# On the Extended Hypergeometric Matrix Functions and Their Applications for the Derivatives of the Extended Jacobi Matrix Polynomial

Authors: Fuli He; Ahmed Bakhet; M. Abdalla; M. Hidan
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

In this paper, we obtain some generating matrix functions and integral representations for the extended Gauss hypergeometric matrix function EGHMF and their special cases are also given. Furthermore, a specific application for the extended Gauss hypergeometric matrix function which includes Jacobi matrix polynomials is constructed.


---

## Body

## 1. Introduction

Generalizations of the classical special functions to matrix setting have become important during the previous years. Special matrix functions appear in solutions for some physical problems. Applications of special matrix functions also grow and become active areas in the recent literature including statistics, Lie groups theory, and differential equations (see, e.g., [1–4] 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 [5–10].Hypergeometric matrix functions are an interesting problem to study from a purely analytic point of view. These functions arise in the study of matrix-valued spherical functions and in the theory of matrix-valued orthogonal polynomials.Moreover, they appear in the practice of various fields of mathematics and engineering, so knowledge of them is necessary for applications of theories associated with these fields.In various areas of applications, generating functions and integral transformations for some families of hypergeometric functions is potentially useful (see [11–16]), especially in situations when these hypergeometric functions are involved in solutions of mathematical, physical, and engineering problems that can be modeled by ordinary and partial differential equations.The main object of this paper is to investigate various properties for the extended Gauss hypergeometric matrix function EGHMF. The generating functions and integral formulas are derived for EGHMF. We also present some special cases of the main results of this work. A specific application for the extended Gauss hypergeometric matrix function which includes Jacobi matrix polynomials is constructed.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 for all ξ∈σA, a matrix A in ℂr×r is a positive stable matrix. In [9, 17–19], 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.Notation 1.
For all A inℂr×r, and(1)A+nI,is invertible for all integersn,then the Pochhammer symbol is defined by [1, 20](2)An=AA+I⋯A+n−1I=ΓA+nIΓ−1A;A0≡I.
By inserting a regularization matrix factore−B/t,B∈ℂr×r. Abul-Dahab and Bakhet [6] have introduced the following generalization of the gamma matrix function.Definition 1.
Let A and B be positive stable matrices inCr×r; then, the generalized Gamma matrix function ΓA,B is defined by(3)ΓA,B=∫0∞tA−Ie−It+B/tdt,tA−I=expA−Ilnt,for B=0 reduces gamma matrix function in [21].
Also, Abdalla and Bakhet [7] considered the extension of Euler’s beta matrix function in the following definition.Definition 2.
Suppose thatA,B, and ℙ are positive stable and commutative matrices in ℂr×r satisfying spectral condition (1); then, the extended beta matrix function ℬA,B;ℙ is defined by(4)ℬA,B;ℙ≔∫01tA−I1−tB−Iexp−ℙt1−tdt.
Hence,(5)ℬA,B;ℙ=ΓA,ℙΓB,ℙΓ−1A+B;ℙ.
Forℙ=0, it obviously reduces to the beta matrix function in [21] by(6)BA,B=∫01tA−I1−tB−Idt.
Forp,q∈ℤ+, we will denote ΓA1…ΓApΓ−1B1…Γ−1Bq by(7)ΓA1,…,ApB1,…,Bq.
Later, Abdalla and Bakhet [8] used ℬA,B;ℙ to extend the Gauss hypergeometric matrix function in the following form:(8)FℙA,B;C;z=ΓCB,C−B×∑n=0∞AnℬB+nI,C−B;ℙznn!.
This matrix power series is seen to converge whenz<1. Also, for ℙ=0, it reduces to the usual Gauss hypergeometric matrix function F21A,B;C;z in [22]:(9)F21A,B;C;z=∑n=0∞AnBnCn−1znn!,where A,B, and C are the matrices in ℂr×r and C satisfying condition (1).Remark 1.
F
2
1
A
,
B
;
C
;
z is the special case of the well-known generalized hypergeometric matrix power series FqpAi;Bj;z defined by [5, 20](10)FqpAi;Bj;z=∑n≥0∏i=1pAin∏j=1qBjn−1znn!.
For commutative matricesAi,1≤i≤p, and for Bj,1≤j≤q, in ℂr×r such that(11)Bj+nIare invertible for all integersn≥0.
Some integral forms of the extended Gauss hypergeometric matrix function proved in [8] are given by(12)FℙA,B,C;z=ΓCB,C−B∫01tB−I1−tC−B−I×1−zt−Aexp−ℙt1−tdt,arg1−z<π,(13)FℙA,B,C;z=2ΓCB,C−B×∫0∞cosh2v−zsinh2v−Asinhv2B−I×coshv2A−C+Iexp−ℙcosh2vcoth2vdv,arg1−z<π,where CB=BC and C, B, and C−B are positive stable.
ForEGHMF, we have the following differential formula [8]:(14)dndznFℙA,B;C;z=AnBnCn−1×FℙA+nI,B+nI;C+nI;z.Definition 3 (see [20, 23]).
Let A and B be positive stable matrices inℂr×r; then, the Jacobi matrix polynomial (JMP) PnA,Bz is defined by(15)PnA,Bz=A+Inn!2F1−nI,A+B+n+1I;A+I;1−z2.


