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

Authors: Chaojun Zou; Mimi Yu; Ahmed Bakhet; Fuli He
Journal: Complexity
 (2021)

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## 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.


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## Body

## 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 Section2, 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,I and 0 will denote the identity matrix and null matrix in ℂr×r, respectively. For a matrix A∈ℂr×r, its spectrum is denoted by σA. We say that if Reξ>0, for all ξ∈σA, a matrix A in ℂr×r is a positive stable matrix, where σA is the set of all eigenvalues of A. In [16, 20], if fz and gz are holomorphic functions in an open set Λ of the complex plane and if A is a matrix in ℂr×r for which σA⊂Λ, then fAgA=gAfA. The logarithmic norm of a matrix A in ℂr×r is defined as (see [17, 21])(1)μA=limh⟶0I+hA−1h=maxz:z∈σA+A∗2.Suppose the numberμ˜A is such that(2)μ˜A=−μ−A=minz:z∈σA+A∗2.For allA in ℂr×r and(3)A+nIis invertible for all integersn,then the Pochhammer symbol is defined by see, e.g., [8, 22]:(4)An=AA+I,…,A+n−1I=ΓA+nIΓ−1A;A0≡I.Definition 1 (see [21]).
LetA be a positive stable matrix in ℂr×r and x be a positive real number. Then, the incomplete Gamma matrix function γA,x and its complement ΓA,x are defined by(5)γA,x=∫0xe−ttA−Idt,tA−I=expA−Ilnt,(6)ΓA,x=∫x∞e−ttA−Idt,tA−I=expA−Ilnt,respectively, which satisfy the following decomposition formula (see [21]):(7)γA,x+ΓA,x=ΓA.By inserting a regularization matrix factore−B/t,B∈ℂr×r, Abul-Dahab and Bakhet [13] have introduced the following generalization of the gamma matrix function.Definition 2.
LetA and B be positive stable matrices in Cr×r; then, the generalized Gamma matrix function ΓA,B is defined by(8)ΓA,B=∫0∞tA−Ie−It+B/tdt,tA−I=expA−Ilnt,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.Definition 3.
Suppose thatA,B, and ℙ are positive stable and commutative matrices in ℂr×r satisfying the spectral condition (3); then, the extended Beta matrix function ℬA,B;ℙ is defined by(9)ℬA,B;ℙ≔∫01tA−I1−tB−Iexp−ℙt1−tdt.Hence,(10)ℬA,B;ℙ=ΓA,ℙΓB,ℙΓ−1A+B;ℙ.Forℙ=0, it obviously reduces to the Beta matrix function in [23, 24] by(11)BA,B=∫01tA−I1−tB−Idt.The Bessel matrix functionJAz of the first kind associate to A is defined in the form (see [21, 25])(12)JAz=∑k=0∞−1kk!Γ−1A+k+1Iz2A+2kI,and the modified Bessel matrix function IAz has been defined in the form(13)IAz=∑k=0∞1k!Γ−1A+k+1Iz2A+2kI,where A is a matrix in ℂr×r satisfying condition (3). We can rewrite the Bessel and modified Bessel matrix functions as(14)JAz=z2AΓ−1A+IF10−;A+I,−z24,(15)IAz=z2AΓ−1A+IF10−;A+I,z24,where F10−;A+I,−z2/4 is a hypergeometric matrix function of 1-denominator [26]:(16)F10−;A+I;−z24=∑n≥0A+In−1n!−z24n,and F10−;A+I,z2/4 is similar.


