# A Note on the Appell Hypergeometric Matrix FunctionF2

Authors: M. Hidan; M. Abdalla
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this article, we introduce some of the mathematical properties of the second Appell hypergeometric matrix functionF2(A, B1, B2, C1, C2; z, w) including integral representations, transformation formulas, and series formulas.


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## Body

## 1. Introduction

Appell defined and studied in [1–3] four kinds of double series of two variables z, w as generalizations of the hypergeometric series:(1)Fα,β,γ;z=∑n=0∞αnβnγn1nzn,where z is a main variable in the unit disk z∈ℂ:z<1,α,β,γ are complex parameters with γ ≠ 0, −1, −2, −3, …, and αn=αα+1α+2…α+n−1n∈ℕ and (α)0 = 1. Here and throughout, let ℂ and ℕ denote the sets of complex numbers and positive integers, respectively, and let ℕ0=ℕ∪0.Appell hypergeometric functionsFs, s = {1, 2, 3, 4} play an important role in mathematical physics in which broad practical applications can be found (see, e.g. [1, 3–7]). In particular, the Appell hypergeometric series F2 arises frequently in various physical and chemical applications ([8–11]). The exact solutions of number of problems in quantum mechanics have been given [6, 7, 9, 12] in terms of Appell’s function F2. For readers, they can find some results of the classical second Appell hypergeometric function F2 in [13–17].On the other hand, many authors [18–25] generalized the hypergeometric series F(α, β, γ; z) by extending parameters α, β, and γ to square matrices A, B, and C in the complex space ℂd×d. Recently, the extension of the classical Appell hypergeometric functions Fs, s = {1, 2, 3, 4}, to the Appell hypergeometric matrix functions has been a subject of intensive studies [26–30]. The purpose of the present work is to study the second Appell hypergeometric matrix function F2A,B1,B2,C1,C2;z,w on the domain z,w∈ℂ2:z+w<1, with square matrix valued parameters A, B1, B2, C1, and C2 in ℂd×d. We investigate some of the mathematical properties of this matrix function and introduce new integral representations, transformation formulas, and summation formulas.


## 2. Some Known Definitions and Results

We begin with a brief review of some definitions and notations. A matrixE is a positive stable matrix in ℂd×d if Re(λ) > 0 for all λ ∈ σ(E), where σ(E) is the set of all eigenvalues of E. I and 0 stand for the identity matrix and the null matrix in ℂd×d, respectively.If Φ(z) and Ψ(z) are holomorphic functions of the complex variable z, which are defined in an open set Ω of the complex plane and E is a matrix in ℂd×d such that σ(E) ⊂ Ω; then, from the properties of the matrix functional calculus [28], it follows that(2)ΦEΨE=ΨEΦE.Hence, ifF in ℂd×d is a matrix for which σ(F) ⊂ Ω and also if EF = FE, then(3)ΦEΨF=ΨFΦE.By application of the matrix functional calculus, forE in ℂd×d, then from [23, 31], the Pochhammer symbol or shifted factorial defined by(4)En=EE+I…E+n−1I=Γ−1EΓE+nI,n∈ℕ,I,n=0,with the condition(5)E+nIis invertible for all integersn∈ℕ0.From (5), it is easy to find that(6)En−k=−1kEnI−E−nIk−1;0≤k≤n,(7)Em+n=EnE+nImorEm+n=EmE+mIn.From [28], one obtains(8)−1kn−k!I=−nkn!I=−nIkn!;0≤k≤n.Definition 1.
(see [31]). If E is a matrix in ℂd×d, such that Re(z) > 0 for all eigenvalues z of E, then Γ(E) is well defined as(9)ΓE=∫0∞τE−Ie−τdτ,τE−I=expE−Ilnτ.Definition 2.
(see [31]). If E and F are positive stable matrices in ℂd×d and EF = FE, then the Beta matrix function is well defined by(10)BE,F=∫01τE−I1−τF−Idτ=Γ−1E+FΓEΓF.Definition 3.
(see[23]). Suppose that N1, N2, and N3 are matrices in ℂd×d, such that N3 satisfies condition (5). Then, the hypergeometric matrix function 2F1(N1, N2; N3; z) is given by(11)F21N1,N2;N3;z=∑n≥0N1nN2nN3n−1n!zn.Definition 4.
IfE is the positive stable matrix in ℂd×d, then the Laguerre-type matrix polynomial is defined by [28](12)LnEz=∑k=0n−1kzkk!n−k!E+InE+Ik−1=E+Inn!F11−nI;E+I;z,n∈ℕ0,where 1F1 is the confluent hypergeometric matrix function (cf. [25]).Definition 5.
(see[28, 32, 33]). Let E and F be positive stable matrices in ℂd×d, then the Jacobi matrix polynomial ℙnE,Fz is defined by(13)ℙnE,Fz=∑k=0nnkΓ−1E+F+n+1IΓ−1F+k+1IΓF+n+1IΓE+F+n+k+1I−1n+k1+zk2kn!,=E+Inn!F21−nI,E+F+n+1I;E+I;1−z2.
Using (6) and (11), we can write the second kind of two complex variables Appell hypergeometric matrix function in the following definition (see [26, 28]).Definition 6.
LetA, B1, B2, C1, and C2 be commutative matrices in ℂd×d with C1 + kI and C2 + kI being invertible for all integers k∈ℕ0. Then, the second Appell hypergeometric matrix function F2A,B1,B2,C1,C2;z,w is defined in the following form:(14)F2A,B1,B2,C1,C2;z,w=∑s1,s2=0∞As1+s2B1s1B2s2C1s1−1C2s2−1zs1ws2s1!s2!=∑s1=0∞As1B1s1C1s1−1F21A+s1I,B2;C2;wzs1s1!,z+w<1.


