On the two variables κ-Appell hypergeometric matrix functions

Abstract

This work aims to introduce $\kappa$-Appell hypergeometric matrix functions ${\mathcal{F}}_{1,\kappa },{\mathcal{F}}_{2,\kappa },{\mathcal{F}}_{3,\kappa }$ and ${\mathcal{F}}_{4,\kappa }$, where $\kappa \in \mathbb{N}$. New relations for these functions, such as the integral representation, important transformation formulas, infinite matrix summation and reduction formulas, are also presented. Finally, new results of the differential formulas associated with $\kappa$-Appell hypergeometric matrix functions ${\mathcal{F}}_{1,\kappa },{\mathcal{F}}_{2,\kappa }$ are derived.

KEYWORDS:

• κ-gamma matrix function
• κ-beta matrix function
• integral representation
• differential formula
• transformation formula

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 15A15
• 33C05
• 33C65
• 33E20
• 33C45

1. Introduction and preliminaries

Special functions play an important role in solving several problems in analysis, mathematical sciences, mathematical physics, engineering [1–5] and many applications because they represent the solutions of many famous differential equations. Specifically, Appell hypergeometric functions exist in several physical, chemical and engineering applications [6–11] and the exact solution of several problems in quantum mechanics is explored using Appell’s function [12,13]. Poul Appell introduced the Appell hypergeometric series ${\mathcal{F}}_1,{\mathcal{F}}_2,{\mathcal{F}}_3\text{ and }{\mathcal{F}}_4$ of two variables that generalize the Gauss hypergeometric series ${}_2{\mathcal{F}}_1$ of one variable in 1880.

Jodar and Cortes [14] investigated gamma and beta matrix function properties. Abdalla et al. [15] defined the extended gamma and beta matrix functions. Rahul Goyal et al. [16] studied the extended gamma and beta matrix functions using a two-parameters Mittag-Leffler matrix function. Mubeen et al. [17] introduced the $\kappa$-gamma and $\kappa$-beta matrix functions. Khammash et al. [18] introduced extended $\kappa$-gamma and $\kappa$-beta matrix functions. Many scholars have analysed these functions in recent years. Ravi Dwivedi et al. [19] defined Appell’s hypergeometric matrix functions and discussed integral representations and regions of convergence. Ravi Dwivedi and Vivek Sahai [20] presented infinite matrix summation formulas involving Appell matrix functions. M. Hidan and M. Abdalla [21] discussed integral representations, transformation formulas, and series formulas for the second Appell hypergeometric matrix function. M. Abdalla et al. [22] explored limit, some differentiation, summation and recursion formulas involving the second Appell function. Abdullah Altin et al. [23] derived the integral representation and demonstrated that matrix differential equations satisfy the first Appell hypergeometric matrix functions. H. Abd-Elmageed et al. [24] discussed generating matrix functions, recursion formulas, contiguous relations, differentiations and series for the first Appell hypergeometric matrix function. Verma and Yadav [25] introduced the incomplete second Appell hypergeometric matrix functions. This study aims to introduce and investigate $\kappa$-Appell hypergeometric matrix functions using the definition of the $\kappa$-Pochhammer symbol matrix and $\kappa$-gamma matrix function as well as analyse integral representations, differential formulas, transformation formulas, and infinite matrix summation formulas. The structure of this paper is organized as follows. We introduce the $\kappa$-Appell hypergeometric matrix functions ${\mathcal{F}}_{1,\kappa },{\mathcal{F}}_{2,\kappa },{\mathcal{F}}_{3,\kappa },\text{ and }{\mathcal{F}}_{4,\kappa }$ using $\kappa$-Pochhammer matrix symbols in Section 2. We provide integral representations for $\kappa$-Appell hypergeometric matrix functions in Section 3. We prove the transformation, infinite matrix summation formulas and some reduction formulas to demonstrate the relations for $\kappa$-hypergeometric matrix functions and $\kappa$-Appell matrix functions in Sections 4 and 5. Differentiation formulas for $\kappa$-Appell hypergeometric matrix functions ${\mathcal{F}}_{1,\kappa }\text{ and }{\mathcal{F}}_{2,\kappa }$ are derived in Section 6.

Throughout the paper, we use of the following notations:

${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$	=	Denotes the vector space of $\mathcal{M}$-square matrices with $\mathcal{M}$ rows and $\mathcal{M}$ columns where entries are complex numbers $\mathbb{C}$.
$\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }},\mathfrak{D}\text{ and }{\mathfrak{D}}^{\mathrm{\prime }}$	=	Denotes the matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$.
$z,w\in \mathbb{C}$	=	Denotes the complex variables.
$\kappa \in {\mathbb{R}}^{+}$	=	Denotes the positive real number.
$m,n\in {\mathbb{N}}^{+}$	=	Denotes the positive natural number.
$\sigma \left(\mathcal{Q}\right)$	=	Denotes the spectrum of the matrix $\mathcal{Q}.$
${}_2^{}{\mathcal{F}}_1\left({\mathcal{Q}}_1,{\mathcal{Q}}_2;{\mathcal{N}}_1;z\right)$	=	Denotes the Gauss hypergeometric matrix function.
${}_2^{}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N};\mathfrak{D};z\right)$	=	Denotes the $\kappa$-hypergeometric matrix functions.
${}_p^{}{\mathcal{F}}_q\left({\mathcal{Q}}_1,\dots ,{\mathcal{Q}}_p;{\mathcal{N}}_1,\dots ,{\mathcal{N}}_q;z\right)$	=	Denotes generalized hypergeometric matrix functions.
$(\mathcal{Q}{)}_m$	=	Denotes the Pochhammer symbol of the matrix argument.
$(\mathcal{Q}{)}_{m,\kappa }$	=	Denotes the Pochhammer $\kappa$-symbol of the matrix argument.
${\mathrm{\Gamma }}_{\kappa }\left(\mathcal{Q}\right)$	=	Denotes the $\kappa$-gamma matrix function.
${\mathfrak{B}}_{\kappa }\left(\mathcal{Q},\mathcal{N}\right)$	=	Denotes the $\kappa$-beta matrix function.
${\mathcal{F}}_1\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)$	=	Denotes the first Appell hypergeometric matrix functions of two variables.
${\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)$	=	Denotes the first $\kappa$-Appell hypergeometric matrix functions of two variables.
${\mathcal{F}}_2\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)$	=	Denotes the second Appell hypergeometric matrix functions of two variables.
${\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)$	=	Denotes the second $\kappa$-Appell hypergeometric matrix functions of two variables.
${\mathcal{F}}_3\left(\mathcal{Q},\mathcal{Q}{}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)$	=	Denotes the third Appell hypergeometric matrix functions of two variables.
${\mathcal{F}}_{3,\kappa }\left(\mathcal{Q},\mathcal{Q}{}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)$	=	Denotes the third $\kappa$-Appell hypergeometric matrix functions of two variables.
${\mathcal{F}}_4\left(\mathcal{Q},\mathcal{N};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)$	=	Denotes the fourth Appell hypergeometric matrix functions of two variables.
${\mathcal{F}}_{4,\kappa }\left(\mathcal{Q},\mathcal{N};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)$	=	Denotes the fourth $\kappa$-Appell hypergeometric matrix functions of two variables.

The generalized hypergeometric matrix functions are defined as follows [19]: (1) $\begin{array}{r}{}_p{\mathcal{F}}_q\left({\mathcal{Q}}_1,\dots ,{\mathcal{Q}}_p;{\mathcal{N}}_1,\dots ,{\mathcal{N}}_q;z\right)=\sum _{m\ge 0}\prod _{i=1}^p\left({\mathcal{Q}}_i\right)m\prod _{i=1}^q[{\left({\mathcal{N}}_j\right)}_m{]}^{-1}\frac{z^m}{m!},p=q+1,|z|<1,\end{array}$(1) where $z\in \mathbb{C},m\in {\mathbb{N}}^{+}\text{ and }{\mathcal{Q}}_i,{\mathcal{N}}_j$ are matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}},1\le i\le p,1\le j\le q.$ Let $p$ and $q$ be finite positive integers, such that ${\mathcal{N}}_j+m\text{I}$ is an invertible matrix for all integrals $m\ge 0$.

