# Some Results on the Extended Hypergeometric Matrix Functions and Related Functions

Authors: Mohamed Abdalla; Sahar Ahmed Idris; Ibrahim Mekawy
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this article, we discuss certain properties for generalized gamma and Euler’s beta matrix functions and the generalized hypergeometric matrix functions. The current results for these functions include integral representations, transformation formula, recurrence relations, and integral transforms.


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## Body

## 1. Introduction

Matrix generalizations of some known classical special functions are important both from the theoretical and applied point of view (see, for example, [1–10]). These new extensions have proved to be very useful in various fields such as physics, engineering, statistics, actuarial sciences, life testing, and telecommunications.In particular, various results of gamma, Euler’s beta, and hypergeometric matrix functions have been presented and investigated (see, e.g., [11–17]). Motivated by investigations of the extended gamma, beta, and Gauss hypergeometric matrix functions given in [14–16, 18, 19], we aim to derive certain properties of matrix generalizations of gamma, Euler’s beta, Gauss, and confluent hypergeometric functions. The results given by many authors [12, 20–24] follow as special cases in this work.The plan of this work is described below:In Section2, we propose to derive some integral representations and recurrence relations for generalizations of gamma and Euler’s beta matrix functions. The various properties for the generalizations of Gauss and confluent hypergeometric matrix functions are investigated in Section 3. Finally, we end up with the conclusion in Section 4.Throughout this paper, letI and 0 denote the identity matrix and null matrix in ℂr×r, respectively. A matrix A in ℂr×r is a positive stable matrix if Reλ>0 for all λ∈σA where σA denotes the set of all eigenvalues of A. In [12], if fz and gz are holomorphic functions in an open set Ω of the complex plane and if A is a matrix in ℂr×r for which σA⊂Ω, then(1)fAgA=gAfA.LetA be a positive stable matrix in ℂr×r; then, the gamma matrix function in [11, 12] is defined by(2)ΓA=∫0∞e−ttA−Idt,tA−I=expA−Ilnt.For positive stable matricesA and B∈ℂr×r, the beta matrix function in [11, 12] is defined by(3)BA,B=∫01tA−I1−tB−Idt.Also, ifA, B,andA+B are positive stable matrices in ℂr×r and AB=BA, then(4)BA,B=ΓAΓBΓ−1A+B.From [12], for a matrix A∈ℂr×r, the matrix version of Pochammer symbol is defined as(5)An=AA+IA+2I,…,A+n−1I;n≥1;A0=I.The