## 2. Generating Functions of the EGHMF

In several areas in applied mathematics and mathematical physics, generating functions play an important role in the investigation of various useful properties of the sequences which they generate. They are used to find certain properties and formulas for numbers and polynomials in a wide variety of research subjects, indeed, in modern combinatorics. One can refer to the extensive work of Srivastava and Manocha [24] for a systematic introduction and several interesting and useful applications of the various methods of obtaining linear, bilinear, bilateral, or mixed multilateral generating functions for a fairly wide variety of sequences of special functions (and polynomials) in one, two, and more variables, among much abundant literature; in this regard, in fact, a remarkable large number of generating functions involving a variety of special functions have been developed by many authors (see, e.g., [13, 25–27]). Here, we present some generating functions involving the following family of the extended Gauss hypergeometric matrix functions:Theorem 1.
LetFℙA,B;C;z be given in (8); then, the following generating function holds true:(16)∑r=0∞ArFℙA+rI,B,C;zwrr!=1−w−AFℙA,B,C;z1−w;z<1,w<1.Proof.
For convenience, let the left-hand side of (16) be denoted by T. Applying the series expression of (8) to T, we obtain(17)T=∑r=0∞ArΓCB,C−B×∑n=0∞A+rInℬB+nI,C−B;ℙznn!wrr!.
By changing the order of summations in (17), we obtain(18)T=ΓCB,C−B∑n=0∞AnℬB+nI,C−B;ℙ×∑r=0∞A+nIrwrr!znn!.
Furthermore, upon using the generalized binomial expansion, we find that the inner sum in (18) yields(19)∑r=0∞A+nIrwrr!=1−w−A+nI;w<1.
Finally, in view of (18) and (19), we obtain the desired result of Theorem 1.
A further generalization of the extended Gauss hypergeometric matrix functions (8) is given in the following definition.Definition 4.
In terms of the extended Gauss hypergeometric matrix function given by (8), we define a sequence Ωnzn∈ℕ0 as follows:(20)Ωnz=ΩnλA,B;C;z=FℙΔλ;A+nI,B,C;z;λ∈ℕ,where, for convenience, Δλ;A abbreviates the array of λ matrix parameters:(21)Aλ,A+Iλ,A+2Iλ,…,A+λ−1Iλ;λ∈ℕ.Remark 2.
In the extended Gauss hypergeometric matrix function occurring in definition (20), it is understood that the matrix parameter A of definition (8) has been replaced by a set of λ parameters which are abbreviated by Δλ;A+nI. The above definition (20) is motivated by the extensive investigation on this subject.
Now, we prove the following result, which provides the generating functions for the extended Gauss hypergeometric matrix functions defined above.Theorem 2.
For eachλ∈ℕ, the following generating function holds true:(22)∑n=0∞A+kInΩk+nλzwnn!=1−w−A+kIΩkλz1−wλ,where z<1,w<1, and k∈ℕ0.Proof.
Using the definitions (20) and (8) and changing the order of summation, the left-hand side ϒ of the result (22) is given by(23)ϒ=∑r=0∞A+kIλrA+k−1Iλr…A+k+λ−1Iλr×ΓCB,C−BℬB+rI,C−B;ℙ×∑n=0∞A+k+rλInwnn!zrr!.
Now, by appealing once again to (19), we easily arrive to the desired result (22) asserted by Theorem 2.Remark 3.
Furthermore, we note the following special cases of generating functions of the EGHMF as follows:(i)
It may be noted that if we setλ=1 and replace A by A−kI in (22), we readily obtain assertion (16) of Theorem 1(ii)
Atℙ=0, we observe that result (16) corresponds with that in [28](iii)
For arbitrary complex numbersα,β,γandη, putting A=α∈ℂ1×1 B=β∈ℂ1×1, C=γ∈ℂ1×1, and ℙ=η∈ℂ1×1 in (16) and (22), we find generating functions for the generalized Gauss hypergeometric function on [13, 27]