Remark 1.1:

We obtain the following Gauss hypergeometric function given the special case of Equation (2), when $p=2$ and $q=1$ [14]: (2) $\begin{array}{r}{}_2{\mathcal{F}}_1\left({\mathcal{Q}}_1,{\mathcal{Q}}_2;{\mathcal{N}}_1;z\right)=\sum _{m\ge 0}\frac{{\left({\mathcal{Q}}_1\right)}_m{\left({\mathcal{Q}}_2\right)}_m({\mathcal{N}}_1{)}_m^{-1}}{m!}{z}^m,\end{array}$(2) where $(\mathcal{Q}{)}_m$ is the Pochhammer symbol of a matrix argument and expressed as (3) $\begin{array}{r}\{\begin{array}{c}{\left(\mathcal{Q}\right)}_m=Q\left(\mathcal{Q}+I\right)\left(\mathcal{Q}+2I\right)\dots \left(\mathcal{Q}+\left(m-1\right)I\right)=\mathrm{\Gamma }\left(\mathcal{Q}+mI\right){\mathrm{\Gamma }}^{-1}\left(\mathcal{Q}\right),m\ge 1,\\ {\left(\mathcal{Q}\right)}_0=I,m=0.\end{array}\end{array}$(3) Let ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$ be a vector space of $\mathcal{M}$-square matrices with $\mathcal{M}$ rows and $\mathcal{M}$ columns where entries are complex numbers $\mathbb{C}$. The $Im\left(z\right)$ and $Re\left(z\right)$ denote the imaginary and real parts of $z\in \mathbb{C}$, respectively. For any matrix $\mathcal{Q}$ in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, $\sigma \left(\mathcal{Q}\right)$ denotes the spectrum of the matrix $\mathcal{Q}$ and $\sigma \left(\mathcal{Q}\right)\text{is}$ the set of all eigenvalues of $\mathcal{Q}$ and defined in [14] as follows: (4) $\begin{array}{r}\alpha \left(\mathcal{Q}\right)=max\left\{Re\left(z\right):z\in \sigma \left(\mathcal{Q}\right)\right\},\text{β}\left(\mathcal{Q}\right)=min\left\{Re\left(z\right):z\in \sigma \left(\mathcal{Q}\right)\right\},\end{array}$(4) where $\alpha \left(\mathcal{Q}\right)$ is referred to as the spectral abscissa of matrix $\mathcal{Q}$ and $\alpha \left(-\mathcal{Q}\right)=-\text{β}\left(\mathcal{Q}\right)$. The square matrix $\mathcal{Q}$ is considered positive stable if and only if $Re\left(z\right)>0$, $\mathrm{\forall }z\in \sigma \left(\mathcal{Q}\right)$.

A matrix norm is the vector norm on ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$. ||$\mathcal{Q}$|| denotes the norm of the matrix $\mathcal{Q}$ and the 2-norm for vectors is defined as follows: (5) $\begin{array}{r}||\mathcal{Q}||=sup\frac{||\mathcal{Q}u||{}_2}{||u||{}_2}=max\left\{\sqrt{\lambda }:\lambda \in \sigma \left({\mathcal{Q}}^{\ast }\mathcal{Q}\right)\right\},\end{array}$(5) where $u$ is any vector in the $\mathcal{M}\text{th}$ complex plane, $||u||{}_2=(u\ast u{)}^{\frac{1}{2}}$ is a Euclidean norm of $u$, and ${\mathcal{Q}}^{\ast }$ denotes the transposed conjugate of $\mathcal{Q}$. $0$ and $I$ stand for the null and identity matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, respectively. Let $\mathcal{Q}$ matrix in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}},\mathrm{\Omega }$ is an open set of the complex plan with $\sigma \left(\mathcal{Q}\right)\subset \mathrm{\Omega }$, if $\psi \left(z\right)\text{ and }\varphi \left(z\right)$ are holomorphic functions of a complex variable $z$, which are defined in $\mathrm{\Omega },$ then from the properties of the matrix functional calculus (see [26,p.558]), it follows that $\psi \left(z\right)\varphi \left(z\right)=\varphi \left(z\right)\psi \left(z\right)$. Furthermore, in [27], if the $\mathcal{G}$ matrix in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, which $\sigma \left(\mathcal{G}\right)\subset \mathrm{\Omega }$, and if $\mathcal{Q}\mathcal{G}=\mathcal{G}\mathcal{Q}$, then $\psi \left(\mathcal{Q}\right)\varphi \left(\mathcal{G}\right)=\varphi \left(\mathcal{G}\right)\psi \left(\mathcal{Q}\right)$.

According to [17], let $(\mathcal{Q}{)}_{m,\kappa }$ be the Pochhammer $\kappa$-symbol of the matrix argument given by (6) $\begin{array}{r}(\mathcal{Q}{)}_{m,\kappa }=\mathcal{Q}\left(\mathcal{Q}+\kappa I\right)\left(\mathcal{Q}+2\kappa I\right)\dots \left(\mathcal{Q}+\left(m-1\right)\kappa I\right)={\mathrm{\Gamma }}_{\kappa }\left(\mathcal{Q}+m\kappa I\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}\right),m\ge 1.\end{array}$(6)

Proposition 1.1:

[28] The following identity holds for any $\kappa \in {\mathbb{R}}^{+},z\in \mathbb{C}$ and $\mathcal{Q}$ be matrix in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$: (7) $\begin{array}{r}\sum _{m\ge 0}(\mathcal{Q}{)}_{m,\kappa }\frac{z^m}{m!}=(1-\kappa z{)}^{\frac{-\mathcal{Q}}{\kappa }},|z|<1,\end{array}$(7) where $(\mathcal{Q}{)}_{m,\kappa }$ is the Pochhammer $\kappa$-symbol of the matrix argument defined in Equation (6).

Definition 1.1:

Let $\mathcal{Q}$ and $\mathcal{N}$ be two positive stable matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$. The integral representation of the $\kappa$-gamma matrix function ${\mathrm{\Gamma }}_{\kappa }\left(\mathcal{Q}\right)$ and the $\kappa$-beta matrix function ${\mathfrak{B}}_{\kappa }\left(\mathcal{Q},\mathcal{N}\right)$ defined by Mubeen et al. [17], for $\kappa >0$, is as follows: (8) $\begin{array}{r}{\mathrm{\Gamma }}_{\kappa }\left(\mathcal{Q}\right)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{\mathcal{Q}-I}{e}^{\frac{-t^{\kappa }}{\kappa }}\text{d}t,{\mathfrak{B}}_{\kappa }\left(\mathcal{Q},\mathcal{N}\right)=\frac{1}{\kappa }\underset{0}{\overset{1}{\int }}{t}^{\frac{\mathcal{Q}}{\kappa }-I}(1-t{)}^{\frac{\mathcal{N}}{\kappa }-I}\text{d}t.\end{array}$(8) Since the reciprocal $\kappa$-gamma function denoted by ${{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(z\right)=\frac{1}{{\mathrm{\Gamma }}_{\kappa }\left(z\right)}$ is an entire function of the complex variable $z$. For any matrix $\mathcal{Q}$ in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$ the image of ${{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(z\right)$ acting on $\mathcal{Q}$, denoted by ${{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}\right)$, is the well-defined matrix for $\kappa >0$. Moreover, according to Mubeen et al. [29], if $\mathcal{Q}$ is a matrix in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$ such that (9) $\begin{array}{r}\mathcal{Q}+n\kappa I\text{ is the invertible matrix for every integer }n\ge 0\text{ and }\kappa >0.\end{array}$(9) Then, by application of the matrix functional calculus,${\mathrm{\Gamma }}_{\kappa }\left(\mathcal{Q}\right)$ is invertible and its inverse coincides with ${{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}\right)$ and (10) $\begin{array}{r}\mathcal{Q}\left(\mathcal{Q}+\kappa I\right)\left(\mathcal{Q}+2\kappa I\right)\dots \left(\mathcal{Q}+\left(m-1\right)\kappa I\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}+m\kappa I\right)={{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}\right),\end{array}$(10) where $m\ge 1\text{ and }\kappa >0$.