Gauss hypergeometric matrix functionF12A,B;C;z is given in [12] as follows:(6)F12A,B;C;z=∑n=0∞AnBnCn−1znn!,where z<1 and A,B,andC∈ℂr×r such that C+nI is invertible for all integer. n≥0.Also, the confluent hypergeometric matrix function is defined by(7)Yz=F11A;B;z=∑n=0∞Ann!Bn−1zn,and it satisfies the following matrix differential equation:(8)zY″+Y′B−zI−AY=0,where AandB∈ℂr×r and B+nI is invertible for every integer n≥0. Furthermore, we have(9)Yz=B−1A,B−A∫01etztA−I1−tB−A−Idt,where A,B,andB−A are positive stable matrices and AB=BA.In the recent paper [14], for any arbitrary parameter p with Rep>0, the matrix generalizations of gamma and Euler’s beta functions are given as follows:(10)ΓX;p=∫0∞exp−t−pttX−Idt,(11)ℬX,Y;p=∫01exp−pt1−ttX−I1−tY−Idt,for p=0, which gives gamma and Euler’s beta matrix functions given by (2) and (3), respectively:(12)ΓA,BX;p=∫0∞F11A;B;−t−pttX−Idt,and(13)ℬA,BX,Y;p=∫01tX−I1−tY−IF11A;B;−pt1−tdt,respectively, where A,B,X,andY are positive stable matrices in ℂr×r and p is any arbitrary parameter with Rep>0. Also, these are matrix versions of gamma and beta functions [23]. The case of A=B in (12) and (13) gives us generalizations of gamma and Euler’s beta matrix functions defined by (10) and (11), respectively.Moreover, author in [14] defined the generalized Gauss and confluent hypergeometric matrix functions as follows:Definition 1.
LetA,B,A∗,B∗, and C∗∈ℂr×r satisfying conditions that A,B,B∗,C∗−B∗,andC∗ are the positive stable matrix, and(14)B∗C∗=C∗B∗,and p be a number with Rep>0. Then, the generalized Gauss hypergeometric matrix function (GGHMF) is defined in [14] by(15)FA,BA∗,B∗;C∗;z;p=∑n=0∞A∗nℬA,BB∗+nI,C∗−B∗;pB−1B∗,C∗−B∗znn!,and the generalized confluent hypergeometric matrix function (GCHMF) in the form(16)F1A,B1B∗;C∗;z;p=∑n=0∞ℬA,BB∗+nI,C∗−B∗;pB−1B∗;C∗−B∗znn!.Remark 1.
ForA=B in (15) and (16), we have(17)FA∗,B∗;C∗;z;p=∑n=0∞A∗nℬB∗+nI,C∗−B∗;pB−1B∗,C∗−B∗znn!,F11B∗,C∗;z;p=∑n=0∞B∗+nI,C∗−B∗;pB−1B∗,C∗−B∗znn!.
Atp=0 in (15), it reduces(18)FA,BA∗,B∗;C∗;z=F12A∗,B∗;C∗;z,and also, at p=0 in (16), we get(19)F1A,B1B∗,C∗;z=F11B∗,C∗,z.Now, let us give the following definition as a generalization of the functions in (15) and (16).Definition 2.
LetA,B,A1∗,…,Ar−1∗,B∗,C∗,andC1∗,…,Cs−1∗∈ℂr×r satisfying conditions that A,B,B∗,C∗−B∗,andC∗ are positive stable matrices, and(20)B∗C∗=C∗B∗,and Ci∗+kI is an invertible matrix for k∈ℕ,i=1,…,s−1, and p be a number with Rep>0. Then, the generalization hypergeometric-type matrix function is defined by(21)FsA,BrA1∗,…,Ar−1∗,B∗;C∗,C1∗,…,Cs−1∗;z;p=∑n=0∞A1∗n,…,Ar−1∗nℬA,BB∗+nI,C∗−B∗;pB−1B∗,C∗−B∗C1∗n−1,…,Cs−1∗n−1znn!,where rands∈ℕ.


