mIjνI+m−1Ij−λϕ−1jj!2I−A−θj×1+nI−Ar−12−nI−A−θj−1.
This completes the proof of Theorem14.Theorem 15.
Letz,λ∈ℂ, Rλ>0, n,m∈ℕ0, and r∈ℕ. Also, let θ,ϑ,ϕ and A be matrices in Mrℂ such that βA>0,ϕ+kI, are invertible for all k∈ℕ0,ϑ,θ+A and θ+A−I satisfies the spectral condition (7). Further, let
(34)g6z=zA−IBnθ,λzI1Bmϑ,ϕzBmϑ,ϕ−z.
Then,(35)G6λ=Lg6z:λ=2A−Iπθ+A−InΓAλ−AI−An−1×8F3−mI,ϑ+m−1I,12θ−I,12ϑ,12A+1−nI,12A−nI,12θ+A+nI,12θ+A+n−1IϑI,12θ+A,12θ+A−I;16λϕ−2.Proof.
Applying the following formula (see [39])
(36)Bmϑ,ϕzBmϑ,ϕ−z=4F1−mI,ϑ+m−1I,12ϑ−I,12ϑϑ−I;4z2ϕ−2.We thus find that
(37)G6λ=LzA−IBnθ,λzI14F1−mI,ϑ+m−1I,12ϑ−I,12ϑϑ−I;4z2ϕ−2=∑s=0n−nIsθ+n−1Iss!−λ−1s×∑j=0m−mIjϑ+m−1Ijj!12ϑ−Ij12ϑjϑ−Ij−14ϕ−2j×LzA−s+1+2jI.
Making use of (15), we observe that
(38)G6λ=∑s=0n−nIsθ+n−1Iss!−λ−1s×∑j=0m−mIjϑ+m−1Ijj!12ϑ−Ij12ϑjϑ−Ij−1×4ϕ−2jλ−A+s−2jIΓA−s−2jI=λ−AΓA∑j=0m−mIjϑ+m−1Ijj!12ϑ−Ij12ϑj×ϑ−Ij−1A2j4λϕ−2j=∑s=0n−nIsθ+n−1Iss!I−A−2jIs−1=λ−A2A−IπΓAΓI−AΓ2I−A−θΓ−1I−A+nIΓ−12I−A−θ−nI×∑j=0m−mIjϑ+m−1Ijj!12ϑ−Ir12ϑj12Ajϑ−Ij−1×12A+Ij12A+1−nIj12A−nIj×12A+θ+nIj12A+θ+n−1Ij12A+Ij−1×12Aj−112A+θj−112A+θ−Ij−1.16λϕ−2j.
Thus, after a simplification, we get the required result (35).Theorem 16.
Letz,λ∈ℂ, Rλ>0, n∈ℕ0, and r∈ℕ. Also, let θ,ϕ and A be matrices in Mrℂ such that βA>0 and ϕ+kI are invertible for all k∈ℕ0. For the function
(39)g7z=zA−IlogzBnθ,ϕz,then, we have
(40)G7λ=Lg7z:λ=λ−AΓA∑s=0n−nIsθ+n−1IsAs×−λϕ−1ss!ψA+sI−logλ,where ψA is the Digamma matrix function defined in (6).Proof.
The proof of this Theorem is quite straight forward as(41)G7λ=∫0∞zA−IlogzBnθ,ϕze−λzdz=∑s=0n−nIsθ+n−1Iks!−ϕ−1s×∫0∞zA+s−1Ilogze−λzdz.Upon using (2,2), we have
(42)ΓA+sI=∫0∞zA+s−1Ie−zdz.
Hence,(43)Γ′A+sI=∫0∞zA+s−1Ie−zlogzdz.
We thus arrive at(44)ΨA+sI=Γ′A+sIΓ−1A+sI=Γ−1A+sI∫0∞zA+s−1Ie−zlogzdz.