The relation between the Pochhammer $\kappa$-symbol of the matrix argument and the $\kappa$-gamma matrix function is expressed as follows [17]: (11) $\begin{array}{r}(\mathcal{Q}{)}_{m,\kappa }={\mathrm{\Gamma }}_{\kappa }\left(\mathcal{Q}+m\kappa I\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}\right),m\ge 1\text{ and }\kappa >0.\end{array}$(11) The matrices $\mathcal{Q}$ and $\mathcal{N}$ are positive stable matrices and commuting matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$ (diagonalizable matrices such that $\mathcal{Q}\mathcal{N}=\mathcal{N}\mathcal{Q}$) so that $\mathcal{Q}+n\kappa I,\mathcal{N}+n\kappa I,$ and $\mathcal{Q}+\mathcal{N}+n\kappa I$ are the invertible matrices for every integer $n\ge 0$and $\kappa >0$. Then, we have [17] (12) $\begin{array}{r}{\mathfrak{B}}_{\kappa }\left(\mathcal{Q},\mathcal{N}\right)={\mathrm{\Gamma }}_{\kappa }\left(\mathcal{Q}\right){\mathrm{\Gamma }}_{\kappa }\left(\mathcal{N}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}+\mathcal{N}\right).\end{array}$(12) Mubeen et al. [17] proved that the generalized $\kappa$-gamma matrix function for any positive stable matrix $\mathcal{Q}$ in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$ is (13) $\begin{array}{r}{\mathrm{\Gamma }}_{\kappa }\left(\mathcal{Q}\right)=\underset{m\to \mathrm{\infty }}{lim}m!{\kappa }^m(m\kappa {)}^{\frac{\mathcal{Q}}{\kappa }-I}(\mathcal{Q}{)}_{m,\kappa }^{-1},\end{array}$(13) where $m\ge 1$ is an integer,$\kappa >0,$ and $I$ is the identity matrix.

Definition 1.2:

Let $\kappa \in {\mathbb{R}}^{+},z\in \mathbb{C},m\in {\mathbb{N}}^{+}\text{ and }\mathcal{Q},\mathcal{N}\text{ and }\mathfrak{D}$ be matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, such that $\mathfrak{D}+m\kappa I$ is the invertible matrix for all integrals $m\ge 0$. Then, we define the $\kappa$-hypergeometric matrix functions as follows [28]: (14) $\begin{array}{r}{}_2{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N};\mathfrak{D};z\right)=\sum _{m\ge 0}\frac{{\left(\mathcal{Q}\right)}_{m,\kappa }{\left(\mathcal{N}\right)}_{m,\kappa }(\mathfrak{D}{)}_{m,\kappa }^{-1}}{m!}{z}^m.\end{array}$(14) Appell hypergeometric matrix functions have been extensively investigated in recent years (see, [19–24,30–35]).

2. κ-Appell hypergeometric matrix function

We introduce the $\kappa$-Appell matrix functions of two variables in this section.

Definition 2.1:

Let $\kappa \in {\mathbb{R}}^{+},z,w\in \mathbb{C},m,n\in {\mathbb{N}}^{+}\text{ and }\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }},\mathfrak{D}$, ${\mathfrak{D}}^{\mathrm{\prime }}$ be matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, such that $\mathfrak{D}+n\kappa I\text{and }{\mathfrak{D}}^{\mathrm{\prime }}+n\kappa I$ are invertible matrices for all integrals $n\ge 0$. We then define the $\kappa$-Appell hypergeometric matrix functions as follows: (15) $\begin{array}{rl}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)& =\sum _{m,n\ge 0}\frac{{\left(\mathcal{Q}\right)}_{m+n,\kappa }{\left(\mathcal{N}\right)}_{m,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{n,\kappa }(\mathfrak{D}{)}_{m+n,\kappa }^{-1}}{m!n!}{z}^m{w}^n,\end{array}$(15) (16) $\begin{array}{rl}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},\mathfrak{D}{}^{\mathrm{\prime }};z,w\right)& =\sum _{m,n\ge 0}\frac{{\left(\mathcal{Q}\right)}_{m+n,\kappa }{\left(\mathcal{N}\right)}_{m,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{n,\kappa }\left(\mathfrak{D}{)}_{m,\kappa }^{-1}\right({\mathfrak{D}}^{\mathrm{\prime }}{)}_{n,\kappa }^{-1}}{m!n!}{z}^m{w}^n,\end{array}$(16) (17) $\begin{array}{rl}{\mathcal{F}}_{3,\kappa }\left(\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)& =\sum _{m,n\ge 0}\frac{{\left(\mathcal{Q}\right)}_{m,\kappa }{\left({\mathcal{Q}}^{\mathrm{\prime }}\right)}_{n,\kappa }{\left(\mathcal{N}\right)}_{m,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{n,\kappa }(\mathfrak{D}{)}_{m+n,\kappa }^{-1}}{m!n!}{z}^m{w}^n,\end{array}$(17) (18) $\begin{array}{rl}{\mathcal{F}}_{4,\kappa }\left(\mathcal{Q},\mathcal{N};\mathfrak{D},\mathfrak{D}{}^{\mathrm{\prime }};z,w\right)& =\sum _{m,n\ge 0}\frac{{\left(\mathcal{Q}\right)}_{m+n,\kappa }{\left(\mathcal{N}\right)}_{m+n,\kappa }\left(\mathfrak{D}{)}_{m,\kappa }^{-1}\right({\mathfrak{D}}^{\mathrm{\prime }}{)}_{n,\kappa }^{-1}}{m!n!}{z}^m{w}^n,\end{array}$(18) where $|z|<\frac{1}{\kappa },|w|<\frac{1}{\kappa };|z|+|w|<\frac{1}{\kappa };|z|<\frac{1}{\kappa },|w|<\frac{1}{\kappa };\sqrt{|z|}+\sqrt{|w|}<\frac{1}{\sqrt{\kappa }}$, respectively.

Remark 2.1:

The special cases of $\kappa$-Appell matrix functions in Definition 2.1 are presented as follows:

(1) When $\kappa =1$, then $\kappa$-Appell matrix functions ${\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)$, ${\mathcal{F}}_{2,\kappa }(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w),{\mathcal{F}}_{3,\kappa }\left(\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right),{\mathcal{F}}_{4,\kappa }\left(\mathcal{Q},\mathcal{N};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)$ reduces to Appell matrix functions ${\mathcal{F}}_1\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right),{\mathcal{F}}_2\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)$, ${\mathcal{F}}_3(\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w),{\mathcal{F}}_4\left(\mathcal{Q},\mathcal{N};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)$ in [19,23,36], as following form (19) $\begin{array}{rl}{\mathcal{F}}_1\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)& =\sum _{m,n\ge 0}\frac{{\left(\mathcal{Q}\right)}_{m+n}{\left(\mathcal{N}\right)}_m{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_n(\mathfrak{D}{)}_{m+n}^{-1}}{m!n!}{z}^m{w}^n,\end{array}$(19) (20) $\begin{array}{rl}{\mathcal{F}}_2\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},\mathfrak{D}{}^{\mathrm{\prime }};z,w\right)& =\sum _{m,n\ge 0}\frac{{\left(\mathcal{Q}\right)}_{m+n}{\left(\mathcal{N}\right)}_m{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_n\left(\mathfrak{D}{)}_m^{-1}\right({\mathfrak{D}}^{\mathrm{\prime }}{)}_n^{-1}}{m!n!}{z}^m{w}^n,\end{array}$(20) (21) $\begin{array}{rl}{\mathcal{F}}_3\left(\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)& =\sum _{m,n\ge 0}\frac{{\left(\mathcal{Q}\right)}_m{\left({\mathcal{Q}}^{\mathrm{\prime }}\right)}_n{\left(\mathcal{N}\right)}_m{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_n(\mathfrak{D}{)}_{m+n}^{-1}}{m!n!}{z}^m{w}^n,\end{array}$(21) (22) $\begin{array}{rl}{\mathcal{F}}_4\left(\mathcal{Q},\mathcal{N};\mathfrak{D},\mathfrak{D}{}^{\mathrm{\prime }};z,w\right)& =\sum _{m,n\ge 0}\frac{{\left(\mathcal{Q}\right)}_{m+n}{\left(\mathcal{N}\right)}_{m+n}\left(\mathfrak{D}{)}_m^{-1}\right({\mathfrak{D}}^{\mathrm{\prime }}{)}_n^{-1}}{m!n!}{z}^m{w}^n,\end{array}$(22) where $|z|,|w|<1;|z|+|w|<1;|z|,|w|<1;\sqrt{|z|}+\sqrt{|w|}<1$, respectively. $\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }},\mathfrak{D}$, ${\mathfrak{D}}^{\mathrm{\prime }}$ are matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, such that $\mathfrak{D}+nI\text{ and }{\mathfrak{D}}^{\mathrm{\prime }}+nI$ are invertible matrices for all integrals $n\ge 0$.