## 2. Properties of Generalizations of Gamma and Beta Matrix Functions

In this section, we drive some properties of generalized gamma and Euler’s beta matrix functions, which are defined by (12) and (13), as follows:Theorem 1.
For the generalized gamma matrix function given by (12), we have the following integral representation:(22)ΓA,BX;p=∫01η−XΓX;pη2B−1A,B−AηA−I1−ηB−A−Idη,where B−A is a positive stable matrix in ℂr×r and AB=BA.Proof.
Using (9) in (12), we have(23)ΓA,BX;p=∫0∞uX−IB−1A,B−A∫01e−ut−pt/utA−I1−tB−A−Idtdu.
Takingv=utandη=t in the above equation, we can write(24)ΓA,BX;p=∫01η−X∫0∞vX−Ie−v−pη2/vdvB−1A,B−AηA−I1−ηB−A−Idη.
Then, by (10), we complete the proof of the theorem.Theorem 2.
The following integral representation for generalized Euler’s beta matrix function in (13) holds well:(25)ℬA,BX,Y,p=∫01ℬX,Y,pttA−I1−tB−A−IB−1A,B−Adt,where B−A is a positive stable matrix in ℂr×r and AB=BA.Proof.
The proof of the theorem is completed similar to Theorem1.Theorem 3.
For generalized Euler’s beta matrix functionℬA,BX,Y;p, we have the next property:(26)∫01pS−IℬA,BX,Y;pdp=BS+X,Y+SΓA,BS,where the matrices S,X,Y,A,andB∈ℂr×r, S+X,Y+S,andX+Y+2S are positive stable matrices, and SX=XSandSY=YS.Proof.
It is enough to use (13), interchange the order of integration, take transformations v=p/t1−t and η=t in (13), and then use (12) in the left-hand side of the above equation, respectively.Theorem 4.
For generalized gamma matrix functionΓA,BX;p, we get(27)ΓA,BX;pΓA,BY;p=4∫0π/2∫0∞rcosθ2X−Irsinθ2Y−I×F11A;B;−r2cos2θ−pr2cos2θF11A;B;−r2sin2θ−pr2sin2θrdrdθ,where AY=YAandBY=YB.Proof.
It follows straightforwardly from (12).Remark 2.
If we takeA=B and p=0 in Theorem 4, we obtain(28)BX,Y=ΓXΓYΓ−1X+Y,where X,Y,andX+Y are positive stable matrices and XY=YX.Theorem 5.
LetXandY be matrices in ℂr×r satisfying conditions that X is a positive stable, 0<Reλ<1 for λ∈σY, and XY=YX. Then, we have(29)ℬA,BX,I−Y;p=∑n=0∞Ynn!ℬA,BX+nI,I;p.Proof.
By (13), we get(30)ℬA,BX,I−Y;p=∫01tX−I1−t−YF11A;B;−pt1−tdt.
On the contrary, if we consider Taylor expansion of1−t−Y at t=0, we can write(31)1−t−Y=∑n=0∞Ynn!tn.
Then, using (13), the theorem can be proved.Remark 3.
ForA=B in Theorem 5, we get(32)ℬX,I−Y;p=∑n=0∞Ynn!ℬX+nI,I;p,and for p=0 in (32), we have(33)BX,I−Y=∑n=0∞Ynn!BX+nI,I.Theorem 6.
The following recurrence relation for the generalized gamma matrix function holds well:(34)d2dp2ΓA,BX+5I;p+p2d2dp2ΓA,BX+3I;p−ddpΓA,BX+2I;B−ddpΓA,BX+3I;p−pddpΓA,BX+I;p+AΓA,BX;p=0.Proof.
From (12), by using the Leibnitz rule, it is easily seen that the left-hand side of the above equation can be written as(35)∫0∞tX−It3+ptd2Zdp2+dZdpBt+t2+pI+AZdt,where Z=F11A;B;−t−p/t. On the one hand, due to (8), we have(36)t3+ptd2Zdp2+dZdpBt+t2+pI+AZ=0.
Thus, the proof is completed.Remark 4.
ForA=B in Theorem 6, we get(37)d2dp2ΓX+5I;p+p2d2dp2ΓX+3I;p−ddpΓX+2I;pA−ddpΓX+3I;p−pddpΓX+I;p+AΓX;p=0.Theorem 7.
Generalized Euler’s beta matrix function verifies the next recurrence relation:(38)pd2dp2ℬA,BX+3I,Y+3I;p+ddpℬA,BX+2I,Y+2I;pB+pddpℬA,BX+I,Y+I;p+AℬA,BX,Y;p=0.Proof.
In view of (13), the left-hand side of above equation is equal to(39)∫0∞tX−I1−tY−Ipt1−td2Zdp2+dZdpBt1−t+pI+AZdt,where Z=F11A;B;−p/t1−t. On the other hand, by using equation (8), we obtain(40)d2Zdp2pt1−t+dZdp1−tBt+pI+ZA=0,which yields the desired result of Theorem 7.Remark 5.
The case ofA=B in Theorem 7 gives(41)pd2dp2ℬX+3I,Y+3I;p+ddpℬX+2I,Y+2I;pA+pddpℬX+I,Y+I;p+AℬX,Y;p=0.


