On the Kampé de Fériet Hypergeometric Matrix Function

Verma, Ashish; Younis, Jihad; Aydi, Hassen

doi:https://doi.org/10.1155/2021/9926176

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 9926176 | https://doi.org/10.1155/2021/9926176

On the Kampé de Fériet Hypergeometric Matrix Function

Ashish Verma,¹Jihad Younis,²and Hassen Aydi^3,4,5

Academic Editor: Sergio Ortobelli Lozza

Received18 Mar 2021

Revised25 Mar 2021

Accepted30 Jul 2021

Published18 Aug 2021

Abstract

In this study, we derive recursion formulas for the Kampé de Fériet hypergeometric matrix function. We also obtain some finite matrix and infinite matrix summation formulas for the Kampé de Fériet hypergeometric matrix function.

1. Introduction

The 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 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 [3–6]. One can refer to various sources [7, 8] for the in-depth study of the hypergeometric functions for several variables.

The theory of generalized matrix special functions has witnessed a rather significant evolution during the last two decades. The 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 [11–13]. In [14], recursion formulas and matrix summation formulas for Srivastava’s triple hypergeometric matrix functions are obtained.

The 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).

2. Preliminaries

Let be the vector space of square matrices with complex entries. For any matrix , its spectrum is the set of eigenvalues of . in is called a positive stable matrix if for all .

The reciprocal gamma function is an entire function of the complex variable . The image of acting on , denoted by , is a well-defined matrix. If is invertible for all integers , then the reciprocal gamma function [15] is defined by , where is the shifted factorial matrix function for given as [16]

being the -square identity matrix. If is a positive stable matrix and , then by [15], we have .

The Gauss hypergeometric matrix function [16] is defined byfor matrices , , and in , such that is invertible for all and .

The Appell matrix functions are defined bywhere , , , , , and are the positive stable matrices in , so that and are invertible for each integer . For regions of convergence of equations (3)–(6), see [17–19].

The Kampé de Fériet hypergeometric matrix function is given as [18, 19]where abbreviates the sequence of matrices , and , , , , , and are the positive stable matrices in , such that , and are invertible for all integers .

Next, we recall the definition of the derivative operatorprovided is differentiable at . Also, for .

In the whole study, is the identity matrix and is a nonnegative integer. In the sequel, consider

Also, we denote

3. 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).

Theorem 1. Let be invertible for each integer . Then, the following recursion formula holds true for the (K de FHMF):

Also, if is invertible for integers , thenwhere , , , , , and are the positive stable matrices in , such that ; ; ; ; ; ; and , and , , and are invertible for each integer .

Proof. In view of equation (7) and the fact thatwe get the following contiguous matrix relation:Replacing with in equation (14), we have the following contiguous matrix relation:Iterating this process -times, we get equation (11). For the proof of equation (12), replace the matrix with in equation (14). As is invertible, we haveIteratively, 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 be invertible for each integer . Then, the following recursion formula holds true for the (K de FHMF):

Also, if is invertible for integers , one writeswhere , , , , , and are the positive stable matrices in , such that ; ; ; ; ; ; and , and , , and are invertible for each integer .

Proof. We prove equation (17) by applying a mathematical induction on . For , the result equation (17) is true due to equation (14). Assume equation (17) is true for , that is,Replacing with in equation (19) and using the contiguous matrix relation equation (14), we getApplying the known relation and (for or ), the above identity can be reduced to the following result:This establishes equation (17) for . Hence, the result equation (17) is true for all values of . The second recursion formula (18) is proved in a similar manner.
Now, we present the recursion formulas for matrices and of the (K de FHMF). The proofs of the following results are omitted.

Theorem 3. Let be invertible for each integer . Then, the following recursion formula is satisfied for the (K de FHMF):

In addition, if is invertible for integers , one getswhere , , , , , and are the positive stable matrices in , such that ; ; ; ; and , and , , and are invertible for each integer .

Theorem 4. Let be invertible for each integer . Then, the following recursion formula is verified for the (K de FHMF):

Also, if is invertible for integers , we getwhere , , , , , and are the positive stable matrices in , such that ; ; ; ; and , and , , and are invertible for each integer .