(2) Definition 2.1 can be rewritten using Equations (6) and (14) as follows: (23) $\begin{array}{rl}& \begin{array}{l}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)=\sum _{m\ge 0}\frac{{\left(\mathcal{Q}\right)}_{m,\kappa }{\left(\mathcal{N}\right)}_{m,\kappa }(\mathfrak{D}{)}_{m,\kappa }^{-1}}{m!}\\ {}_2{\mathcal{F}}_{1,k}\left(\mathcal{Q}+m\kappa I,{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D}+m\kappa I;w\right)z^m,\end{array}\end{array}$(23) (24) $\begin{array}{rl}& \begin{array}{l}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},\mathfrak{D}{}^{\mathrm{\prime }};z,w\right)=\sum _{m\ge 0}\frac{{\left(\mathcal{Q}\right)}_{m,\kappa }{\left(\mathcal{N}\right)}_{m,\kappa }(\mathfrak{D}{)}_{m,\kappa }^{-1}}{m!}\\ {}_2{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q}+m\kappa I,{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D}{}^{\mathrm{\prime }};w\right)z^m,\end{array}\end{array}$(24) (25) $\begin{array}{rl}& \begin{array}{l}{\mathcal{F}}_{3,\kappa }\left(\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)=\sum _{m\ge 0}\frac{{\left(\mathcal{Q}\right)}_{m,\kappa }{\left(\mathcal{N}\right)}_{m,\kappa }(\mathfrak{D}{)}_{m,\kappa }^{-1}}{m!}\\ {}_2{\mathcal{F}}_{1,\kappa }\left({\mathcal{Q}}^{\mathrm{\prime }},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D}+m\kappa I;w\right)z^m,\end{array}\end{array}$(25) (26) $\begin{array}{rl}& \begin{array}{l}{\mathcal{F}}_{4,\kappa }\left(\mathcal{Q},\mathcal{N};\mathfrak{D},\mathfrak{D}{}^{\mathrm{\prime }};z,w\right)=\sum _{m\ge 0}\frac{{\left(\mathcal{Q}\right)}_{m,\kappa }{\left(\mathcal{N}\right)}_{m,\kappa }(\mathfrak{D}{)}_{m,\kappa }^{-1}}{m!}\\ {}_2{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q}+m\kappa I,\mathcal{N}+m\kappa I;\mathfrak{D}{}^{\mathrm{\prime }};w\right)z^m,\end{array}\end{array}$(26) where $|z|<\frac{1}{\kappa },|w|<\frac{1}{\kappa };|z|+|w|<\frac{1}{\kappa };|z|<\frac{1}{\kappa },|w|<\frac{1}{\kappa };\sqrt{|z|}+\sqrt{|w|}<\frac{1}{\sqrt{\kappa }},$ respectively.

Theorem 2.1:

Relationships between $\kappa$-Appell hypergeometric matrix functions and original Appell hypergeometric matrix functions are expressed as follows: (27) $\begin{array}{rl}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)& ={\mathcal{F}}_1\left(\mathcal{Q}/\kappa ,\mathcal{N}/\kappa ,{\mathcal{N}}^{\mathrm{\prime }}/\kappa ;\mathfrak{D}/\kappa ;\kappa z,\kappa w\right),\end{array}$(27) (28) $\begin{array}{rl}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},\mathfrak{D}{}^{\mathrm{\prime }};z,w\right)& ={\mathcal{F}}_2\left(\mathcal{Q}/\kappa ,\mathcal{N}/\kappa ,{\mathcal{N}}^{\mathrm{\prime }}/\kappa ;\mathfrak{D}/\kappa ,{\mathfrak{D}}^{\mathrm{\prime }}/\kappa ;\kappa z,\kappa w\right),\end{array}$(28) (29) $\begin{array}{rl}{\mathcal{F}}_{3,\kappa }\left(\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)& ={\mathcal{F}}_3\left(\mathcal{Q}/\kappa ,{\mathcal{Q}}^{\mathrm{\prime }}/\kappa ,\mathcal{N}/\kappa ,{\mathcal{N}}^{\mathrm{\prime }}/\kappa ;\mathfrak{D}/\kappa ;\kappa z,\kappa w\right),\end{array}$(29) (30) $\begin{array}{rl}{\mathcal{F}}_{4,\kappa }\left(\mathcal{Q},\mathcal{N};\mathfrak{D},\mathfrak{D}{}^{\mathrm{\prime }};z,w\right)& ={\mathcal{F}}_4\left(\mathcal{Q}/\kappa ,\mathcal{N}/\kappa ;\mathfrak{D}/\kappa ,{\mathfrak{D}}^{\mathrm{\prime }}/\kappa ;\kappa z,\kappa w\right),\end{array}$(30)

Proof. These relationships are direct consequences of the definition of the $\kappa$-Appell hypergeometric matrix functions and property $(\mathcal{Q}{)}_{n,\kappa }={\kappa }^n{\left(\frac{\mathcal{Q}}{\kappa }\right)}_n$.

3. Integral representation

The integral representations for $\kappa$-Appell hypergeometric matrix functions ${\mathcal{F}}_{1,\kappa }$, ${\mathcal{F}}_{2,\kappa },{\mathcal{F}}_{3,\kappa }\text{ and }{\mathcal{F}}_{4,\kappa }$, where $\kappa \in {\mathbb{R}}^{+}$, are provided in this section.

Theorem 3.1:

Let $\kappa \in {\mathbb{R}}^{+},z,w\in \mathbb{C},m,n\in {\mathbb{N}}^{+}\text{ and }\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }}\text{ and }\mathfrak{D}$ be matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, such that $\mathcal{Q}\mathcal{N}=\mathcal{N}\mathcal{Q},{\mathcal{N}}^{\mathrm{\prime }}\mathfrak{D}=\mathfrak{D}{\mathcal{N}}^{\mathrm{\prime }}.\text{ Furthermore}$, $\mathfrak{D}+n\kappa I$ is an invertible matrix for all integrals $n\ge 0$. Suppose that $\mathcal{Q},\mathfrak{D}and\mathfrak{D}-\mathcal{Q}$ are positively stable, and $\mathcal{Q}\mathfrak{D}=\mathfrak{D}\mathcal{Q},\mathcal{Q}{\mathcal{N}}^{\mathrm{\prime }}={\mathcal{N}}^{\mathrm{\prime }}\mathcal{Q}$, then the integral representations of ${\mathcal{F}}_{1,\kappa }$ for $|z|<\frac{1}{\kappa },|w|<\frac{1}{\kappa }$ are defined as follows (31) $\begin{array}{r}\begin{array}{l}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)\\ =\frac{1}{\kappa }{\mathrm{\Gamma }}_{\kappa }\left(\mathfrak{D}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathfrak{D}-\mathcal{Q}\right)\times \underset{0}{\overset{1}{\int }}{t}^{\frac{\mathcal{Q}}{\kappa }-I}(1-t{)}^{\frac{\mathfrak{D}-\mathcal{Q}}{\kappa }-I}\\ (1-\kappa zt{)}^{\frac{-\mathcal{N}}{\kappa }}(1-\kappa wt{)}^{\frac{-\mathcal{N}{}^{\mathrm{\prime }}}{\kappa }}\text{d}t.\end{array}\end{array}$(31)

Proof

. Let $\mathcal{Q},\mathfrak{D}\text{ and }\mathfrak{D}-\mathcal{Q}$ be positive stable, $\mathcal{Q}\mathfrak{D}=\mathfrak{D}\mathcal{Q}\text{ and }\mathcal{Q}{\mathcal{N}}^{\mathrm{\prime }}={\mathcal{N}}^{\mathrm{\prime }}\mathcal{Q}$. We then obtain the following formula using Equations (7) and (12): (32) $\begin{array}{r}\begin{array}{l}(\mathcal{Q}{)}_{m+n,\kappa }[{\left(\mathfrak{D}\right)}_{m+n,\kappa }{]}^{-1}={\mathrm{\Gamma }}_{\kappa }\left(\mathfrak{D}\right){\mathrm{\Gamma }}_{\kappa }\left(\mathcal{Q}+\left(m+n\right)\kappa I\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathfrak{D}+\left(m+n\right)\kappa I\right)\\ =\frac{{\mathrm{\Gamma }}_{\kappa }\left(\mathfrak{D}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathfrak{D}-\mathcal{Q}\right)}{\kappa }\times \underset{0}{\overset{1}{\int }}{t}^{\frac{\mathcal{Q}+\left(m+n\right)kI}{\kappa }-I}(1-t{)}^{\frac{\mathfrak{D}-\mathcal{Q}}{\kappa }-I}\text{d}t.\end{array}\end{array}$(32) We also obtain the following formula using Equations (7), (15), and (32): $\begin{array}{rl}& \begin{array}{l}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)=\sum _{m,n\ge 0}\frac{{\mathrm{\Gamma }}_{\kappa }\left(\mathfrak{D}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathfrak{D}-\mathcal{Q}\right)}{\kappa }\\ \times \underset{0}{\overset{1}{\int }}{t}^{\frac{\mathcal{Q}}{\kappa }+\left(m+n-1\right)I}(1-t{)}^{\frac{\mathfrak{D}-\mathcal{Q}}{\kappa }-I}\text{d}t\times \sum _{m,n\ge 0}\frac{{\left(\mathcal{N}\right)}_{m,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{n,\kappa }}{m!n!}{z}^m{w}^n\end{array}\end{array}$ (33) $\begin{array}{rl}& =\frac{1}{\kappa }{\mathrm{\Gamma }}_{\kappa }\left(\mathfrak{D}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathfrak{D}-\mathcal{Q}\right)\times \underset{0}{\overset{1}{\int }}{t}^{\frac{\mathcal{Q}}{\kappa }-I}(1-t{)}^{\frac{\mathfrak{D}-\mathcal{Q}}{\kappa }-I}(1-\kappa zt{)}^{\frac{-\mathcal{N}}{\kappa }}(1-\kappa wt{)}^{\frac{-{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }}\text{d}t.\end{array}$(33)

Remark 3.1:

We obtain the integral representation of Appell’s ${\mathcal{F}}_1$ in ([23,36]) when $\kappa \to 1$ in Theorem 3.1.