## 3. Integral Representations for the EGHMF

Integral formulas with such special matrix functions such as the beta matrix functions and the hypergeometric matrix functions are used in solving numerous applied problems. Hence, their demonstrated applications and several generalizations of integral transforms with hypergeometric matrix functions have been actively investigated. Here, by means of the extended beta matrix functionℬA,B;ℙ given in (4), we introduce some new generalized integral formulas for the EGHMF in this section.Theorem 3.
Forα∈ℂ, the extended Gauss hypergeometric matrix function satisfies the following integral relations:(i)
∫0∞zα−1FℙA,B;C;−zdz=ΓCB,C−BBA−αI,αIℬB−αI,C−B;ℙ(ii)
∫0∞exp−zzα−1FℙA,B;C;zdz=ΓCB,C−B∑n=0∞ℬB+nI,C−B;ℙαnΓαAn/n!(iii)
∫011−z2α−1FℙA,B;C;1−z/2dz=ΓCB,C−B∑n=0∞22α+n−2An×Bα+nI,αIℬB+nI,C−B;ℙ1/2n/n!,whereA,B,C, and C−B are the positive stable matrices in ℂr×rProof.
(i)
ReplacingFℙA,B;C;−z by its integral representation (12) and changing the order of integration, we get
(24)
∫
0
∞
z
α
−
1
F
ℙ
A
,
B
;
C
;
−
z
d
z
=
Γ
C
B
,
C
−
B
×
∫
0
1
t
B
−
I
1
−
t
C
−
B
−
I
exp
−
ℙ
t
1
−
t
∫
0
∞
z
α
−
1
1
+
t
z
−
A
d
t
d
z
.
Now, if we integrate with respect toz using the properties of beta matrix function and substitute u=tz, we will have
(25)
∫
0
∞
z
α
−
1
F
ℙ
A
,
B
;
C
;
−
z
d
z
=
Γ
C
B
,
C
−
B
∫
0
1
t
B
−
I
1
−
t
C
−
B
−
I
×
exp
−
ℙ
t
1
−
t
d
t
∫
0
∞
u
t
α
−
1
1
+
u
−
A
d
u
t
=
Γ
C
B
,
C
−
B
B
A
−
α
I
,
α
I
ℬ
B
−
M
,
C
−
B
;
ℙ
,
which is the required result ini.
(ii)
Direct calculations using (12) yield
(26)
F
ℙ
A
,
B
;
C
;
z
=
Γ
C
B
,
C
−
B
×
∫
0
1
t
B
−
I
1
−
t
C
−
B
−
I
1
−
z
t
−
A
exp
−
ℙ
t
1
−
t
d
t
.
Since1−zt−A=∑n=0∞Anztn/n!, then we have
(27)
∫
0
∞
exp
−
z
z
α
−
1
F
ℙ
A
,
B
;
C
;
z
d
z