## 3. Generalized Incomplete Gamma Matrix Function

Definition 4.
LetA and B be positive stable matrices in ℂr×r and x be a positive real number. Then, the generalized incomplete Gamma matrix function γA,B;x and its complement ΓA,B;x are defined by(17)γA,B;x≔∫0xtA−Ie−It+B/tdt,tA−I=expA−Ilnt,(18)ΓA,B;x≔∫x∞tA−Ie−It+B/tdt,tA−I=expA−Ilnt.
TakingB=0 in (17) and (18), we get the results as [21](19)γA,0;x=γA;x,(20)ΓA,0;x=ΓA;x,where γA;x and ΓA;x are defined in (5) and (6).Theorem 1.
LetA and B be positive stable matrices in ℂr×r; then, each of the following properties holds true:(i)
γA,B;x+ΓA,B;x=2expA/2lnBKA2B(ii)
∫x∞tA−Iexp−αtI+B/tdt=α−AΓA,αB;αx;α>0Proof.
(i)
The following is obtained from Definition4 and using equation (18)(ii)
The left-hand side equals(21)∫x∞tA−Iexp−αtI+Btdt.
Substitutingt=τ/α and dt=dτ/α, α>0, we get that left-hand side becomes(22)α−A∫αx∞τA−Iexp−τI+αBτdτ=α−AΓA,αB;αx,which 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 functionΓA,B;x satisfies the following properties:(i)
ΓA+I,B;x=AΓA,B;x+BΓA−I,B;x+xAexp−xI+B/x(ii)
ΓA+I;x=AΓA;x+xAexpx(iii)
∂/∂BΓA,B;x=−ΓA−I,B;x(iv)
ΓA,B;x=∑0∞ΓA−nI;x−Bn/n!x>0Proof.
(i)
Let us defineft≔e−tI+Bt−1Ht−x, where(23)Ht−x=1,t>x,0,t<x,whereHt−x is the Heaviside step function; using the Mellin transform of ft, we obtain(24)ΓA,B;x≔Mft;A.The differentiation offt is given by(25)f′t=−I+Bt−2ft+e−tI+Bt−1δt−x,whereδ is the Dirac delta function.From the relation,(26)Mf′t;A=−A−IMftA−I,and between the Mellin transform of a function and derivative, we see that(27)−A−IΓA−I,B;x=−ΓA,B;x+BΓA−2I;B;x+xA−Iexp−xI+Bx−1.ReplacingA by A+I in (27), we get the proof of (i).(ii)
This follows from (i) when we putB=0.(iii)
From the definition of the generalized incomplete Gamma matrix function, we have(28)∂∂BΓA,B;x=∂∂B∫x∞tA−Ie−It+B/tdt=−∫x∞tA−2Ie−It+B/tdt=−ΓA−I,B;x.(iv)
Replacinge−B/t in (18) by its series representation yields the series(29)ΓA,B;x=∑n=0∞−Bnn!∫x∞tA−n+1Idt,which is exactly (iv).Theorem 3.
For the generalized incomplete Gamma matrix functionΓA,B;x, we have the following integral:(30)∫0tΓA,B;1xexpB−Cxdx=B−C−1exp−tCΓA,B;1texptB−ΓA,C;1texptC.Proof.
According to (18), we have(31)ΓA,B;1t=∫1/t∞τA−Ie−τI+B/τdτ.
Substitutingτ=1/ζ in (31), we obtain(32)ΓA,B;1t=∫0te−Bζe−I/ζζ−A+Idζ.
Multiplying both the sides in (32) by eBt, we find that(33)etBΓA,B;1t=∫0teBt−ζe−I/ζζ−A+Idζ,which can be written in the convolution operator form as(34)etBΓA,B;1t=etB∗e−I/tt−A+I.
Taking the convolution operator of both the side in (34) with eCt and using the associative property of convolution, it follows that(35)etC∗etBΓA,B;1t=etC∗etB∗e−I/tt−A+I.
However,(36)etC∗etB=B−C−1etB−etC,from (35) and (36) and using (34), we obtain(37)etC∫0tΓA,B;1xexpB−Cxdx=B−C−1×ΓA,B;1texptB−ΓA,C;1texptC.
The multiplication of both sides in (37) by e−Ct yields the proof of Theorem 3.