## 3. Properties of the GGHMF and GCHMF

In this section, we give some of the main results of the GGHMF and GCHMF as follows.Theorem 8.
For the GGHMFFA,BA∗,B∗;C∗;z;p, the following integral form holds true:(42)FA,BA∗,B∗;C∗;z;p=∫011−tz−A∗tB∗−I1−tC∗−B∗−IB−1B∗,C∗−B∗F11A;B;−pt1−tdt,where arg1−z<π.Proof.
If we take(43)ℬpA,BB∗+nI,C∗−B∗=∫01tB∗+nI−I1−tC∗−B∗−IF11A;B;−p/t1−tdt,in (15) and use the following relation:(44)∑n=0∞A∗nn!tzn=1−tz−A∗,then we arrive at the required result.Corollary 1.
If we apply the substitutiont=sin2v in (42), we derive(45)FA,BA∗,B∗;C∗;z;p=2∫0π/21−zsin2v−A∗sin2B∗−Ivcos2C∗−2B∗−Iv×B−1B∗,C∗−B∗F11A;B;−psin2vcos2vdv,and applying the substitution t=u/1+u in (42), we obtain(46)FpA,BA∗,B∗;C∗;z=∫0∞1+u1−z−A∗1+uA∗−C∗uB∗−I×1F1A;B;−2p−pu−pu−1B−1B∗,C∗−B∗du,where A∗C∗=C∗A∗.Theorem 9.
The following integral form for GGHMF holds true:(47)FA,BA∗,B∗;C∗;z;p=∫01FA∗,B∗;C∗;z;pttA−I1−tB−A−IB−1A,B−Adt,where B−A is a positive stable matrix in ℂr×r and AB=BA.Proof.
From (15) and Theorem 2, we have the desired relation.Theorem 10.
The GCHMFF1A,B;p1B∗;C∗;z has the next representation:(48)F1A,B1B∗;C∗;z;p=∫01tB∗−I1−tC∗−B∗−IB−1B∗,C∗−B∗F11A;B;−pt1−teztdt.Proof.
One can easily prove the theorem similar to Theorem7.Corollary 2.
If we taket=1−u in (48), we derive(49)F1A,B;p1B∗;C∗;z=ez∫011−uB∗−IuC∗−B∗−IB−1B∗,C∗−B∗F11A;B;−pu1−ue−zudu.Theorem 11.
For the GGHMF witharg1−z<π, then the following transformation formula holds true:(50)FA,BA∗,B∗;C∗;z;p=1−z−A∗FA,BA∗,C∗−B∗;C∗;zz−1;p.Proof.
In (42), by writing 1−t instead of t and using the following equation(51)1−z1−t−A∗=1−z−A∗1+z1−zt−A∗,we obtain(52)FA,BA∗,B∗;C∗;z;p=1−z−A∗FA,BA∗,C∗−B∗;C∗;zz−1;p.Remark 6.
ForA=B in Theorem 11, one can easily obtain(53)FA∗,B∗;C∗;z;p=1−z−A∗FA∗,C∗−B∗;C∗;zz−1;p,and also, for p=0,(54)F12A∗,B∗;C∗;z=1−z−A∗F12A∗,C∗−B∗;C∗;zz−1,and it is satisfied which is given in [25].Corollary 3.
From Theorem11, the GGHMF FA,BA∗,B∗;C∗;z;p satisfies the transformation formula:(55)FA,BA∗,B∗;C∗;1−1z;p=zA∗FA,BA∗,C∗−B∗;C∗;1−z;p,where argz<π.Remark 7.
In the case ofA=B in Corollary 3, we have(56)FA∗,B∗;C∗;1−1z;p=zA∗FA∗,C∗−B∗;C∗;1−z;p,and also, if we get p=0, we find(57)F12A∗,B∗;C∗;1−1z=zA∗F12A∗,C∗−B∗;C∗;1−z.Corollary 4.
For the GGHMF, the following transformation formula holds true:(58)FA,BA∗,B∗;C∗;zz+1;p=1+zA∗FA,BA∗,C∗−B∗;C∗;−z;p,where arg1+z<π.Theorem 12.
The GGHMFFA,BA∗,B∗;C∗;z;p verifies the recurrence relation:(59)BB∗+3I;C∗−B∗+3Ipd2dp2FA,BA∗,B∗+3I;C∗+6I;z;p−BB∗+2I;C∗−B∗+2IBddpFA,BA∗,B∗+2I;C∗+4I;z;p−BB∗+I;C∗−B∗+IpddpFA,BA∗,B∗+I;C∗+2I;z;p+AFA,BA∗,B∗;C∗;z;p=0.Proof.
By using relation (42), we can write the left-hand side of (59) in the following form:(60)∫01tB∗−I1−tC∗−B∗−Ipt1−td2Zdp2+dZdpBt1−t+pI+AZ1−tz−A∗dt,where Z=F11A;B;−p/t1−t. On the other hand, it follows from (8):(61)d2Zdp2pt1−t+dZdp1−tBt+pI+ZA=0,which proves the theorem.Theorem 13.
For the GCHMFF1A,B1B∗;C∗;z;p, the following recurrence relation holds true:(62)BB∗+3I;C∗−B∗+3Ipd2dp2F1A,B1B∗+3I;C∗+6I;z;p−BB∗+2I;C∗−B∗+2IBddpF1A,B1B∗+2I;C∗+4I;z;p−BB∗+I;C∗−B∗+IpddpF1A,B1B∗+I;C∗+2I;z;p+AF1A,B1B∗;C∗;z;p=0.Proof.