=
Γ
C
B
,
C
−
B
∫
0
∞
∫
0
1
t
B
−
I
1
−
t
C
−
B
−
I
exp
−
z
z
α
−
1
×
exp
−
ℙ
t
1
−
t
∑
n
=
0
∞
A
n
n
!
z
t
n
d
t
d
z
=
Γ
C
B
,
C
−
B
∫
n
=
0
∞
A
n
1
n
!
∫
0
∞
∫
0
1
t
B
+
n
−
1
I
1
−
t
C
−
B
−
I
×
exp
−
z
z
α
+
n
−
1
exp
−
ℙ
t
1
−
t
d
t
d
z
=
Γ
C
B
,
C
−
B
∑
n
=
0
∞
A
n
1
n
!
∫
0
∞
exp
−
z
z
α
+
n
−
1
d
z
×
∫
0
1
t
B
+
n
−
1
I
1
−
t
C
−
B
−
I
exp
−
ℙ
t
1
−
t
d
t
=
Γ
α
Γ
C
B
,
C
−
B
∑
n
=
0
∞
A
n
α
n
ℬ
B
+
n
I
,
C
−
B
;
ℙ
1
n
!
.
Thus, we get the desired assertion (ii) of Theorem3.
(iii)
By using (12), it follows that
(28)
F
ℙ
A
,
B
;
C
;
1
−
z
2
=
Γ
C
B
,
C
−
B
×
∫
0
1
t
B
−
I
1
−
t
C
−
B
−
I
∑
n
=
0
∞
A
n
n
!
1
−
z
2
n
t
n
exp
−
ℙ
t
1
−
t
d
t
.
Thus,
(29)
∫
0
1
1
−
z
2
α
−
1
F
ℙ
A
,
B
;
C
;
1
−
z
2
d
z
=
Γ
C
B
,
C
−
B
∫
0
1
∫
0
1
1
−
z
2
α
−
1
t
B
−
I
1
−
t
C
−
B
−
I
×
exp
−
ℙ
t
1
−
t
∑
n
=
0
∞
A
n
n
!
1
−
z
2
n
t
n
d
t
d
z
.
Now, we putz=cosθ, 1−z=2sin2θ/2, and 1+z=2cos2θ/2 in (29); we obtain(30)∫011−z2α−1FℙA,B;C;1−z2dz=ΓCB,C−B∑n=0∞An2nn!×∫0π/2∫012sin2θ2n4sin2θ2cos2θ2α−1×sinθtB+n−1I1−tC−B−Iexp−ℙt1−tdθdt.
Hence, we easily arrive to the desired result (iii) asserted by Theorem3:(31)∫011−z2α−1FℙA,B;C;1−z2dz=ΓCB,C−B∑n=0∞An2nn!×∫0π/2∫0122α+n−1sin2α+2n−1θ2×cos2α−1θ2tB+n−1I1−tC−B−Iexp−ℙt1−tdθ2dt=ΓCB,C−B∑n=0∞22α+n−2AnBα+nI,αI×ℬB+nI,C−B;ℙ1/2nn!,which proves the assertion iii of Theorem 3.Remark 4.
It is worthy of note that the special cases can be obtained from formulasi, ii, and ii of Theorem 3 as follows:(1)
Takingℙ=0, we find some integral representations for the Gauss hypergeometric matrix function (GHMF) (cf. [10, 20, 22])(2)
Furthermore, choosinga,b,c,andη∈ℂ and setting A=a∈ℂ1×1,B=b∈ℂ1×1,C=c∈ℂ1×1 and ℙ=η∈ℂ1×1 in Theorem 3, we obtain some integral representations for Gauss hypergeometric function (cf. [14])


