A Note on the Appell Hypergeometric Matrix Function F 2

where z is a main variable in the unit disk z ∈ C: |z|< 1 { }, α, β, c are complex parameters with c ≠ 0, − 1, − 2, − 3, . . ., and (α)n � α(α+1)(α+2) . . .(α+ n − 1) (n ∈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 functions Fs, 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]). (e 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, c; z) by extending parameters α, β, and c to square matrices A, B, and C in the complex space C. 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].(e purpose of the present work is to study the second Appell hypergeometric matrix function F2(A, 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 C. We investigate some of the mathematical properties of this matrix function and introduce new integral representations, transformation formulas, and summation formulas.

On the other hand, many authors [18][19][20][21][22][23][24][25] generalized the hypergeometric series F(α, β, c; z) by extending parameters α, β, and c to square matrices A, B, and C in the complex space C d×d . Recently, the extension of the classical Appell hypergeometric functions F s , s � {1, 2, 3, 4}, to the Appell hypergeometric matrix functions has been a subject of intensive studies [26][27][28][29][30]. e purpose of the present work is to study the second Appell hypergeometric matrix function F 2 (A, B 1 , B 2 , C 1 , C 2 ; z, w) on the domain (z, w) ∈ C 2 : |z| + |w| < 1 , with square matrix valued parameters A, B 1 , B 2 , C 1 , and C 2 in C d×d . We investigate some of the mathematical properties of this matrix function and introduce new integral representations, transformation formulas, and summation formulas.

Some Known Definitions and Results
We begin with a brief review of some definitions and no- is the set of all eigenvalues of E. I and 0 stand for the identity matrix and the null matrix in C d×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 C d×d such that σ(E) ⊂ Ω; then, from the properties of the matrix functional calculus [28], it follows that Hence, if F in C d×d is a matrix for which σ(F) ⊂ Ω and also if EF � FE, then

(3)
By application of the matrix functional calculus, for E in C d×d , then from [23,31], the Pochhammer symbol or shifted factorial defined by with the condition E + nI is invertible for all integers n ∈ N 0 .
From (5), it is easy to find that From [28], one obtains Definition 1 (see [31]). If E is a matrix in C d×d , such that Re(z) > 0 for all eigenvalues z of E, then Γ(E) is well defined as Definition 2 (see [31]). If E and F are positive stable matrices in C d×d and EF � FE, then the Beta matrix function is well defined by Definition 3 (see [23]). Suppose that N 1 , N 2 , and N 3 are matrices in C d×d , such that N 3 satisfies condition (5). en, the hypergeometric matrix function 2 F 1 (N 1 , N 2 ; N 3 ; z) is given by Definition 4. If E is the positive stable matrix in C d×d , then the Laguerre-type matrix polynomial is defined by [28] where 1 F 1 is the confluent hypergeometric matrix function (cf. [25]).
Definition 5 (see [28,32,33]). Let E and F be positive stable matrices in C d×d , then the Jacobi matrix polynomial 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. Let A, B 1 , B 2 , C 1 , and C 2 be commutative matrices in C d×d with C 1 + kI and C 2 + kI being invertible for all integers k ∈ N 0 . en, the second Appell hypergeometric matrix function F 2 (A, B 1 , B 2 , C 1 , C 2 ; z, w) is defined in the following form:

Main Results
In this section, we investigate some of the main properties of the second Appell hypergeometric matrix function F 2 (A, B 1 , B 2 , C 1 , C 2 ; z, w) such as integral representations, transformation formulas, and summation formulas

Integral Representations
Theorem 1. Let A, C 1 , and C 2 be positive stable matrices in C d×d . en, for |z| + |w| < 1, then the function F 2 (A, B 1 , B 2 , C 1 , C 2 ; z, w) defined in (14) can be represented in the following integer forms: 2 Mathematical Problems in Engineering 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 which completes proof relation (16).

Transformation Formulas
Theorem 2. For the matrix function F 2 (A, B 1 , B 2 , C 1 , C 2 ; z, w), we have the following transformations: where A, B 1 , B 2 , C 1 , and C 2 are commutative matrices in C d×d with C 1 + kI and C 2 + kI being invertible for all integer k ∈ N 0 , and B 1 , B 2 , C 1 , C 2 , C 1 − B1, and C 2 − B 2 are positively stable.
Proof. We will prove only (18) since the others can be proved similarly. Using matrix Kummer's first formula (cf. [8]), in (15), we have Substituting t � (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. Let F 2 (A, B, B′, C 1 , C 2 ; z, w) be given in (14). e following formulas hold true: where 2 F 1 is the Gauss hypergeometric matrix function defined in (11).

Mathematical Problems in Engineering
Proof.
where A and C are positively stable in C d×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 μ n�0 (C + I) n n! F 2 (A, − nI, − nI, C + I, C + I; z, w) By interchanging the order of summation and integration and applying the following formula [28]: 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 function F 2 ; it follows that where F 4 is the four Appell's matrix function defined in [27][28][29].
Proof. To prove (32), we require formula (19) and the relations (12); thus, we have is completes the proof of eorem 5.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.