## 3. Main Results

In this section, we investigate some of the main properties of the second Appell hypergeometric matrix functionF2(A, B1, B2, C1, C2; z, w) such as integral representations, transformation formulas, and summation formulas


### 3.1. Integral Representations

Theorem 1.
LetA, C1, and C2 be positive stable matrices in ℂd×d. Then, for |z| + |w| < 1, then the function F2(A, B1, B2, C1, C2; z, w) defined in (14) can be represented in the following integer forms:(15)F2A,B1,B2,C1,C2;z,w=Γ−1A×∫0∞uA−Ie−uF11B1;C1;zuF11B2;C2;wudu,(16)F2A,−mI,−nI,C1+I,C2+I;z,w=m!n!C1+Im−1C2+In−1Γ−1A×∫0∞uA−Ie−uLmC1zuLnC2wudu.Proof.
Replacing the Pochhammer symbol (A)m+n in definition (14) by its integral representation which is obtained from (5) and (9), we get the desired result (15).
Using integral formula (15) and the relation given in (12), we have(17)F2A,−mI,−nI,C1+I,C2+I;z,w=Γ−1A∫0∞uA−Ie−uF11−mI;C1+I;zuF11−nI;C2+I;wudu=m!n!C1+Im−1C2+In−1Γ−1A×∫0∞uA−Ie−uLmC1zuLnC2wudu,which completes proof relation (16).


### 3.2. Transformation Formulas

Theorem 2.
For the matrix functionF2(A, B1, B2, C1, C2; z, w), we have the following transformations:(18)F2A,B1,B2,C1,C2;z,w=1−z−AF2A,C1−B1,B2,C1,C2;−z1−z,w1−w,(19)F2A,B1,B2,C1,C2;z,w=1−w−AF2A,B1,C2−B2,C1,C2;z1−w,ww−1,(20)F2A,B1,B2,C1,C2;z,w=1−z−w−AF2A,C1−B1,C2−B2,C1,C2;−z1−z−w,−w1−z−w,where A, B1, B2, C1, and C2 are commutative matrices in ℂd×d with C1 + kI and C2 + kI being invertible for all integer k∈ℕ0, and B1, B2, C1, C2, C1 − B1, and C2 − B2 are positively stable.Proof.
We will prove only (18) since the others can be proved similarly. Using matrix Kummer’s first formula (cf. [8]),(21)F11B;C;z=ezF11C−B;C;−z,in (15), we have(22)F2A,B1,B2,C1,C2;z,w=Γ−1A×∫0∞uA−Ie−1−zuF11C1−B1;C1;zuF11B2;C2;wudu.
Substitutingt = (1 − z)u into (22), we obtain formula (18).
Now, connections with the Gauss hypergeometric matrix function is considered by the following theorem:Theorem 3.
LetF2(A, B, B′, C1, C2; z, w) be given in (14). The following formulas hold true:(23)F2A,B1,B2,C1,C2;0,w=F21A,B2;C2;w,(24)F2A,B1,B2,C1,C2;z,0=F21A,B1;C1;z,(25)F2A,0,B2,C1,C2;z,w=F21A,B2;C2;z,(26)F2A,B1,0,C1,C2;z,w=F21A,B1;C1;z,(27)F2A,0,B2,C1,C2;z,0=1−z−AF21A,B2;C2;z1−z,(28)F2A,B1,0,C1,C2;0,w=1−w−AF21A,B1;C1;w1−w,where 2F1 is the Gauss hypergeometric matrix function defined in (11).Proof.
The proof of (23)–(26) is a direct consequence of definition (27). The relation (27) is obtained setting C1 = B1 in (18) and then using (25). Similarly, the relation (28) is derived setting C2 = B2 in (19) and then using (26).