## 4. Extended Incomplete Beta Matrix Function

Definition 5.
LetAandB be positive stable and commuting matrices in ℂr×r satisfying the spectral condition (3) and x be a positive real number; then, the incomplete Beta matrix function BxA,B is defined in the form(38)BxA,B≔∫0xtA−I1−tB−Idt,0<x<1.Now, we consider some properties of the incomplete Beta matrix function; we have the following theorem.Theorem 4.
The incomplete Beta matrix functionBxA,B satisfies the following properties:(i)
BxA,B=BA,B−B1−xB,A(ii)
BxA,B+B1−xB,A=BA,B(iii)
BxA,B=BxA+I,B+BxA,B+IProof.
(i)
the right-hand side of (i), we obtain(39)∫1−x1uB−I1−uA−Idu.Puttingu=1−t, we have(40)∫0xtA−I1−tB−Idt=BxA,B.(ii)
(ii) can obviously be obtained from (i).(iii)
the right-hand side of (iii), we obtain(41)∫0xtA1−tB−I+tA−I1−tBdt,which, after simple algebraic manipulation, yields(42)∫0xtA−I1−tB−Idt=BxA,B.Definition 6.
LetA,B, and ℙ be positive stable and commuting matrices in ℂr×r satisfying the spectral condition (3) and x be a positive real number. Then, the extended incomplete Beta matrix function ℬxA,B;ℙ is defined in the form(43)ℬxA,B;ℙ:=∫0xtA−I1−tB−Iexp−ℙt1−tdt,0<x<1.Theorem 5.
The extended incomplete Beta matrix functionℬxA,B;ℙ satisfies the following integral representations:(i)
ℬxA,B;ℙ=2∫0Tsinu2A−Icosu2B−Iexp−ℙsec2ucsc2udu,0<T=sin−1x≤π/2(ii)
ℬxA,B;ℙ=e−2ℙ∫0TuA−I1+u−A+Bexp−ℙu+u−1du,0<T=x/1−x<∞Proof.
All cases are straightforward. In particular, (i) follows when we use the transformationt=sin2u in (43). The transformation t=u/1+u 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 functionℬxA,B;ℙ satisfies the following properties:(i)
ℬxA+I,B;ℙ+ℬxA,B+I;ℙ=ℬxA,B;ℙ(ii)
ℬA,B;ℙ−ℬ1−xB,A;ℙ=ℬxA,B;ℙ(iii)
ℬxA,B;ℙ+ℬ1−xB,A;ℙ=ℬA,B;ℙ(iv)
ℬxA,I−B;ℙ=∑n=0∞Bn/n!,ℬxA+nI,I;ℙ(v)
ℬxA,B;ℙ=∑n=0∞ℬxA+nI,B+I;ℙProof.
(i)
From the left-hand side, we obtain(44)∫0xtA−I1−tB+tA1−tB−Iexp−ℙt1−tdt,which, after simple algebraic manipulation, yields(45)∫0xtA−I1−tB−Iexp−ℙt1−tdt=ℬxA,B;ℙ.(ii)
From the left-hand side, we obtain(46)∫1−x1uB−I1−uA−Iexp−ℙt1−tdu.Puttingu=1−t, we have(47)∫0xtA−I1−tB−Iexp−ℙt1−tdt=ℬxA,B;ℙ.(iii)
It is obvious that (iii) can be obtained from (ii).(iv)
Notice that(48)1−t−B=∑n=0∞Bnn!tn,t<1,ℬxA,I−B;ℙ=∫0xtA−I1−tI−B−Iexp−ℙt1−tdt=∫0xtA−I∑n=0∞Bnn!tnexp−ℙt1−tdt=∑n=0∞Bnn!∫0xtAn−1Iexp−ℙt1−tdt=∑n=0∞Bnn!ℬxA+nI,I;ℙ.Thus, it is asserted by (iv).(v)
Replacing1−tB−I in (43) by its series representation,(49)1−tB−I=1−tB∑n=0∞tn,we obtain(50)ℬxA,B;ℙ=∫0x1−tB∑n=0∞tA+n−1Iexp−ℙt1−tdt.
Interchanging the order of the integration and the summation and using (43) yields the desired result (v).Remark 1.
IfB=A and x=1/2 in (iii) of Theorem 5, we find that(51)ℬ1/2A,A;ℙ=12ℬA,A;ℙ,which can be further written in terms of the Whittaker matrix function (see [14]) to give(52)ℬ1/2A,A;ℙ=π2−A+IℙA−I/2e−2ℙW−A/2,A/24ℙ,μℙ>0.
In particular, whenA=1/2I, we see that(53)ℬ1/212I,12I;ℙ=π22ℙ,μℙ>0.