Theorem 3.2:

Let $\kappa \in {\mathbb{R}}^{+},z,w\in \mathbb{C},\text{ and }\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }},\mathfrak{D}\text{ and }{\mathfrak{D}}^{\mathrm{\prime }}$ be matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, such that $\mathfrak{D}+n\kappa I$ and ${\mathfrak{D}}^{\mathrm{\prime }}+n\kappa I$ are invertible matrices for all integrals $n\ge 0$. Suppose that $\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }},\mathfrak{D}$ and ${\mathfrak{D}}^{\mathrm{\prime }}$ are commutative matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}},\text{then}{\mathcal{N}}^{\mathrm{\prime }}\mathfrak{D}=\mathfrak{D}{\mathcal{N}}^{\mathrm{\prime }}$ and ${\mathcal{N}}^{\mathrm{\prime }}{\mathfrak{D}}^{\mathrm{\prime }}=\mathfrak{D}{}^{\mathrm{\prime }}{\mathcal{N}}^{\mathrm{\prime }}$. Suppose that ${\mathcal{N}}^{\mathrm{\prime }},{\mathfrak{D}}^{\mathrm{\prime }}\text{ and }{\mathcal{N}}^{\mathrm{\prime }}-{\mathfrak{D}}^{\mathrm{\prime }}$ are positive stable for every integral $n\ge 0$, then the integral representation of ${\mathcal{F}}_{2,\kappa }$ is defined as follows $\begin{array}{rl}& \begin{array}{l}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)\\ =\frac{{\mathrm{\Gamma }}_{\kappa }\left(\mathfrak{D}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{N}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathfrak{D}-\mathcal{N}\right){\mathrm{\Gamma }}_{\kappa }\left({\mathfrak{D}}^{\mathrm{\prime }}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left({\mathcal{N}}^{\mathrm{\prime }}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left({\mathfrak{D}}^{\mathrm{\prime }}-{\mathcal{N}}^{\mathrm{\prime }}\right)}{{\kappa }^2}\end{array}\end{array}$ (34) $\begin{array}{rl}& \times \underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{t}^{\frac{\mathcal{N}}{\kappa }-I}{s}^{\frac{{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }-I}(1-t{)}^{\frac{\mathfrak{D}-\mathcal{N}}{\kappa }-I}(1-s{)}^{\frac{{\mathfrak{D}}^{\mathrm{\prime }}-{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }-I}(1-\kappa zt-\kappa ws{)}^{-\frac{\mathcal{Q}}{\kappa }}\text{d}t\text{d}s.\end{array}$(34)

Proof.

The following equation is according to the definition of the $\kappa$-Pochhammer matrix symbol and using Equation (7): (35) $\begin{array}{r}\begin{array}{l}(\mathcal{N}{)}_{m,\kappa }({\mathcal{N}}^{\mathrm{\prime }}{)}_{n,\kappa }\left(\mathfrak{D}{)}_{m,\kappa }^{-1}\right({\mathfrak{D}}^{\mathrm{\prime }}{)}_{n,\kappa }^{-1}={\mathrm{\Gamma }}_{\kappa }\left(\mathcal{N}+m\kappa I\right){\mathrm{\Gamma }}_{\kappa }^{-1}\left(\mathcal{N}\right)\\ {\mathrm{\Gamma }}_{\kappa }\left({\mathcal{N}}^{\mathrm{\prime }}+n\kappa I\right){\mathrm{\Gamma }}_{\kappa }^{-1}\left({\mathcal{N}}^{\mathrm{\prime }}\right){\mathrm{\Gamma }}_{\kappa }^{-1}\left(\mathfrak{D}+n\kappa I\right)\\ {\mathrm{\Gamma }}_{\kappa }\left(\mathfrak{D}\right){\mathrm{\Gamma }}_{\kappa }^{-1}\left({\mathfrak{D}}^{\mathrm{\prime }}+n\kappa I\right){\mathrm{\Gamma }}_{\kappa }\left({\mathfrak{D}}^{\mathrm{\prime }}\right).\end{array}\end{array}$(35) We can rewrite Equation (35) in the following form: (36) $\begin{array}{r}\begin{array}{l}(\mathcal{N}{)}_{m,\kappa }(\mathfrak{D}{)}_{m,\kappa }^{-1}={\mathfrak{B}}_{\kappa }(\mathcal{N}+m\kappa I,\mathfrak{D}-\mathcal{N}\left){\mathrm{\Gamma }}_{\kappa }\right(\mathfrak{D}\left){{\mathrm{\Gamma }}_{\kappa }}^{-1}\right(\mathcal{N}\left){{\mathrm{\Gamma }}_{\kappa }}^{-1}\right(\mathfrak{D}-\mathcal{N})\\ (\mathcal{N}{}^{\mathrm{\prime }}{)}_{n,\kappa }({\mathfrak{D}}^{\mathrm{\prime }}{)}_{n,\kappa }^{-1}={\mathfrak{B}}_{\kappa }(\mathcal{N}{}^{\mathrm{\prime }}+n\kappa I,{\mathfrak{D}}^{\mathrm{\prime }}-\mathcal{N}{}^{\mathrm{\prime }}\left){\mathrm{\Gamma }}_{\kappa }\right({\mathfrak{D}}^{\mathrm{\prime }}\left){{\mathrm{\Gamma }}_{\kappa }}^{-1}\right(\mathcal{N}{}^{\mathrm{\prime }}\left){{\mathrm{\Gamma }}_{\kappa }}^{-1}\right({\mathfrak{D}}^{\mathrm{\prime }}-\mathcal{N}{}^{\mathrm{\prime }}).\end{array}\end{array}$(36) We then obtain the following equation using Equation (36) in Equation (16): $\begin{array}{l}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)\\ =\frac{{\mathrm{\Gamma }}_{\kappa }\left(\mathfrak{D}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{N}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathfrak{D}-\mathcal{N}\right){\mathrm{\Gamma }}_{\kappa }\left({\mathfrak{D}}^{\mathrm{\prime }}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left({\mathcal{N}}^{\mathrm{\prime }}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left({\mathfrak{D}}^{\mathrm{\prime }}-{\mathcal{N}}^{\mathrm{\prime }}\right)}{{\kappa }^2}\\ \times \underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{t}^{\frac{\mathcal{N}}{\kappa }-I}{s}^{\frac{{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }-I}(1-t{)}^{\frac{\mathfrak{D}-\mathcal{N}}{\kappa }-I}(1-s{)}^{\frac{{\mathfrak{D}}^{\mathrm{\prime }}-{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }-I}\text{d}t\text{d}s\times \sum _{m,n\ge 0}(\mathcal{Q}{)}_{m+n,\kappa }\frac{{\left(zt\right)}^m{\left(ws\right)}^n}{m!n!}\\ =\frac{{\mathrm{\Gamma }}_{\kappa }\left(\mathfrak{D}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{N}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathfrak{D}-\mathcal{N}\right){\mathrm{\Gamma }}_{\kappa }\left({\mathfrak{D}}^{\mathrm{\prime }}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left({\mathcal{N}}^{\mathrm{\prime }}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left({\mathfrak{D}}^{\mathrm{\prime }}-{\mathcal{N}}^{\mathrm{\prime }}\right)}{{\kappa }^2}\\ \times \underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{t}^{\frac{\mathcal{N}}{\kappa }-I}{s}^{\frac{{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }-I}(1-t{)}^{\frac{\mathfrak{D}-\mathcal{N}}{\kappa }-I}(1-s{)}^{\frac{{\mathfrak{D}}^{\mathrm{\prime }}-{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }-I}\times (1-\kappa zt-\kappa ws{)}^{\frac{-\mathcal{Q}}{\kappa }}\text{d}t\text{d}s.\end{array}$

Remark 3.2:

We obtain the integral representation of the second Appell hypergeometric matrix function ${\mathcal{F}}_2$ in [19] when $\kappa \to 1$.