### 3.3. Some Summation Formulas

We now present the summation formulas behavior of the second Appell hypergeometric matrix functionF2A,B1,B2,C1,C2;z,w by the following results.Theorem 4.
The following finite summation formula holds true:(29)∑n=0μC+Inn!F2A,−nI,−nI,C+I,C+I;z,w=A−I−1C+Iμ+1z−wμ!F2A−I,−μI,−μ+1I,C+I,C+I;z,w+z⇌w,where A and C are positively stable in ℂd×d and z ⇌ w indicates the presence of a second term that originates from the first by interchanging z and w.Proof.
Using (16), we find that(30)∑n=0μC+Inn!F2A,−nI,−nI,C+I,C+I;z,w=Γ−1A∑n=0μn!C+In−1∫0∞uA−Ie−uLμCzuLnCwudu.
By interchanging the order of summation and integration and applying the following formula [28]:(31)∑n=0μn!C+In−1LnCzLnCw=μ+1!C+Iμ−1z−w−1LμCzLμ+1Cw−Lμ+1CzLμCw,and then taking into consideration (16), we obtain formula (29).
To extend this theorem, we propose to obtain some more formulas centering around the Appell’s matrix functionF2; it follows thatTheorem 5.
Suppose thatA and B are positively stable in ℂd×d such that B satisfies spectral condition (5), with t<1,zt/1−w1−t<1 and w/1−w1−t<1. The following generating matrix function holds true:(32)∑n=0∞B+Iμ+nn!F2A,−nI,−n+μI;B+I,B+I;z,wtn=B+Iμ1−w−A1−t−B+1+μI×F4A,B+1+μI;B+I,B+I;−zt1−w1−t;−w1−w1−t.where F4 is the four Appell’s matrix function defined in [27–29].Proof.
To prove (32), we require formula (19) and the relations (12); thus, we have(33)∑n=0∞B+Iμ+nn!F2A,−nI,−n+μI;B+I,B+I;z,wtn=1−w−A×∑n=0∞B+Iμ+nn!F2A,−nI;B+μ+n+1I;B+I,B+I;z1−w;ww−1=1−w−A∑s,n=0∞∑r=0nAs+rB+Ir−1B+Is−1B+Iμ+n+r×tnn−r!s!r!−z1−wr−w1−ws=1−w−A∑s,r=0∞1s!r!As+rB+Ir−1B+Is−1B+Iμ+s+r×−zt1−wr−w1−ws∑n=0∞B+1+μ+s+rInn!=B+Iμ1−w−A1−t−B+1+μI×F4A,B+1+μI;B+I,B+I;−zt1−w1−t;−w1−w1−t.
This completes the proof of Theorem5.
Puttingμ = 0 and then using the following formula,(34)F4A,B;B,B;z,w=1−z−w−AF21A2,12A+I;B;4zw1−z−w2.
Thus, (32) reduces to(35)∑n=0∞B+Iμ+nn!F2A,−nI,−n+μI;B+I,B+I;z,wtn=1−w−A×∑n=0∞B+Iμ+nn!F2A,−nI;B+μ+n+1I;B+I,B+I;z1−w;ww−1=1−tA−B+I1−1−z−wt−AF21A2,12A+I;B+I;4zwt1−1−z−wt2.
Now, we shall see that (32) and (35) enable us to obtain some new formulas. By (29), we have(36)∑n=0∞A−1B+In+1z−wn!F2A−I,−nI,−n+1I;B+I,B+I;z,w+z⇌wtn=∑n=0∞∑r=0nB+Ikk!F2A,−kI;−kI;B+I,B+I;z,wtn.
Using (35), we arrive at(37)∑n=0∞B+In+1n!F2A−I,−nI,−n+1I;B+I,B+I;z,w+z⇌wtn=F2A,−kI;−kI;B+I,B+I;z,wtn=Az−w1−tA−B+2I1−1−z−wt−A×F21A2,12A+I;B+I;4zwt1−1−z−wt2.
With the help of a generating function for Jacobi matrix polynomials (see [28, 32, 33]),(38)∑n=0∞B+C+InB+In−1ℙnB,Cztn=1−t−B+C+I2F112B+C+I,12B+C+2I;B+I;2tz−11−t2.
We rewrite (35) as(39)∑n=0∞B+In+1n!F2B+C,−nI,−n+1I;B+I,B+I;z,w+z⇌wtn=B+C+Iz−w1−tC−I∑r=0∞B+C+IrB+Ir−1×ℙrB,C1−w1−z+zw1−z−w1−z−wrtr,which yields(40)ℙnB,C1−w1−z+zw1−z−w=B+InB+C+IB+C+In−1×z−w−11−z−w−n∑r=0nC−Irn−r!B+Ir+1r!×F2B+C,−rI,−r+1I;B+I,B+I;z,w+z⇌w.