## 4. An Application of the Computation ofm Derivatives of the Extended Jacobi Matrix Polynomial

The Jacobi matrix polynomial and their special cases play important roles in approximation theory and its applications [20]. In this section, the extended Jacobi matrix polynomial ispresented and prove the following theorems for the mth derivatives of extended Jacobi matrix polynomials. Using the definition of the extended Gauss hypergeometric matrix functions EGHMF to define the extended matrix Jacobi polynomial and their special cases.Definition 5.
Let A, B, andℙ be positive stable matrices in ℂr×r whose eigenvalues, z, all satisfy Rez>−1. For any positive integer n, the nth extended Jacobi matrix polynomial is(32)PnA,B;ℙz=A+Inn!F21ℙ−nI,A+B+n+1I;A+I;1−z2.Theorem 4.
Let A, B, andℙ be positive stable matrices in ℂr×r. For the derivatives of extended Jacobi matrix polynomial, we find(33)DmPnA,B;ℙz=2−mA+B+n+1ImPn−mA+mI,B+mI;ℙz,where z<1 and D=d/dz.Proof.
Using (14) and (32) with the parameters A=−nI, B=A+B+n+1I, and C=A+I, we get(34)DmF21ℙ−nI,A+nI;A+I2;1−z2=−12m−nmA+B+n+1ImA+Im−1F21ℙ−n+mI,A+n+mI;A+;A+m+1I;1−z2,from (34) and multiplying by A+In/n!, we obtain(35)DmA+Inn!F21ℙ−nI,A+nI;A+I2;1−z2=A+Inn!−12m−nmA+B+n+1ImA+Im−1F21ℙ−n+mI,A+n+mI;A+;A+m+1I;1−z2.
Now making use of the extended of Jacobi matrix polynomial (32), we find(36)DmPnA,B;ℙz=A+Inn!−12m−nmA+B+n+1ImA+Im−1F21ℙ−n+mI,A+n+mI;A+;A+m+1I;1−z2.
By using (36), we get(37)DmPnA,B;ℙz=−12mΓ−n+mΓ−nn!A+B+n+1Im×Γ−1A+m+1IΓA+n+1I×F21ℙ−n+mI,A+n+mI;A+;A+m+1I;1−z2.
Further simplification yields(38)DmPnA,B;ℙz=2−mA+B+n+1ImΓ−1A+m+1IΓA+n+1In−m!×F21ℙ−n+mI,A+n+mI;A+;A+m+1I;1−z2=2−mA+B+n+1ImPn−mA+mI,B+mI;ℙz.
This completes the proof of Theorem4.Theorem 5.
LetA, B, C, D, and ℙ are positive stable matrices in ℂr×r and suppose that(39)PnC,D;ℙz=∑m=0nδnmA,B,C,DPmA,B;ℙz.Then,(40)δnmA,B,C,D=C+D+n+1IkC+m+1In−mm−n!×Γ−1A+B+2m+1IΓA+B+m+1I×F32−n+mI,C+D+m+n+1I,A+m+1I;C+m+1I,A+B+2m+1I;1.Proof.
Substitution of (40) into the RHS of (39) and making use of the extended of Jacobi matrix polynomial (32), we find(41)∑m=0nδnmA,B,C,DPmA,B;ℙz=C+D+n+1IkC+m+1In−mm−n!×Γ−1A+B+2m+1IΓA+B+m+1I×3F2−n+mI,C+D+m+n+1I,A+m+1I;C+m+1I,A+B+2m+1I;1Γ−1A+IΓA+m+1Im!F21ℙ−mI,A+B+m+1I;A+I;1−z2,since(42)F23−nI,A,B;C,D;1=Cn−1C−A−Bn−1C−AnC−Bn,and we have(43)∑m=0nδnmA,B,C,DPmA,B;ℙz=∑m=0nΓ−1A+B+2m+1IC+D+n+1IkΓA+B+m+1Im−n!×−A+D+n+m+1I−1−nI+Dn−mC−An−mΓ−1A+B+m+1IΓ−1−B+mIΓA+m+1Im!∑r=0∞Γ−m+rΓ−mℬA+B+m+r+1I,−B−mI;ℙ1−z/2rr!.
Expanding (43) and collecting similar terms, we obtain(44)∑m=0nδnmA,B,C,DPmA,B;ℙz=Γ−1C+D+n+1IΓ−1−D+nIΓC+n+1In!,∑r=0∞−nIrℬC+D+n+r+1I,−D−nI;ℙ1−z/2rr!=C+In!F21ℙ−nI,C+D+n+1I;C+I;1−z2=PnC,D;ℙz.
This completes the proof of Theorem5.
Finally, for the definition of extended of Jacobi matrix polynomial, we consider some of the extended special matrix polynomial as follows.Definition 6.
Let A andℙ be positive stable matrices in ℂr×r whose eigenvalues, z, all satisfy Rez>−1. For any positive integer n, the nth extended ultraspherical matrix polynomials are(45)CnA;ℙz=2Ann!F21ℙ−nI,A+nI;A+I2;1−z2.