Theorem 3.3:

Let $\kappa \in {\mathbb{R}}^{+},z,w\in \mathbb{C},\text{ and }\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }},\mathfrak{D},\text{ and }{\mathfrak{D}}^{\mathrm{\prime }}$ be matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, such that $\mathcal{Q}+n\kappa I\mathfrak{D}+n\kappa I$ and $\mathfrak{D}{}^{\mathrm{\prime }}+n\kappa I$ are invertible matrices for all integrals $n\ge 0$. Suppose that ${\mathcal{N}}^{\mathrm{\prime }}$ and $\mathfrak{D}$ are commutative matrices in ${\mathbb{C}}^{\mathcal{N}\times \mathcal{N}}$, then ${\mathcal{N}}^{\mathrm{\prime }}\mathfrak{D}=\mathfrak{D}{\mathcal{N}}^{\mathrm{\prime }}$. The integral representation of ${\mathcal{F}}_{2,\kappa }$ for $|z|+|w|<\frac{1}{\kappa }$ is defined as follows: (37) $\begin{array}{r}\begin{array}{l}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},\mathfrak{D}{}^{\mathrm{\prime }};z,w\right)={\mathrm{\Gamma }}_{\kappa }^{-1}\left(\mathcal{Q}\right)\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\frac{t^{\kappa }}{\kappa }}{t}^{\mathcal{Q}-I}\\ {}_1{\mathcal{F}}_{1,\kappa }\left(\mathcal{N};\mathfrak{D};z{t}^{\kappa }\right){}_1{\mathcal{F}}_{1,\kappa }\left({\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D}{}^{\mathrm{\prime }};w{t}^{\kappa }\right)\text{d}t.\end{array}\end{array}$(37)

Proof.

Replacing the integral representation of the $\kappa$-Pochhammer matrix symbol, which is attained from (8) and (11) in place of $(\mathcal{Q}{)}_{m+n,\kappa }$ in Equation (16), we get (38) $\begin{array}{r}\begin{array}{l}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)\\ ={\mathrm{\Gamma }}_{\kappa }^{-1}\left(\mathcal{Q}\right)\sum _{m,n\ge 0}\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\frac{t^{\kappa }}{\kappa }}{t}^{\mathcal{Q}+\left(m+n\right)\kappa I-I}\text{d}t\frac{{\left(\mathcal{N}\right)}_{m,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{n,\kappa }\left(\mathfrak{D}{)}_{m,\kappa }^{-1}\right({\mathfrak{D}}^{\mathrm{\prime }}{)}_{n,\kappa }^{-1}}{m!n!}{z}^m{w}^n.\end{array}\end{array}$(38) This equation can be rewritten as follows: (39) $\begin{array}{r}\begin{array}{l}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)\\ ={\mathrm{\Gamma }}_{\kappa }^{-1}\left(\mathcal{Q}\right)\left[\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\frac{t^{\kappa }}{\kappa }}{t}^{\mathcal{Q}-I}\left(\sum _{m\ge 0}\frac{{\left(\mathcal{N}\right)}_{m,\kappa }(\mathfrak{D}{)}_{m,\kappa }^{-1}}{m!}{\left(z{t}^{\kappa }\right)}^m\right)\left(\sum _{n\ge 0}\frac{{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{n,\kappa }({\mathfrak{D}}^{\mathrm{\prime }}{)}_{n,\kappa }^{-1}}{n!}{\left(w{t}^{\kappa }\right)}^n\right)\text{d}t\right]\\ ={\mathrm{\Gamma }}_{\kappa }^{-1}\left(\mathcal{Q}\right)\left[\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\frac{t^{\kappa }}{\kappa }}{t}^{\mathcal{Q}-I}{}_1{\mathcal{F}}_{1,\kappa }\left(\mathcal{N};\mathfrak{D};z{t}^{\kappa }\right){}_1{\mathcal{F}}_{1,\kappa }\left({\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D}{}^{\mathrm{\prime }};w{t}^{\kappa }\right)\text{d}t\right].\end{array}\end{array}$(39)

Remark 3.3:

When $\kappa \to 1,$ then we get the integral representation of the second Appell hypergeometric matrix function ${\mathcal{F}}_2$ in [21].

Theorem 3.4:

Let $\kappa \in {\mathbb{R}}^{+},z,w\in \mathbb{C},\text{ and }\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }},\text{ and }\mathfrak{D}$ be matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, such that $\mathfrak{D}+n\kappa I$ is the invertible matrix for all integrals $n\ge 0$. Suppose that $\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }}$ and $\mathfrak{D}$ are commutative matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, then the integral representations of ${\mathcal{F}}_{3,\kappa }$ is defined as follows: (40) $\begin{array}{r}\begin{array}{l}{\mathcal{F}}_{3,\kappa }(\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w)=\frac{{\mathrm{\Gamma }}_{\kappa }\left(\mathfrak{D}\right){\mathrm{\Gamma }}_{\kappa }^{-1}\left(\mathcal{N}\right){\mathrm{\Gamma }}_{\kappa }^{-1}\left({\mathcal{N}}^{\mathrm{\prime }}\right){\mathrm{\Gamma }}_{\kappa }^{-1}(\mathfrak{D}-\mathcal{N}-{\mathcal{N}}^{\mathrm{\prime }})}{{\kappa }^2}\\ \times \int \underset{D}{\int }{t}^{\frac{\mathcal{N}}{\kappa }-I}{s}^{\frac{{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }-I}(1-\kappa zt{)}^{-\frac{\mathcal{Q}}{\kappa }}(1-\kappa ws{)}^{-\frac{{\mathcal{Q}}^{\mathrm{\prime }}}{\kappa }}(1-t-s{)}^{\frac{\mathfrak{D}-\mathcal{N}-{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }-I}dtds,\end{array}\end{array}$(40) where $D=\left\{t\ge 0,s\ge 0,t+s\le 1\right\}.$

Remark 3.4:

When $\kappa \to 1,$ then we get the integral representation of the Appell hypergeometric matrix function ${\mathcal{F}}_3$ in [19].

Theorem 3.5:

Let $\kappa \in {\mathbb{R}}^{+},z,w\in \mathbb{C},\text{ and }\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }},\text{ and }\mathfrak{D}$ be matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, such that $\mathfrak{D}+n\kappa I$ is invertible matrix for all integrals $n\ge 0$. Suppose that $\mathcal{B},{\mathcal{B}}^{\mathrm{\prime }}$ and $\mathfrak{D}$ are commutative matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, then the integral representations of ${\mathcal{F}}_{4,\kappa }$ is defined as follows: (41) $\begin{array}{r}\begin{array}{l}{\mathcal{F}}_{4,\kappa }\left(\mathcal{Q},\mathcal{N};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)=\frac{{\kappa }^{\frac{\mathcal{Q}+\mathcal{N}}{\kappa }}{{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{N}\right)}{{\kappa }^2}\\ \underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{\frac{\mathcal{Q}}{\kappa }-I}{s}^{\frac{\mathcal{N}}{\kappa }-I}{e}^{-t-s}{}_0{\mathcal{F}}_{1,\kappa }\left(-;\mathfrak{D};{\kappa }^{2}zts\right){}_0{\mathcal{F}}_{1,\kappa }\left(-;{\mathfrak{D}}^{\mathrm{\prime }};{\kappa }^{2}wts\right)\text{d}t\text{d}s.\end{array}\end{array}$(41) Formulas in Equations (40) and (41) can be proven in a similar way; hence, the details are omitted.

4. Transformation formulas

We intend to derive transformation formulas involving the $\kappa$-Appell functions ${\mathcal{F}}_{1,\kappa }$ and ${\mathcal{F}}_{2,\kappa }$ in the matrix setting.