## 3.1. Integral Representations

Theorem 1.
LetA, C1, and C2 be positive stable matrices in ℂd×d. Then, for |z| + |w| < 1, then the function F2(A, B1, B2, C1, C2; z, w) defined in (14) can be represented in the following integer forms:(15)F2A,B1,B2,C1,C2;z,w=Γ−1A×∫0∞uA−Ie−uF11B1;C1;zuF11B2;C2;wudu,(16)F2A,−mI,−nI,C1+I,C2+I;z,w=m!n!C1+Im−1C2+In−1Γ−1A×∫0∞uA−Ie−uLmC1zuLnC2wudu.Proof.
Replacing the Pochhammer symbol (A)m+n in definition (14) by its integral representation which is obtained from (5) and (9), we get the desired result (15).
Using integral formula (15) and the relation given in (12), we have(17)F2A,−mI,−nI,C1+I,C2+I;z,w=Γ−1A∫0∞uA−Ie−uF11−mI;C1+I;zuF11−nI;C2+I;wudu=m!n!C1+Im−1C2+In−1Γ−1A×∫0∞uA−Ie−uLmC1zuLnC2wudu,which completes proof relation (16).


## 3.2. Transformation Formulas

Theorem 2.
For the matrix functionF2(A, B1, B2, C1, C2; z, w), we have the following transformations:(18)F2A,B1,B2,C1,C2;z,w=1−z−AF2A,C1−B1,B2,C1,C2;−z1−z,w1−w,(19)F2A,B1,B2,C1,C2;z,w=1−w−AF2A,B1,C2−B2,C1,C2;z1−w,ww−1,(20)F2A,B1,B2,C1,C2;z,w=1−z−w−AF2A,C1−B1,C2−B2,C1,C2;−z1−z−w,−w1−z−w,where A, B1, B2, C1, and C2 are commutative matrices in ℂd×d with C1 + kI and C2 + kI being invertible for all integer k∈ℕ0, and B1, B2, C1, C2, C1 − B1, and C2 − B2 are positively stable.Proof.
We will prove only (18) since the others can be proved similarly. Using matrix Kummer’s first formula (cf. [8]),(21)F11B;C;z=ezF11C−B;C;−z,in (15), we have(22)F2A,B1,B2,C1,C2;z,w=Γ−1A×∫0∞uA−Ie−1−zuF11C1−B1;C1;zuF11B2;C2;wudu.
Substitutingt = (1 − z)u into (22), we obtain formula (18).
Now, connections with the Gauss hypergeometric matrix function is considered by the following theorem:Theorem 3.
LetF2(A, B, B′, C1, C2; z, w) be given in (14). The following formulas hold true:(23)F2A,B1,B2,C1,C2;0,w=F21A,B2;C2;w,(24)F2A,B1,B2,C1,C2;z,0=F21A,B1;C1;z,(25)F2A,0,B2,C1,C2;z,w=F21A,B2;C2;z,(26)F2A,B1,0,C1,C2;z,w=F21A,B1;C1;z,(27)F2A,0,B2,C1,C2;z,0=1−z−AF21A,B2;C2;z1−z,(28)F2A,B1,0,C1,C2;0,w=1−w−AF21A,B1;C1;w1−w,where 2F1 is the Gauss hypergeometric matrix function defined in (11).Proof.
The proof of (23)–(26) is a direct consequence of definition (27). The relation (27) is obtained setting C1 = B1 in (18) and then using (25). Similarly, the relation (28) is derived setting C2 = B2 in (19) and then using (26).


