A Note on the Appell Hypergeometric Matrix Function <i>F</i><sub>2</sub>

Hidan, M.; Abdalla, M.

doi:https://doi.org/10.1155/2020/6058987

Mathematical Problems in Engineering

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Volume 2020 | Article ID 6058987 | https://doi.org/10.1155/2020/6058987

A Note on the Appell Hypergeometric Matrix Function F₂

M. Hidan¹and M. Abdalla^1,2

Guest Editor: Shilpi Jain

Received05 Feb 2020

Accepted27 Feb 2020

Published31 Mar 2020

Abstract

In this article, we introduce some of the mathematical properties of the second Appell hypergeometric matrix function F₂(A, B₁, B₂, C₁, C₂; z, ) including integral representations, transformation formulas, and series formulas.

1. Introduction

Appell defined and studied in [1–3] four kinds of double series of two variables z, as generalizations of the hypergeometric series:where z is a main variable in the unit disk are complex parameters with γ ≠ 0, −1, −2, −3, …, and and (α)₀ = 1. Here and throughout, let and denote the sets of complex numbers and positive integers, respectively, and let .

Appell hypergeometric functions F_s, 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 . 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 [26–30]. The purpose of the present work is to study the second Appell hypergeometric matrix function on the domain , with square matrix valued parameters A, B₁, B₂, C₁, and C₂ in . 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 matrix E is a positive stable matrix in 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 , 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 such that σ(E) ⊂ Ω; then, from the properties of the matrix functional calculus [28], it follows that

Hence, if F in is a matrix for which σ(F) ⊂ Ω and also if EF = FE, then

By application of the matrix functional calculus, for E in , then from [23, 31], the Pochhammer symbol or shifted factorial defined bywith the condition

From (5), it is easy to find that

From [28], one obtains

Definition 1. (see [31]). If E is a matrix in , 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 and EF = FE, then the Beta matrix function is well defined by

Definition 3. (see[23]). Suppose that N₁, N₂, and N₃ are matrices in , such that N₃ satisfies condition (5). Then, the hypergeometric matrix function ₂F₁(N₁, N₂; N₃; z) is given by

Definition 4. If E is the positive stable matrix in , then the Laguerre-type matrix polynomial is defined by [28]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 , then the Jacobi matrix polynomial is defined byUsing (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, B₁, B₂, C₁, and C₂ be commutative matrices in with C₁ + kI and C₂ + kI being invertible for all integers . Then, the second Appell hypergeometric matrix function is defined in the following form:

3. Main Results

In this section, we investigate some of the main properties of the second Appell hypergeometric matrix function F₂(A, B₁, B₂, C₁, C₂; z, ) such as integral representations, transformation formulas, and summation formulas

3.1. Integral Representations

Theorem 1. Let A, C₁, and C₂ be positive stable matrices in . Then, for |z| + || < 1, then the function F₂(A, B₁, B₂, C₁, C₂; z, ) defined in (14) can be represented in the following integer forms:

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 havewhich completes proof relation (16).

3.2. Transformation Formulas

Theorem 2. For the matrix function F₂(A, B₁, B₂, C₁, C₂; z, ), we have the following transformations:where A, B₁, B₂, C₁, and C₂ are commutative matrices in with C₁ + kI and C₂ + kI being invertible for all integer , 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]),in (15), we haveSubstituting 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 F₂(A, B, B′, C₁, C₂; z, ) be given in (14). The following formulas hold true: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 function by the following results.

Theorem 4. The following finite summation formula holds true:where A and C are positively stable in and z ⇌ w indicates the presence of a second term that originates from the first by interchanging z and .

Proof. Using (16), we find thatBy 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 F₂; it follows that

Theorem 5. Suppose that A and B are positively stable in such that B satisfies spectral condition (5), with and . The following generating matrix function holds true: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 haveThis completes the proof of Theorem 5.
Putting μ = 0 and then using the following formula,Thus, (32) reduces toNow, we shall see that (32) and (35) enable us to obtain some new formulas. By (29), we haveUsing (35), we arrive atWith the help of a generating function for Jacobi matrix polynomials (see [28, 32, 33]),We rewrite (35) aswhich yields

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

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

Acknowledgments

The authors wish to acknowledge the approval and the support of this research study from the Deanship of Scientific Research in King Khalid University, Abha, Saudi Arabia under grant (R.G.P-79-41).

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Copyright © 2020 M. Hidan and M. Abdalla. 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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