Theorem 4.1:

For $\kappa \in {\mathbb{R}}^{+},z,w\in \mathbb{C},\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }}$ and $\mathfrak{D}$ are commutative matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$ with $\mathfrak{D}+n\kappa I$ is invertible matrix for all integrals $n\ge 0$. Suppose that $\mathcal{Q},\mathfrak{D}\text{ and }\mathfrak{D}-\mathcal{Q}$ are positive stable, and $\mathcal{Q}\mathfrak{D}=\mathfrak{D}\mathcal{Q},\mathcal{Q}{\mathcal{N}}^{\mathrm{\prime }}={\mathcal{N}}^{\mathrm{\prime }}\mathcal{Q}$. For the matrix function ${\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)$, we have the following transformations: (42) $\begin{array}{rl}& \begin{array}{l}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)=(1-\kappa z{)}^{\frac{-\mathcal{N}}{\kappa }}(1-\kappa w{)}^{\frac{-{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }}\\ {\mathcal{F}}_{1,\kappa }\left(\mathfrak{D}-\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};\frac{-z}{1-\kappa z},\frac{-w}{1-\kappa w}\right),\end{array}\end{array}$(42) (43) $\begin{array}{rl}& \begin{array}{l}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)=(1-\kappa z{)}^{\frac{-\mathcal{Q}}{\kappa }}\\ {\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathfrak{D}-\mathcal{N}-{\mathcal{N}}^{\mathrm{\prime }},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};\frac{-z}{1-\kappa z},-\frac{z-w}{1-\kappa z}\right),\end{array}\end{array}$(43) (44) $\begin{array}{rl}& \begin{array}{l}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)=(1-\kappa w{)}^{\frac{-\mathcal{Q}}{\kappa }}\\ {\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},\mathfrak{D}-\mathcal{N}-{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};-\frac{w-z}{1-\kappa w},\frac{-w}{1-\kappa w}\right),\end{array}\end{array}$(44) where $\left|\frac{z}{1-\kappa z}\right|<\frac{1}{\kappa },\left|\frac{w}{1-\kappa w}\right|<\frac{1}{\kappa }$, $|z|<\frac{1}{\kappa },|w|<\frac{1}{\kappa };\left|\frac{z}{1-\kappa z}\right|<\frac{1}{\kappa },\left|\frac{z-w}{1-\kappa z}\right|<\frac{1}{\kappa }$, $|z|<\frac{1}{\kappa },|w|<\frac{1}{\kappa }$; $\left|\frac{w-z}{1-\kappa w}\right|<\frac{1}{\kappa },\left|\frac{w}{1-\kappa w}\right|<\frac{1}{\kappa }$, $|z|<\frac{1}{\kappa },|w|<\frac{1}{\kappa }$, respectively.

Proof.

We proved only Equation (42), since the others Equations (43) and (44) can be proven similarly.

Substituting $t:=1-t_1$ in the Equations (31) and (12) to obtain the following equation: (45) $\begin{array}{r}\begin{array}{l}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)=\frac{{\mathrm{\Gamma }}_{\kappa }\left(\mathfrak{D}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathfrak{D}-\mathcal{Q}\right)}{\kappa }(1-\kappa z{)}^{\frac{-\mathcal{N}}{\kappa }}(1-\kappa w{)}^{\frac{-{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }}\\ \times \underset{0}{\overset{1}{\int }}(1-t_1{)}^{\frac{\mathcal{Q}}{\kappa }-I}{t_1}^{\frac{\mathfrak{D}-\mathcal{Q}}{\kappa }-I}\times {\left(1+\frac{\kappa z{t}_1}{1-\kappa z}\right)}^{\frac{-\mathcal{N}}{\kappa }}{\left(1+\frac{\kappa w{t}_1}{1-\kappa w}\right)}^{\frac{-{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }}\text{d}{t}_1\\ =(1-\kappa z{)}^{\frac{-\mathcal{N}}{\kappa }}(1-\kappa w{)}^{\frac{-{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }}\times {\mathcal{F}}_{1,\kappa }\left(\mathfrak{D}-\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};\frac{-z}{1-\kappa z},\frac{-w}{1-\kappa w}\right).\end{array}\end{array}$(45) Furthermore, we can prove Equations (43) and (44) using the same argument proof of Equation (42) by substituting $t:=\frac{t_1}{1-\kappa z+\kappa t_{1}z}$ and $t:=\frac{t_1}{1-\kappa w+\kappa t_{1}w}$ in Equation (31) and Equation (12) to obtain Equations (43) and (44), respectively.

Theorem 4.2:

Let $\kappa \in {\mathbb{R}}^{+}$, $z,w\in \mathbb{C},m,n\in {\mathbb{N}}^{+}\text{ and }\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }}\text{ and }\mathfrak{D}$ be matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, such that $\mathcal{Q}\mathcal{N}=\mathcal{N}\mathcal{Q},{\mathcal{N}}^{\mathrm{\prime }}\mathfrak{D}=\mathfrak{D}{\mathcal{N}}^{\mathrm{\prime }}\text{ and }\mathfrak{D}+n\kappa I$ is invertible matrix for all integrals $n\ge 0$. Suppose that $\mathcal{Q},\mathfrak{D}and\mathfrak{D}-\mathcal{Q}$ are positive stable and commuting matrices then ${\mathfrak{B}}_{\kappa }\left(\mathcal{Q},\mathfrak{D}-\mathcal{Q}\right)={\mathfrak{B}}_{\kappa }\left(\mathfrak{D}-\mathcal{Q},\mathcal{Q}\right)$. For ${\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right),$ we have transformations as follows: (46) $\begin{array}{rl}& \begin{array}{l}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)\\ =(1-\kappa z{)}^{\frac{\mathfrak{D}-\mathcal{Q}-\mathcal{N}}{\kappa }}(1-\kappa w{)}^{\frac{-{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }}\times {\mathcal{F}}_{1,\kappa }\left(\mathfrak{D}-\mathcal{Q},\mathfrak{D}-\mathcal{N}-{\mathcal{N}}^{\mathrm{\prime }},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,-\frac{w-z}{1-\kappa w}\right),\end{array}\end{array}$(46) (47) $\begin{array}{rl}& \begin{array}{l}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)\\ =(1-\kappa z{)}^{\frac{-\mathcal{N}}{\kappa }}(1-\kappa w{)}^{\frac{\mathfrak{D}-\mathcal{Q}-{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }}\times {\mathcal{F}}_{1,\kappa }\left(\mathfrak{D}-\mathcal{Q},\mathcal{N},\mathfrak{D}-\mathcal{N}-{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};-\frac{w-z}{1-\kappa z},w\right),\end{array}\end{array}$(47)

Proof.

The first relation can be proven by setting $t:=\frac{t_1}{1-\kappa z+\kappa z{t}_1}$ and $t_1:=1-t_2$ in Equation (31) and using Equation (12), we get (48) $\begin{array}{r}\begin{array}{l}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)\\ =\frac{{\mathrm{\Gamma }}_{\kappa }\left(\mathfrak{D}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathfrak{D}-\mathcal{Q}\right)}{\kappa }(1-\kappa z{)}^{\frac{-\mathcal{Q}}{\kappa }}\times \underset{0}{\overset{1}{\int }}(1-t_2{)}^{\frac{\mathcal{Q}}{\kappa }-I}{t_2}^{\frac{\mathcal{C}-\mathcal{Q}}{\kappa }-I}\\ {\left(1+\frac{\kappa z-\kappa z{t}_2}{1-\kappa z}\right)}^{\frac{\mathcal{N}+{\mathcal{N}}^{\mathrm{\prime }}-\mathfrak{D}}{\kappa }}\times {\left(1+\frac{\kappa z-\kappa z{t}_2-\kappa w+\kappa w{t}_2}{1-\kappa z}\right)}^{\frac{-{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }}\text{d}{t}_2\\ =\frac{{\mathrm{\Gamma }}_{\kappa }\left(\mathfrak{D}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathcal{Q}\right){{\mathrm{\Gamma }}_{\kappa }}^{-1}\left(\mathfrak{D}-\mathcal{Q}\right)}{\kappa }(1-\kappa z{)}^{\frac{\mathfrak{D}-\mathcal{Q}-\mathcal{N}}{\kappa }}(1-\kappa w{)}^{\frac{-{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }}\\ \times \underset{0}{\overset{1}{\int }}(1-t_2{)}^{\frac{\mathcal{Q}}{\kappa }-I}{t_2}^{\frac{\mathfrak{D}-\mathcal{Q}}{\kappa }-I}(1-\kappa z{t}_2{)}^{\frac{\mathcal{N}+{\mathcal{N}}^{\mathrm{\prime }}-\mathfrak{D}}{\kappa }}\times {\left(1+\frac{\kappa w{t}_2-\kappa z{t}_2}{1-\kappa w}\right)}^{\frac{-{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }}\text{d}{t}_2\\ =(1-\kappa z{)}^{\frac{\mathfrak{D}-\mathcal{Q}-\mathcal{N}}{\kappa }}(1-\kappa w{)}^{\frac{-{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }}\times {\mathcal{F}}_{1,\kappa }\left(\mathfrak{D}-\mathcal{Q},\mathfrak{D}-\mathcal{N}-{\mathcal{N}}^{\mathrm{\prime }},\mathcal{N}{}^{\mathrm{\prime }};\mathfrak{D};z,-\frac{w-z}{1-\kappa w}\right).\end{array}\end{array}$(48) The second relation is proven using the same argument proof as the first relation by substituting $t:=\frac{t_1}{1-\kappa w+\kappa t_{1}w}$ and $t_1:=1-t_2$ in Equation (31) and using Equation (12).