## 3.3. Some Summation Formulas

We now present the summation formulas behavior of the second Appell hypergeometric matrix functionF2A,B1,B2,C1,C2;z,w by the following results.Theorem 4.
The following finite summation formula holds true:(29)∑n=0μC+Inn!F2A,−nI,−nI,C+I,C+I;z,w=A−I−1C+Iμ+1z−wμ!F2A−I,−μI,−μ+1I,C+I,C+I;z,w+z⇌w,where A and C are positively stable in ℂd×d and z ⇌ w indicates the presence of a second term that originates from the first by interchanging z and w.Proof.
Using (16), we find that(30)∑n=0μC+Inn!F2A,−nI,−nI,C+I,C+I;z,w=Γ−1A∑n=0μn!C+In−1∫0∞uA−Ie−uLμCzuLnCwudu.
By interchanging the order of summation and integration and applying the following formula [28]:(31)∑n=0μn!C+In−1LnCzLnCw=μ+1!C+Iμ−1z−w−1LμCzLμ+1Cw−Lμ+1CzLμCw,and then taking into consideration (16), we obtain formula (29).
To extend this theorem, we propose to obtain some more formulas centering around the Appell’s matrix functionF2; it follows thatTheorem 5.
Suppose thatA and B are positively stable in ℂd×d such that B satisfies spectral condition (5), with t<1,zt/1−w1−t<1 and w/1−w1−t<1. The following generating matrix function holds true:(32)∑n=0∞B+Iμ+nn!F2A,−nI,−n+μI;B+I,B+I;z,wtn=B+Iμ1−w−A1−t−B+1+μI×F4A,B+1+μI;B+I,B+I;−zt1−w1−t;−w1−w1−t.where F4 is the four Appell’s matrix function defined in [27–29].Proof.
To prove (32), we require formula (19) and the relations (12); thus, we have(33)∑n=0∞B+Iμ+nn!F2A,−nI,−n+μI;B+I,B+I;z,wtn=1−w−A×∑n=0∞B+Iμ+nn!F2A,−nI;B+μ+n+1I;B+I,B+I;z1−w;ww−1=1−w−A∑s,n=0∞∑r=0nAs+rB+Ir−1B+Is−1B+Iμ+n+r×tnn−r!s!r!−z1−wr−w1−ws=1−w−A∑s,r=0∞1s!r!As+rB+Ir−1B+Is−1B+Iμ+s+r×−zt1−wr−w1−ws∑n=0∞B+1+μ+s+rInn!=B+Iμ1−w−A1−t−B+1+μI×F4A,B+1+μI;B+I,B+I;−zt1−w1−t;−w1−w1−t.
This completes the proof of Theorem5.
Puttingμ = 0 and then using the following formula,(34)F4A,B;B,B;z,w=1−z−w−AF21A2,12A+I;B;4zw1−z−w2.
Thus, (32) reduces to(35)∑n=0∞B+Iμ+nn!F2A,−nI,−n+μI;B+I,B+I;z,wtn=1−w−A×∑n=0∞B+Iμ+nn!F2A,−nI;B+μ+n+1I;B+I,B+I;z1−w;ww−1=1−tA−B+I1−1−z−wt−AF21A2,12A+I;B+I;4zwt1−1−z−wt2.
Now, we shall see that (32) and (35) enable us to obtain some new formulas. By (29), we have(36)∑n=0∞A−1B+In+1z−wn!F2A−I,−nI,−n+1I;B+I,B+I;z,w+z⇌wtn=∑n=0∞∑r=0nB+Ikk!F2A,−kI;−kI;B+I,B+I;z,wtn.
Using (35), we arrive at(37)∑n=0∞B+In+1n!F2A−I,−nI,−n+1I;B+I,B+I;z,w+z⇌wtn=F2A,−kI;−kI;B+I,B+I;z,wtn=Az−w1−tA−B+2I1−1−z−wt−A×F21A2,12A+I;B+I;4zwt1−1−z−wt2.
With the help of a generating function for Jacobi matrix polynomials (see [28, 32, 33]),(38)∑n=0∞B+C+InB+In−1ℙnB,Cztn=1−t−B+C+I2F112B+C+I,12B+C+2I;B+I;2tz−11−t2.
We rewrite (35) as(39)∑n=0∞B+In+1n!F2B+C,−nI,−n+1I;B+I,B+I;z,w+z⇌wtn=B+C+Iz−w1−tC−I∑r=0∞B+C+IrB+Ir−1×ℙrB,C1−w1−z+zw1−z−w1−z−wrtr,which yields(40)ℙnB,C1−w1−z+zw1−z−w=B+InB+C+IB+C+In−1×z−w−11−z−w−n∑r=0nC−Irn−r!B+Ir+1r!×F2B+C,−rI,−r+1I;B+I,B+I;z,w+z⇌w.



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