### 4.1. Special Cases

Upon assigning particular values to the parameters and variables, we interestingly get a range of special cases for (32) and (45) as discussed below:(i)
SettingA=0 in (45), we find(46)Tn0;ℙz=F21ℙ−nI,nI;I2;1−z2,whereTn0;ℙz is called the extended Chebyshev matrix polynomial of the first kind.(ii)
PuttingA=I in (45), we have(47)Un0;ℙz=1n+11−nz2F21ℙ−n+1I,n+1I;3I2;1−z2,whereUn0;ℙz is called the extended Chebyshev matrix polynomial of the second kind.(iii)
Furthermore, takingA=I/2 in (45), we get(48)Pn0;ℙz=F21ℙ−nI,n+1I;I;1−z2,wherePn0;ℙz is called the extended Legendre matrix polynomial.(iv)
Forℙ=0 in (32), we get of Jacobi matrix polynomial in (15).(v)
Takingℙ=0 and putting {A=α∈ℂ1×1, and B=β∈ℂ1×1, } in (32), we get Jacobi polynomial as follows:(49)Pnα,βz=α+1nn!F21−n,1+α+β+n;α+1;121−z.


## 4.1. Special Cases

Upon assigning particular values to the parameters and variables, we interestingly get a range of special cases for (32) and (45) as discussed below:(i)
SettingA=0 in (45), we find(46)Tn0;ℙz=F21ℙ−nI,nI;I2;1−z2,whereTn0;ℙz is called the extended Chebyshev matrix polynomial of the first kind.(ii)
PuttingA=I in (45), we have(47)Un0;ℙz=1n+11−nz2F21ℙ−n+1I,n+1I;3I2;1−z2,whereUn0;ℙz is called the extended Chebyshev matrix polynomial of the second kind.(iii)
Furthermore, takingA=I/2 in (45), we get(48)Pn0;ℙz=F21ℙ−nI,n+1I;I;1−z2,wherePn0;ℙz is called the extended Legendre matrix polynomial.(iv)
Forℙ=0 in (32), we get of Jacobi matrix polynomial in (15).(v)
Takingℙ=0 and putting {A=α∈ℂ1×1, and B=β∈ℂ1×1, } in (32), we get Jacobi polynomial as follows:(49)Pnα,βz=α+1nn!F21−n,1+α+β+n;α+1;121−z.



---

*Source: 4268361-2020-03-24.xml*