The recursion formulas for are obtained by replacing , , and in Theorems 3 and 4.

Next, we state the recursion formulas for the matrix of the (K de FHMF).

Theorem 5. Let be invertible for each integer . Then, the following recursion formula holds true for the (K de FHMF):where , , , , , and are the positive stable matrices in , such that ; ; ; ; ; ; and , and , , and are invertible for each integer .

Proof. Applying the definition of the (K de FHMF) and the fact thatthe following contiguous matrix relation is obtained:With the help of this contiguous matrix relation to the (K de FHMF) with the matrix for -times, we get equation (26).
Next, we give recursion formulas for the (K de FHMF) , . The proof is omitted for the following result.

Theorem 6. Let be invertible for each integer . Then, the following recursion formula holds true for the (K de FHMF):where , , , , , and are the positive stable matrices in , such that ; ; ; ; and , and , , and are invertible for each integer .

The recursion formulas for are obtained by replacing , , and in Theorem 6.

4. 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. These formulas are matrix analogues for some summation formulas of double hypergeometric functions [8]. The ^th derivative on of the (K de FHMF) is obtained as follows:where , , , , , and are the positive stable matrices in , such that ; ; ; and , and , , and are invertible for each integer .

By using the generalized Leibnitz formula,and equation (30), we derive the following finite matrix summation formulas of the (K de FHMF).

Theorem 7. Let , , , , , and be the positive stable matrices in , such that ; ; ; ; and , and , , and are invertible for all integers . Then, the following finite matrix summation formulas hold for 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 haveWe used equation (30) and some simplification in the second equality. Next, we combine with the variable in the (K de FHMF) and apply the derivative operator -times on to get the following result:Equating the above two relations leads to equation (32).

Theorem 8. Let , , , , , and be the positive stable matrices in , such that ; ; ; ; and , and , , and are invertible for all integers . Then, the following finite matrix summation formulas of the (K de FHMF) hold true:where is an invertible matrix for and .

Proof. Applying the derivative operator on , -times, gives the formula as explained in the proof of Theorem 7. We omit the details.
Applying a derivative operator and some transformations, we can get the finite matrix summation formulas of the (K de FHMF) as follows.

Theorem 9. Let , , , , , and be the positive stable matrices in , such that ; ; ; and , and , , and are invertible for each integer . Then, the following finite matrix summation formulas of the (K de FHMF) hold true:where , , and is an invertible matrix for in (36); is an invertible matrix in equation (37), .

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 -times directly and equating with the above equality gives equation (36) after some simplifications. Next, applying the operator onand proceeding as in the proof of equation (36) gives the result equation (37).

5. 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).

Theorem 10. Let , , , , , and be the positive stable matrices in , such that , , and are invertible for each integer . Then, the following infinite summation formulas of the (K de FHMF) hold true:where ;where .

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) is written asUsing the identity,and after simplifications, the right side of equation (40) is obtained. This ends the proof of equation (40). The identity equation (41) is proved in a similar manner.

Theorem 11. Let , , , , , and be the positive stable matrices in , such that ; ; ; and , and , , and are invertible for each integer . Then, the infinite summation formulas of the (K de FHMF) hold true:

Proof:. From the definition of the (K de FHMF) and the transformation , the right side of equation (45) is expressed asReplace by in the above result. After some simplifications, we haveUsing the relationin the inner summation, finishes the proof of equation (45).

Theorem 12. Let , , , , , and be the positive stable matrices in , such that , , and are invertible for each integer . Then, the infinite summation formulas of the (K de FHMF) hold true:where , .

Proof:. From the definition of the (K de FHMF), the left side of equation (49) can be expressed asTaking , changing the summation order and simplifying, we getEvaluating the inner -series in the above equation by equation (44),and simplifying, we get the right side of this theorem. This completes the proof.

Theorem 13. Let , , , , , and be the positive stable matrices in , such that , , and are invertible for all integers . Then, the infinite summation formulas of the (K de FHMF) hold true:where , .

Proof:. It is similar to Theorem 12. We omit the details.

6. 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 [17–19].

Data Availability

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

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Copyright © 2021 Ashish Verma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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