On the Matrix Versions of Incomplete Extended Gamma and Beta Functions and Their Applications for the Incomplete Bessel Matrix Functions

Abstract

In this paper, we first introduce the incomplete extended Gamma and Beta functions with matrix parameters; then, we establish some different properties for these new extensions. Furthermore, we give a specific application for the incomplete Bessel matrix function by using incomplete extended Gamma and Beta functions; at last, we construct the relation between the incomplete confluent hypergeometric matrix functions and incomplete Bessel matrix function.

1. Introduction

In many areas of applied mathematics, various types of special functions have become essential tools for scientists and engineers. The continuous development of mathematical physics, probability theory, and other areas has led to new classes of special functions and their extensions and generalizations (see [1–7]). Generalizations of the classical special functions to matrix setting have become important during last years. Special matrix functions appear in solutions for some physical problems. Applications of special matrix functions also grow and have become active areas in recent literature including statistics, Lie groups theory, and differential equations (see, e.g., [8–11] and elsewhere). New extensions of some of the well-known special matrix functions such as gamma matrix function, beta matrix function, and Gauss hypergeometric matrix function have been extensively studied in recent papers [12–19]. Our main purpose in this paper is to obtain an extension of the incomplete Gamma and Beta matrix functions and will be introduced as application to incomplete Bessel functions with matrix coefficients.

The structure of this paper is as follows. In Section 2, we give basic definitions and preliminaries that are needed in the subsequent sections. In Section 3, we define the generalized incomplete Gamma function with matrix coefficients and study to some properties of the generalized incomplete Gamma matrix function. In Section 4, we present the Beta matrix functions and consider some properties of the incomplete Beta matrix function. Finally, in Section 5, we consider application of the incomplete Bessel matrix function by using incomplete Beta and incomplete confluent hypergeometric matrix function.

2. Preliminaries and Basic Definitions

Throughout this paper, and will denote the identity matrix and null matrix in , respectively. For a matrix , its spectrum is denoted by . We say that if , for all , a matrix in is a positive stable matrix, where is the set of all eigenvalues of . In [16, 20], if and are holomorphic functions in an open set of the complex plane and if is a matrix in for which , then . The logarithmic norm of a matrix in is defined as (see [17, 21])

Suppose the number is such that

For all in andthen the Pochhammer symbol is defined by see, e.g., [8, 22]:

Definition 1 (see [21]). Let be a positive stable matrix in and be a positive real number. Then, the incomplete Gamma matrix function and its complement are defined byrespectively, which satisfy the following decomposition formula (see [21]):

By inserting a regularization matrix factor , Abul-Dahab and Bakhet [13] have introduced the following generalization of the gamma matrix function.

Definition 2. Let and be positive stable matrices in ; then, the generalized Gamma matrix function is defined byfor reduces gamma matrix function in [23].

Also, Abdalla and Bakhet [14] considered the extension of Euler’s beta matrix function in the following definition.

Definition 3. Suppose that , and are positive stable and commutative matrices in satisfying the spectral condition (3); then, the extended Beta matrix function is defined by

Hence,

For , it obviously reduces to the Beta matrix function in [23, 24] by

The Bessel matrix function of the first kind associate to is defined in the form (see [21, 25])and the modified Bessel matrix function has been defined in the formwhere is a matrix in satisfying condition (3). We can rewrite the Bessel and modified Bessel matrix functions aswhere is a hypergeometric matrix function of 1-denominator [26]:and is similar.

3. Generalized Incomplete Gamma Matrix Function

Definition 4. Let and be positive stable matrices in and be a positive real number. Then, the generalized incomplete Gamma matrix function and its complement are defined byTaking in (17) and (18), we get the results as [21]where and are defined in (5) and (6).

Theorem 1. Let and be positive stable matrices in ; then, each of the following properties holds true:(i)(ii)

Proof. (i)The following is obtained from Definition 4 and using equation (18)(ii)The left-hand side equalsSubstituting and , , we get that left-hand side becomeswhich is the right-hand side.

For the properties of the generalized incomplete Gamma matrix function, we have these results.

Theorem 2. The generalized incomplete Gamma matrix function satisfies the following properties:(i)(ii)(iii)(iv)

Proof. (i)Let us define , where where is the Heaviside step function; using the Mellin transform of , we obtain The differentiation of is given by where is the Dirac delta function. From the relation, and between the Mellin transform of a function and derivative, we see that Replacing by in (27), we get the proof of (i).(ii)This follows from (i) when we put .(iii)From the definition of the generalized incomplete Gamma matrix function, we have(iv)Replacing in (18) by its series representation yields the serieswhich is exactly (iv).