## 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.
LetA be matrix in ℂr×r, satisfying condition (3); then, the incomplete confluent hypergeometric matrix function (ICHMF) of 1-denominator is defined in the form(54)F1,x0−;C;z=I+z∑k=0∞2Ik−1ℬxk+1I,Czkk!.By using integral representation of the incomplete beta matrix function given by (7), then we can obtain the integral representation of the ICHMF as(55)F1,x0−;C;z=I+z∑k=0∞2Ik−1zkk!∫0xtk1−tC−Idt.=I+z∫0t1−tC−IF10−;2I;tzdt.Now, we give definitions of an incomplete Bessel matrix function (IBMF) by using ICHMF.Definition 8.
LetA be matrix in ℂr×r, satisfying condition (3); then, the incomplete Bessel matrix function IBMF is defined in the form(56)JA,xz=z2AΓ−1A+IF1,x0−;A+I,−z24,where F1,x0−;A+I,−z2/4 is incomplete hypergeometric matrix function of 1-denominator. By using equation (55), we can easily obtain series representations for the IBMF as(57)JA,xz=z2AΓ−1A+II−z24∑k=0∞−1kℬxk+1I,A+Iz/22k2kk!,and the integral representations are found from (54) and (55) to obtain(58)JA,xz=z2AΓ−1A+II−z2∫0xt−1/21−tAJIztdt.Remark 2.
(i)
Ifu=zt is used in (58), we have(59)JA,xz=z2AΓ−1A+II−∫0zx1−u2z2AJIudu.(ii)
PuttingA=0 in (59), we have(60)J0,xz=I−∫0zxJIudu.(iii)
By differentiating both sides of (60) with respect to z, we get the following elegant relation between the IBMF and the BMF as follows:(61)J0,x′z=JIzx.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.
LetA be matrix in ℂr×r; we have the relation between the ICHMF and CHMF as follows:(62)∫01F1,x0−;A;zdt=F10−;A+I;Z.Proof.
By using (54) and integrating both sides over the unit interval, we obtain(63)∫01F1,x0−;A;zdt=I+z∫01∑k=0∞2Ik−1ℬxk+1I,Azkk!dt,since the integral of the unity over the unit interval is again one. So, as before, interchanging the integral sign with the summation sign gives(64)∫01F1,x0−;A;zdt=I+z∫01∑k=0∞zk2Ik−1k!∫01ℬxk+1I,Adt.
Now, by using properties of the integral in compleat Beta matrix function, we find that(65)ℬxk+1I,A=ℬk+1I,A+I,(66)∫01F1,x0−;A;zdt=I+z∑k=0∞2Ik−1ℬxk+1I,A+Izkk!,which is the required proof.Theorem 8.
LetA be matrix in ℂr×r, and we have the relation between the IBMF and BMF as follows:(67)∫01JA,xzdt=2A+IzJA+Iz.Proof.
From definition of an incomplete Bessel matrix function, we have(68)∫01JA,xzdtz2AΓ−1A+I∫01F1,x0−;A+I,−z24.
By (62) and (68), it is written as(69)∫01JA,xzdt=z2AΓ−1A+I∫01F10−;A+2I,−z24.
By properties of Bessel matrix function, we obtain(70)F10−;A+2I,−z24=z2−A+IΓA+2IJA+Iz.
Hence, 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. TakingB=0 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 A=αI and B=βI. If A=αI and B=0, we have Gamma and Beta functions.



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