Therefore, we get(45)ΨA+sI=λA+sIΓ−1A+sI∫0∞zA+s−1Ie−λzlogλzdz=λA+sIΓ−1A+sI∫0∞zA+s−1Ie−λzlogλ+logzdz=λA+sIΓ−1A+sI∫0∞zA+s−1Ie−λzlogλdz+λA+sIΓ−1A+sI∫0∞zA+s−1Ie−λzlogzdz=logλ+λA+sIΓ−1A+sI∫0∞zA+s−1Ie−λzlogzdz.
We thus have(46)∫0∞zA+s−1Ie−λzlogzdz=λ−A+sIΓA+sIΨA+sI−logλ.
From the above equations, we get the required result as follows:(47)G7λ=∑s=0n−nIsθ+n−1IsAss!−λϕ−1s×λ−AΓAΨA+sI−logλ=λ−AΓA∑s=0n−nIsθ+n−1IsAss!×−λϕ−1sΨA+sI−logλ.Theorem 17.
Letz,λ∈ℂ, Rλ>0, n,m,q∈ℕ0, and r∈ℕ. Also, let θ,ϕ,E,D and A be matrices in Mrℂ such that βA>o, and ϕ+kI are invertible for all k∈ℕ0. Further, let
(48)g8z=z2A−ImFqE;D;z2Bnθ,ϕz2.
Then,(49)G8λ=Lg8:λ=22A−IπΓAΓA+12λ−2A×∑k=0n1k!−nIkθ+n−1IkAkA+12k−4λ2ϕ−1k×m+2FqE,A+kI,A+k+12I;D;4λ−2,where mFqE;D;z is the generalized hypergeometric type matrix functions defined in (10) such that Reλ>0 if m<q−1 and Reλ>∣βA∣ if m=q−1.Proof.
Using Definitions (10) and (12) and upon using (15), we obtain
(50)G8λ=∑k=0n1k!−nIkθ+n−1Ik4ϕ−1k×∑r=0∞∏i=1mEir∏j=1qDjr−11k!Lz2A−1−2k−2rI=∑k=0n1k!−nIkθ+n−1Ik4ϕ−1k×∑r=0∞∏i=1mEir∏j=1qDjr−11k!×λ−2A−2k+2rIΓ2A+2k+2rI.
Thus, after a simplification, we obtain the result (49) in Theorem 3.11.Theorem 18.
Letz,υ,σ,λ∈ℂ, Rλ>0,Rυ>−1,Rσ>0, n,m∈ℕ0, and r∈ℕ. Also, let θ be matrix in Mrℂ such that βA>0 and ϕ+kI are invertible for all k∈ℕ0. For the function
(51)g9z=zυ/2Jυ2σz1/2Bnθ,λz1.Then, we have
(52)G9λ=Lg9z:λ=συ/2θ+υIn1−υnλ−υ+1×2F21+υ−m,θ+n+υI1+υ,θ+υI;−σλ,where Jυz is the Bessel function of the first kind of order υ defined by (see, e.g., [38, 41, 43])
(53)Jυz=∑s=0∞−1ss!Γ1+υ+sz2υ+2s.Proof.
According to (12) and (53) and upon sing (15), it follows that
(54)G9λ=Lzυ/2Jυ2σz1/2Bnθ,λz1=∑m=0∞−1mσm+υ/2m!Γ1+υ+m×∑k=0n−nIkθ+n−1Ikk!−λ−1kLzυ2+υ2−k+m=συ/2∑m=0∞−σmm!Γ1+υ+m×∑k=0n−nIkθ+n−1Ikk!−λ−1kΓ1+υ+m−k.λυ−m+k−1=συ/2λυ−1∑m=0∞−σmλ−mΓ1+υ+mm!Γ1+υ+m×∑k=0n−nIkθ+n−1Ik−υ+mkk!=συ/2λ−υ−1θ+υIn−υn∑m=0∞1+υ−nmυ+nI+θmθ+υIm−1m!1+υm−σλm.
This completes the proof of Theorem18.