Theorem 3. For the generalized incomplete Gamma matrix function , we have the following integral:

Proof. According to (18), we haveSubstituting in (31), we obtainMultiplying both the sides in (32) by , we find thatwhich can be written in the convolution operator form asTaking the convolution operator of both the side in (34) with and using the associative property of convolution, it follows thatHowever,from (35) and (36) and using (34), we obtainThe multiplication of both sides in (37) by yields the proof of Theorem 3.

4. Extended Incomplete Beta Matrix Function

Definition 5. Let be positive stable and commuting matrices in satisfying the spectral condition (3) and be a positive real number; then, the incomplete Beta matrix function is defined in the form

Now, we consider some properties of the incomplete Beta matrix function; we have the following theorem.

Theorem 4. The incomplete Beta matrix function satisfies the following properties:(i)(ii)(iii)

Proof. (i)the right-hand side of (i), we obtain Putting , we have(ii)(ii) can obviously be obtained from (i).(iii)the right-hand side of (iii), we obtain which, after simple algebraic manipulation, yields

Definition 6. Let , and be positive stable and commuting matrices in satisfying the spectral condition (3) and be a positive real number. Then, the extended incomplete Beta matrix function is defined in the form

Theorem 5. The extended incomplete Beta matrix function satisfies the following integral representations:(i)(ii)

Proof. All cases are straightforward. In particular, (i) follows when we use the transformation in (43). The transformation in (43) yields (ii).

Then, we consider some properties of the extended incomplete Beta matrix function, and we get the following theorem.

Theorem 6. The extended incomplete Beta matrix function satisfies the following properties:(i)(ii)(iii)(iv)(v)

Proof. (i)From the left-hand side, we obtain which, after simple algebraic manipulation, yields(ii)From the left-hand side, we obtain Putting , we have(iii)It is obvious that (iii) can be obtained from (ii).(iv)Notice that Thus, it is asserted by (iv).(v)Replacing in (43) by its series representation,we obtainInterchanging the order of the integration and the summation and using (43) yields the desired result (v).

Remark 1. If and in (iii) of Theorem 5, we find thatwhich can be further written in terms of the Whittaker matrix function (see [14]) to giveIn particular, when , we see that

5. Incomplete Bessel Matrix Function

In this section, we obtain the application of the incomplete Bessel matrix function (IBMF). First, we give some definitions.

Definition 7. Let be matrix in , satisfying condition (3); then, the incomplete confluent hypergeometric matrix function (ICHMF) of 1-denominator is defined in the form

By using integral representation of the incomplete beta matrix function given by (7), then we can obtain the integral representation of the ICHMF as

Now, we give definitions of an incomplete Bessel matrix function (IBMF) by using ICHMF.

Definition 8. Let be matrix in , satisfying condition (3); then, the incomplete Bessel matrix function is defined in the formwhere is incomplete hypergeometric matrix function of 1-denominator. By using equation (55), we can easily obtain series representations for the IBMF asand the integral representations are found from (54) and (55) to obtain

Remark 2. (i)If is used in (58), we have(ii)Putting in (59), we have(iii)By differentiating both sides of (60) with respect to , we get the following elegant relation between the IBMF and the BMF as follows:

The following two theorems give the integrals of the ICHMF and the IBMF over the unit interval as a direct elegant relation with the complete version of both of them, respectively.

Theorem 7. Let be matrix in ; we have the relation between the ICHMF and CHMF as follows:

Proof. By using (54) and integrating both sides over the unit interval, we obtainsince the integral of the unity over the unit interval is again one. So, as before, interchanging the integral sign with the summation sign givesNow, by using properties of the integral in compleat Beta matrix function, we find thatwhich is the required proof.

Theorem 8. Let be matrix in , and we have the relation between the IBMF and BMF as follows:

Proof. From definition of an incomplete Bessel matrix function, we haveBy (62) and (68), it is written asBy properties of Bessel matrix function, we obtainHence, substitution of (70) in (69) completes the proof of Theorem 8 desired.

6. Conclusions

Special matrix functions play an important role in certain parts of mathematics and physics over the last years. In this paper, we have introduced the incomplete extended Gamma and Beta functions with matrix parameters; then, we establish some properties for these new extension; furthermore, we give a special application for the incomplete Bessel matrix function by using incomplete extended Gamma and Beta functions; at last, we construct the relation between the incomplete confluent hypergeometric matrix functions and incomplete Bessel matrix function. Taking in (17) and (18), we get the results in [21].

Also, by using the identity matrix, we can find the scalar case incomplete extended Gamma and Beta functions in [1, 27–30] as and . If and , we have Gamma and Beta functions.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by Educational Science “13th Five Year Plan” Project Foundation of Hunan Province (XJK19BXX005) and Natural Science Foundation of Hunan Province (2020JJ4105).