It is enough to make similar calculations as in Theorem11.Next, the integral transforms for the GGHMFFA,BA∗,B∗;C∗;z;p are given as follows:Theorem 14.
The following beta matrix transform formula holds true:(63)BFA,BP+Q,B∗;C∗;yz;p:P,Q=BP,QFA,BP,B∗;C∗;x;p,where A,B,B∗,C∗,P,Q,andP+Q are positive stable matrices and commutative in ℂr×r with Rep≥0,x<1, and the beta matrix transform of fz is defined as follows [25]:(64)Bfz:P,Q=∫01zP−I1−zQ−Ifzdz,where P and Q are positive stable matrices in ℂr×r.Proof.
Using relation (64) and applying (15) to the beta matrix transform of (63), we have(65)∫01zP−I1−zQ−IFA,BP+Q,B∗,C∗,xzdz∫01zP−I1−zQ−I∑n=0∞P+QnℬA,BB∗+nI,C∗−B∗;pB−1B∗,C∗−B∗xznn!dz.
By interchanging the order of integration and summation with (4), we obtain(66)∫01zP−I1−zQ−IFA,BP+Q,B∗;C∗;xz;pdz=∑n=0∞P+QnℬA,BB∗+nI,C∗−B∗;pB−1B∗,C∗−B∗ΓP+nIΓQΓ−1P+Q+nIynn!=BP,Q∑n=0∞PnℬA,BB∗+nI,C∗−B∗;pB−1B∗,C∗−B∗xnn!,which, according to (15), yields our desired result (63). This completes the proof of Theorem 13.Theorem 15.
IfRes>0,Rep≥0,M∈ℂr×r, and x/s<1, then the Laplace transform holds true:(67)ℒzM−IFA,BA∗,B∗,C∗,xz;p=s−MΓMFA,B1A∗,M,B∗;C∗;xs;p,where the Laplace transform of fz is defined as follows [26]:(68)ℒfz=∫0∞e−szfzdz.Proof.
From definition (68) and (15), we find that(69)∫0∞zM−Ie−szFA,BA∗,B∗;C∗;xz;pdz=∫0∞zM−Ie−sz∑n=0∞AnℬA,BB∗+In,C∗−B∗;pB−1B∗,C∗−B∗xznn!dz.
By interchanging the order of integration and summation and using Laplace transform, we have(70)∫0∞zl−1e−szFA,BA∗,B∗,C∗,xz;pdz=∑n=0∞AnℬA,BB∗,+nI,C∗−B∗;pB−1B∗,C∗−B∗sM+nIΓM+nIxnn!,which, upon using (21), yields our desired result (67).Theorem 16.
Ifρandδ∈ℂ, Rep≥0, and w/δ<1, then the following Whittaker transform formula hold true:(71)∫0∞tρ−1e−δt/2Wλ,μδtFA,BA∗,B∗;C∗,wt;pdt=δ−ρΓ1/2+μ+ρΓ1/2−μ+ρΓ1−λ+ρ×F1A,B2A∗,12+μ+ρI,12−μ+ρI,B∗;C∗,1−λ+ρI;wδ;p.Proof.
Starting withδt=v in L.H.S of (71), we obtain(72)T=∫0∞vδρ−1e−v/2Wλ,μν∑n=0∞A∗nB−1B∗,C∗−B∗ℬA,βB∗+nI,C∗−B∗;pwνnδnδn!dv.
By interchanging the order of integration and summation, we obtain(73)T=δ−ρ∑n=0∞A∗nBB∗,C∗−B∗ℬA,BB∗+nI,C∗−B∗;pwnδnn!×∫0∞νp+n−1e−ν/2Wλ,μνdν.
Using the following integral form (cf. [26]),(74)∫0∞tν−1e−t/2Wλ,μtdt=Γ1/2+μ+νΓ1/2−μ+νΓ1/2−λ+ν,ℛν±μ>−12,and (73) becomes the following equation:(75)T=δ−ρ∑n=0∞A∗nB−1B∗,C∗−B∗ℬA,BB∗+nI,C∗−B∗;p×wnδnn!Γ1/2+μ+ρ+nΓ1/2−μ+ρ+nΓ1−λ+ρ+n,and relation (15) evidently leads us to the required result.


## 4. Conclusion

This manuscript is a continuation of the recent papers [14–16, 18, 19]. In the current paper, we introduced new properties of extension of the gamma and beta matrix function. We also introduced new extensions of the Gauss hypergeometric matrix function and confluent hypergeometric matrix function. Then, we discussed certain properties of these extended matrix functions such as the integral representations, transformation formulae, recurrence relations, and integral transforms. In addition, some interesting special cases of our main results are archived.



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