On the Kampé de Fériet Hypergeometric Matrix Function

Department of Mathematics, Veer Bahadur Singh Purvanchal University, Jaunpur, India Department of Mathematics, Aden University, Aden, Yemen Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa China Medical University Hospital, China Medical University, Taichung 40402, Taiwan


Introduction
e theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Symbolic computation and engineering problems usually recognize the majority of special functions. Recently, there has been a surge in the study of recursion formulas for multivariable hypergeometric functions. Recursion formulas for the Appell function F 2 have been investigated by Opps et al. [1], followed by Wang [2], who presented the recursion relations for all Appell functions. Furthermore, recursion formulas for variant multivariable hypergeometric functions were presented in [3][4][5][6]. One can refer to various sources [7,8] for the in-depth study of the hypergeometric functions for several variables. e theory of generalized matrix special functions has witnessed a rather significant evolution during the last two decades. e reasons of interest have a manifold motivation. Restricting ourselves to the applicative field, we note that for some physical problems, the use of new classes of matrix special functions provided solutions hardly achievable with conventional analytical and numerical means. Special matrix functions appear in the literature related to statistics [9], Lie theory [10], and more recently in connection with the matrix version of Laguerre, Hermite, and Legendre differential equations and the corresponding polynomial families [11][12][13]. In [14], recursion formulas and matrix summation formulas for Srivastava's triple hypergeometric matrix functions are obtained. e study is organized in the following manner. In Section 2, we list basic definitions that are needed in the sequel. In Section 3, we obtain recursion formulas for the Kampé de Fériet hypergeometric matrix function (its abbreviation is K de FHMF). In Section 4, we present finite matrix summation formulas for the (K de FHMF) by applying a derivative operator. Finally, in Section 5, we establish infinite matrix summation formulas for the (K de FHMF).

Preliminaries
Let C r×r be the vector space of r square matrices with complex entries. For any matrix H ∈ C r×r , its spectrum σ(H) is the set of eigenvalues of H. H in C r×r is called a positive stable matrix if R(λ) > 0 for all λ ∈ σ(H). e reciprocal gamma function Γ − 1 (θ) � 1/Γ(θ) is an entire function of the complex variable θ. e image of Γ − 1 (θ) acting on H, denoted by Γ − 1 (H), is a well-defined matrix. If H + ℓI is invertible for all integers ℓ ≥ 0, then the reciprocal gamma function [15] is defined by where (H) ℓ is the shifted factorial matrix function for H ∈ C r×r given as [16] (H) ℓ � I, ℓ � 0, I being the r-square identity matrix. If H ∈ C r×r is a positive stable matrix and ℓ ≥ 1, then by [15], we have e Gauss hypergeometric matrix function [16] is defined by 2 for matrices A, B, and C in C r×r , such that C + kI is invertible for all k ≥ 0 and |x| ≤ 1. e Appell matrix functions are defined by where A, A ′ , B, B ′ , C, and C ′ are the positive stable matrices in C r×r , so that C + kI and C ′ + kI are invertible for each integer k ≥ 0. For regions of convergence of equations (3)-(6), see [17][18][19].
e Kampé de Fériet hypergeometric matrix function is given as [18,19] where A abbreviates the sequence of matrices A 1 , . . . , A m 1 , and A i , B i , C i , D i , E i , and F i are the positive stable matrices in C r×r , such that D i + kI, E i + kI and F i + kI are invertible for all integers k ≥ 0. Next, we recall the definition of the derivative operator for k � 0, 1, 2, . . .. In the whole study, I is the identity matrix and s is a nonnegative integer. In the sequel, consider A + sI � A 1 + sI, A 2 + sI, . . . , A m 1 + sI, Also, we denote

Recursion Formulas for the Kampé de Fériet Hypergeometric Matrix Function (K de FHMF)
In this section, we obtain the recursion formulas for the (K de FHMF). where and D i + kI, E i + kI, and F i + kI are invertible for each integer k ≥ 0.
Proof. In view of equation (7) and the fact that we get the following contiguous matrix relation: Replacing A i with A i + I in equation (14), we have the following contiguous matrix relation:

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Iterating this process s-times, we get equation (11). For the proof of equation (12), replace the matrix A i with A i − I in equation (14).
Iteratively, we get equation (12). Using contiguous matrix relations equations (14) and (16), we get the following forms of the recursion formulas for the (K de FHMF). □ Theorem 2. Let A i + sI, i � 1, . . . , m 1 be invertible for each integer s ≥ 0. en, the following recursion formula holds true for the (K de FHMF): Proof. We prove equation (17) by applying a mathematical induction on s. For s � 1, the result equation (17) is true due to equation (14). Assume equation (17) is true for s � t, that is, Replacing A i with A i + I in equation (19) and using the contiguous matrix relation equation (14), we get Applying the known relation n k + n k − 1 � n + 1 k and n k � 0 (for k> n or k < 0), the above identity can be reduced to the following result: is establishes equation (17) for s � t + 1. Hence, the result equation (17) is true for all values of s. e second recursion formula (18) is proved in a similar manner. Now, we present the recursion formulas for matrices B i and C i of the (K de FHMF). e proofs of the following results are omitted. where Also, if B i − kI is invertible for integers k ≤ s, we get  Next, we state the recursion formulas for the matrix D i of the (K de FHMF).   Proof. Applying the definition of the (K de FHMF) and the fact that

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the following contiguous matrix relation is obtained: With the help of this contiguous matrix relation to the (K de FHMF) with the matrix D i − sI for s-times, we get equation (26).
Next, we give recursion formulas for the (K de FHMF) E i , F i . e proof is omitted for the following result.

Finite Matrix Summation Formulas for the Kampé de Fériet Hypergeometric Matrix Function by a Derivative Operator
In this section, we obtain the finite matrix summation formulas for the (K de FHMF) by a derivative operator. ese formulas are matrix analogues for some summation formulas of double hypergeometric functions [8]. e p th derivative on y of the (K de FHMF) is obtained as follows: and D i F j � F j D i , and D i + kI, E i + kI, and F i + kI are invertible for each integer k ≥ 0. By using the generalized Leibnitz formula, and equation (30), we derive the following finite matrix summation formulas of the (K de FHMF).
Proof. From definition of the (K de FHMF) and the generalized Leibnitz formula for differentiation of a product of two functions, we have We used equation (30) and some simplification in the second equality. Next, we combine y C i +(p− 1)I with the variable y in the (K de FHMF) and apply the derivative operator p-times on y to get the following result: Equating the above two relations leads to equation (32).

Proof.
Applying the derivative operator on Applying a derivative operator and some transformations, we can get the finite matrix summation formulas of the (K de FHMF) as follows.

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D i E j � E j D i ; and D i F j � F j D i , and D i + kI, E i + kI, and F i + kI are invertible for each integer k ≥ 0. en, the following finite matrix summation formulas of the (K de FHMF) hold true: where Proof. We first prove identity equation (36). From the definition of the (K de FHMF) and the generalized Leibnitz formula for differentiation of a product of two functions, we obtain the following result: Now, using the derivative operator on the (K de FHMF) for r-times directly and equating with the above equality gives equation (36) after some simplifications. Next, applying the operator D r y on and proceeding as in the proof of equation (36) gives the result equation (37).

Infinite Summation Formulas for the Kampé de Fériet Hypergeometric Matrix Function
In this section, we will establish the infinite summation formulas of the (K de FHMF).
Proof: We shall prove equation (40). We apply the definition of the (K de FHMF) and transformation, to get that the left side of equation (40) x m y n m!n! . (43) Using the identity, and after simplifications, the right side of equation (40) Replace m + k by l in the above result. After some simplifications, we have ∞ l,n�0 Using the relation in the inner summation, finishes the proof of equation ( where A j B i � B i A j , i � 1, . . . , n 1 .

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Proof: From the definition of the (K de FHMF), the left side of equation (49) can be expressed as Taking k � m + l, changing the summation order and simplifying, we get Evaluating the inner 1 F 0 -series in the above equation by equation (44), and simplifying, we get the right side of this theorem. is completes the proof.
where A j B i � B i A j , i � 1, . . . , n 1 .
Proof: It is similar to eorem 12. We omit the details. □

Conclusion
We have investigated recursion formulas and some finite matrix and infinite matrix summation formulas involving the (K de FHMF). We remark that by specializing the sequence of matrices in the (K de FHMF), we can deduce recursion formulas and finite matrix and infinite matrix summation formulas for some Appell matrix functions [17][18][19].

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.