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

College of Economics and Management, Changsha Normal University, Changsha 410148, China College of Mechanical and Electrical Engineering, Hunan Agricultural University, Changsha 410128, China School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt School of Mathematics and Statistics, Central South University, Changsha 410083, China


Introduction
In many areas of applied mathematics, various types of special functions have become essential tools for scientists and engineers. e 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][2][3][4][5][6][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][9][10][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][13][14][15][16][17][18][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. e 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.

Preliminaries and Basic Definitions
roughout this paper, I and 0 will denote the identity matrix and null matrix in C r×r , respectively. For a matrix A ∈ C r×r , its spectrum is denoted by σ(A). We say that if Re(ξ) > 0, for all ξ ∈ σ(A), a matrix A in C r×r is a positive stable matrix, where σ(A) is the set of all eigenvalues of A. In [16,20], if f(z) and g(z) are holomorphic functions in an open set Λ of the complex plane and if A is a matrix in C r×r e logarithmic norm of a matrix A in C r×r is defined as (see [17,21]) (1) Suppose the number μ(A) is such that For all A in C r×r and A + nI is invertible for all integers n, then the Pochhammer symbol is defined by see, e.g., [8,22]: Definition 1 (see [21]). Let A be a positive stable matrix in C r×r and x be a positive real number. en, the incomplete Gamma matrix function c(A, x) and its complement Γ(A, x) are defined by (6) respectively, which satisfy the following decomposition formula (see [21]): By inserting a regularization matrix factor e − (B/t) , B ∈ C r×r , Abul-Dahab and Bakhet [13] have introduced the following generalization of the gamma matrix function.
Definition 2. Let A and B be positive stable matrices in C r×r ; then, the generalized Gamma matrix function Γ(A, B) is defined by for B � 0 reduces gamma matrix function in [23].
Also, Abdalla and Bakhet [14] considered the extension of Euler's beta matrix function in the following definition. Hence, For P � 0, it obviously reduces to the Beta matrix function in [23,24] by e Bessel matrix function J A (z) of the first kind associate to A is defined in the form (see [21,25]) and the modified Bessel matrix function I A (z) has been defined in the form where A is a matrix in C r×r satisfying condition (3). We can rewrite the Bessel and modified Bessel matrix functions as where 0 F 1 (− ; A + I, (− z 2 /4)) is a hypergeometric matrix function of 1-denominator [26]:

Theorem 1.
Let A and B be positive stable matrices in C r×r ; then, each of the following properties holds true: Proof (i) e following is obtained from Definition 4 and using equation (18) Substituting t � τ/α and dt � dτ/α, α > 0, we get that lefthand side becomes which is the right-hand side.

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

Theorem 2.
e generalized incomplete Gamma matrix function Γ(A, B; x) satisfies the following properties: where H(t − x) is the Heaviside step function; using the Mellin transform of f(t), we obtain e differentiation of f(t) is given by where δ is the Dirac delta function. From the relation, (26) and between the Mellin transform of a function and derivative, we see that Replacing A by A + I in (27), we get the proof of (i). (ii) is follows from (i) when we put B � 0.

(iii) From the definition of the generalized incomplete
Gamma matrix function, we have (iv) Replacing e − B/t in (18) by its series representation yields the series which is exactly (iv).

Complexity
Proof. According to (18), we have Substituting τ � 1/ζ in (31), we obtain Multiplying both the sides in (32) by e Bt , we find that which can be written in the convolution operator form as Taking the convolution operator of both the side in (34) with e Ct and using the associative property of convolution, it follows that e tC * e tB Γ A, B; 1 t � e tC * e tB * e (− I/t) t − (A+I) .

Extended Incomplete Beta Matrix Function
Definition 5. Let A and B be positive stable and commuting matrices in C r×r satisfying the spectral condition (3) and x be a positive real number; then, the incomplete Beta matrix function B x (A, B) is defined in the form Now, we consider some properties of the incomplete Beta matrix function; we have the following theorem. (39) Putting u � 1 − t, 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 A, B, and P be positive stable and commuting matrices in C r×r satisfying the spectral condition (3) and x be a positive real number. en, the extended incomplete Beta matrix function B x (A, B; P) is defined in the form Theorem 5. e extended incomplete Beta matrix function B x (A, B; P) satisfies the following integral representations: Proof. All cases are straightforward. In particular, (i) follows when we use the transformation t � sin 2 u in (43). e transformation t � u/1 + u in (43) yields (ii). □ en, we consider some properties of the extended incomplete Beta matrix function, and we get the following theorem. 4 Complexity which, after simple algebraic manipulation, yields (ii) From the left-hand side, we obtain Putting u � 1 − t, we have (iii) It is obvious that (iii) can be obtained from (ii).
(iv) Notice that us, it is asserted by (iv). (v) Replacing (1 − t) B− I in (43) by its series representation, we obtain Interchanging the order of the integration and the summation and using (43) yields the desired result (v). □ Remark 1. If B � A and x � 1/2 in (iii) of eorem 5, we find that which can be further written in terms of the Whittaker matrix function (see [14]) to give In particular, when A � 1/2I, we see that

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

Complexity
Definition 7. Let A be matrix in C r×r , satisfying condition (3); then, the incomplete confluent hypergeometric matrix function (ICHMF) of 1-denominator is defined in the form 0 F 1,x (− ; C; z) � I + z ∞ k�0 (2I) k − 1 B x ((k + 1)I, C)z k k! . (54) 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.