## 4. Inverse Laplace Type Integrals of Functions InvolvingBnP,Qz

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.
Letz,λ,σ∈ℂ, Rλ>1/2∣Rσ∣, n∈ℕ0, and r∈ℕ. Also, let A be matrix in Mrℂ such that βA>0. If
(55)G10λ=ΓAλ+12σ−ABnA−n+1I,1λ+1/2σ−σ.
Then,(56)g10z=zA−Iexp−12σz1−σzn.Proof.
It is sufficient to find Laplace transform ofg10z(57)G10λ=LzA−Iexp−12σz1−σzn=LzA−Iexp−12σz1F0−n−;σz=∑k=0n−nIkσkk!LzA−1−kIexp−12σz=∑k=0n−nIkσkk!ΓA+kIλ+12σ−A+kI=ΓAλ+12σ−A∑k=0n−nIkAkk!σλ+1/2σk,
This finalizes the proof of Theorem19.Theorem 20.
Letz,λ,σ∈ℂ, Rλ>0,Rσ>0, n∈ℕ0, and r∈ℕ. Also, let A be matrix in Mrℂ such that βA+nI>0. Further, let
(58)G11λ=−1nσ12A+nIλ−A+2n+1Iexp−σλzBnI−A−2nI,σλ.
Then,(59)g11z=zA2+nIJυ2σz1/2.Proof.
By invoking to (15) and (53), we consider
(60)G11λ=LzA2+nIJυ2σz1/2=∑r=0∞Γ−1A+1+rI−σrσA/2r!LzA+n+rI=σA/2Γ−1A+I∑r=0∞−rA+Ir−1r!ΓA+r+n+1Iλ−A+r+n+1I=σA/2A+Inλ−A+n+1Iexp−σλ∑r=0n−nIrA+Ir−1r!σλr=σA2+nIA+Inλ−A+2n+1Iexp−σλ∑r=0n−nIrA+Ir−1r!σλr−n.
Puttingn−r=k, we obtain
(61)G11λ=−1nσA2+nIλ−A+2n+1Iexp−σλ×∑k=0n−nIk−A+nIkk!−λσk=−1nσ12A+nIλ−A+2n+1Iexp−σλzBnI−A−2nI,σλ.
This finalizes the proof of Theorem20.The remaining results, which are given in the following theorems, can also be proven in a similar way. So we prefer to omit the details.Theorem 21.
Letz,λ∈ℂ, Rλ>0, n∈ℕ0, and r∈ℕ. Also, let θ and ϕ be matrices in Mrℂ such that ϕ+kI are invertible for all k∈ℕ0. Further, let
(62)G12λ=−ϕnλθ+2n−2IΓ2I−θBn2I−θ−2nI,ϕ−λIλ−n.
Then,(63)g12z=z−θ+n−1IBnθ,ϕz−1.Theorem 22.
Letz,λ∈ℂ, Rλ>0, n∈ℕ0, and r∈ℕ. Also let θ and ϕ be matrices in Mrℂ such that ϕ+kI are invertible for all k∈ℕ0. Further, let
(64)G13λ=1λ2F0−n,θ−n+1I−;λϕ−1.
Then,(65)g13z=Bnθ,ϕz−1.Theorem 23.
Letz,λ,μ∈ℂ, Rλ>Rμ>0, n∈ℕ0, and r∈ℕ. Also let θ and ϕ be matrices in Mrℂ such that ϕ+kI are invertible for all k∈ℕ0. Further, let
(66)G14λ=λ−μ−12F0−n,θ−1−nI−;λ−μϕ−1.
Then,(67)g14z=expμzBnθ,ϕz−1.


## 5. Conclusion

In fact, this work is a continuation of the recent paper by Abdalla [44]. In the current manuscript, the authors introduced various Laplace integral formulas of generalized Bessel matrix polynomials with certain elementary matrix functions, Binomial matrix functions exponential function, logarithmic function, generalized hypergeometric matrix functions, and Bessel function of the first kind. We also presented inverse Laplace transforms of generalized Bessel matrix polynomials with some functions. It is obvious that the results presented here which are involved in certain matrices in Mrℂ may reduce to yield the corresponding scalar ones when r=1. Furthermore, the results derived in this article yields to many special cases; the interested reader may be referred to (see, e.g., [1, 7, 45]).A remarkably large number of Laplace transforms and inverse Laplace transforms involving a variety of functions and polynomials have been presented (see, e.g., [45, pp. 129--299]). In this connection, we tried to give matrix versions of those outcomes for Laplace transforms and inverse Laplace formulas involving a variety of functions and polynomials (see, [45, pp. 129--299]).



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