Theorem 4.3:

For $\kappa \in {\mathbb{R}}^{+},z,w\in \mathbb{C}$, $\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }},\mathfrak{D}$, and ${\mathfrak{D}}^{\mathrm{\prime }}$ are commutative matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, such that $\mathfrak{D}+n\kappa I$ and ${\mathfrak{D}}^{\mathrm{\prime }}+n\kappa I$ are invertible matrices for all integrals $n\ge 0.$ Suppose that $\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }},\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }},\mathfrak{D}-\mathcal{N}$, and ${\mathfrak{D}}^{\mathrm{\prime }}-{\mathcal{N}}^{\mathrm{\prime }}$ are positively stable, then the matrix function ${\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)$ have transformations as follows: (49) $\begin{array}{rl}& {\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)\\ & =(1-\kappa z{)}^{\frac{-\mathcal{Q}}{\kappa }}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathfrak{D}-\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};\frac{-z}{1-\kappa z},\frac{w}{1-\kappa z}\right),\end{array}$(49) (50) $\begin{array}{rl}& {\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)\\ & =(1-\kappa w{)}^{\frac{-\mathcal{Q}}{\kappa }}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathfrak{D}}^{\mathrm{\prime }}-{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};\frac{z}{1-\kappa w},\frac{-w}{1-\kappa w}\right),\end{array}$(50) (51) $\begin{array}{rl}& \begin{array}{l}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)\\ =(1-\kappa z-\kappa w{)}^{\frac{-\mathcal{Q}}{\kappa }}{\mathcal{F}}_{2,k}\left(\mathcal{Q},\mathfrak{D}-\mathcal{N},{\mathfrak{D}}^{\mathrm{\prime }}-{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};\frac{-z}{1-\kappa z-\kappa w},\frac{-w}{1-\kappa z-\kappa w}\right).\end{array}\end{array}$(51)

Proof.

In the first relation we taking $t:=1-t_1$, for the second relation $s:=1-s_1$ and, finally, for the third relation $t:=1-t_1,s:=1-s_1$ together in the double integral in Equation (34), we find Equations (49)–(51), respectively. These complete the proof.

Connections with some reduction formulas for Appell matrix functions ${\mathcal{F}}_{1,\kappa }$ and ${\mathcal{F}}_{2,\kappa }$ in terms of the ${}_2{\mathcal{F}}_{1,\kappa }$ generalized hypergeometric matrix function presented in the next theorem.

Theorem 4.4:

let $\kappa \in {\mathbb{R}}^{+},z,w\in \mathbb{C}$, then ${\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)$ and ${\mathcal{F}}_{2,\kappa }(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w)$ be given in Definition 2.1 have the special cases given as follows (52) $\begin{array}{rl}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)& =(1-\kappa z{)}^{\frac{-\mathcal{Q}}{\kappa }}{}_2{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},{\mathcal{N}}^{\mathrm{\prime }};\mathcal{N}+{\mathcal{N}}^{\mathrm{\prime }};-\frac{z-w}{1-\kappa z}\right),\end{array}$(52) (53) $\begin{array}{rl}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)& =(1-\kappa w{)}^{\frac{-\mathcal{Q}}{\kappa }}{}_2{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N};\mathcal{N}+{\mathcal{N}}^{\mathrm{\prime }};-\frac{w-z}{1-\kappa w}\right),\end{array}$(53) (54) $\begin{array}{rl}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)& =(1-\kappa z{)}^{\frac{-\mathcal{Q}}{\kappa }}{}_2{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},{\mathcal{N}}^{\mathrm{\prime }};{\mathfrak{D}}^{\mathrm{\prime }};\frac{w}{1-\kappa z}\right),\end{array}$(54) (55) $\begin{array}{rl}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)& =(1-\kappa w{)}^{\frac{-\mathcal{Q}}{\kappa }}{}_2{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N};\mathfrak{D};\frac{z}{1-\kappa w}\right),\end{array}$(55) where ${}_2{\mathcal{F}}_{1,\kappa }$ is the generalized hypergeometric matrix function defined in [28].

Proof.

We obtain the relations in Equations (52) and (53) if we set $\mathfrak{D}=\mathcal{N}+{\mathcal{N}}^{\mathrm{\prime }}$ in Equations (43) and (44), respectively. Similarly, we set $\mathfrak{D}=\mathcal{N}$ and ${\mathfrak{D}}^{\mathrm{\prime }}={\mathcal{N}}^{\mathrm{\prime }}$ in Equations (49) and (50) to get the relations in Equations (54) and (55), respectively.

5. Infinite matrix summation formulas

In this section, we obtain certain infinite matrix summation formulas involving $\kappa$-Appell matrix functions. The infinite matrix summation formulas involving the $\kappa$-Appell matrix function ${\mathcal{F}}_{1,\kappa }$are listed in the following theorem.

Theorem 5.1:

Let $\kappa \in {\mathbb{R}}^{+},z,w\in \mathbb{C},\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }}$ and $\mathfrak{D}$ be commutative matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$such that $\mathfrak{D}+n\kappa I$ is invertible matrix for all integral $n\ge 0$ with $\left|\frac{z}{1-\kappa \text{t}}\right|<\frac{1}{\kappa },\left|\frac{w}{1-\kappa \text{t}}\right|<\frac{1}{\kappa }$, and $|t|<\frac{1}{\kappa }$. Then the following infinite matrix summation formulas involving the $\kappa$-Appell matrix function ${\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)$ hold: (56) $\begin{array}{r}\begin{array}{l}\sum _{r\ge 0}(\mathcal{Q}{)}_{r,\kappa }\frac{t^r}{r!}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q}+r\kappa I,\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)\\ =(1-\kappa t{)}^{\frac{-\mathcal{Q}}{k}}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};\frac{z}{1-\kappa \text{t}},\frac{w}{1-\kappa \text{t}}\right),\end{array}\end{array}$(56) where $\left|\frac{z}{1-\kappa \text{t}}\right|<\frac{1}{\kappa },\left|\frac{w}{1-\kappa \text{t}}\right|<\frac{1}{\kappa },$ and $|t|<\frac{1}{\kappa }$. (57) $\begin{array}{r}\begin{array}{l}\sum _{r\ge 0}(\mathcal{Q}{)}_{r,\kappa }\frac{{\left(\kappa t\right)}^r}{r!}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q}+r\kappa I,\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }}+r\kappa I;\mathfrak{D}+r\kappa I;z,w\right)({\mathcal{N}}^{\mathrm{\prime }}{)}_{r,\kappa }(\mathfrak{D}{)}_{r,\kappa }^{-1}\\ ={\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w+\kappa \text{t}\right),\end{array}\end{array}$(57) where $|z|<\frac{1}{\kappa },|w+\kappa \text{t}|<\frac{1}{\kappa }$, $|t|<\frac{1}{\kappa }$ and ${\mathcal{N}}^{\mathrm{\prime }}\mathfrak{D}=\mathfrak{D}\mathcal{N}{}^{\mathrm{\prime }}.$

Proof.

By using Equation (15) in Equation (56), we obtain the following formula: (58) $\begin{array}{r}\begin{array}{l}(1-\kappa t{)}^{\frac{-\mathcal{Q}}{\kappa }}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};\frac{z}{1-\kappa \text{t}},\frac{w}{1-\kappa \text{t}}\right)\\ =\sum _{m,n\ge 0}(1-\kappa t{)}^{\frac{-\left(\mathcal{Q}+\left(m+n\right)\kappa I\right)}{\kappa }}\frac{{\left(\mathcal{Q}\right)}_{m+n,\kappa }{\left(\mathcal{N}\right)}_{m,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{n,\kappa }(\mathfrak{D}{)}_{m+n,\kappa }^{-1}}{m!n!}{z}^m{w}^n.\end{array}\end{array}$(58) Using the following identity in Equation (58), we get $\begin{array}{rl}& \sum _{r\ge 0}(\mathcal{Q}+\left(m+n\right)\kappa I{)}_{r,\kappa }\frac{t^r}{r!}=(1-\kappa t{)}^{\frac{-\left(\mathcal{Q}+\left(m+n\right)\kappa I\right)}{\kappa }},|t|<\frac{1}{\kappa },\end{array}$ $\begin{array}{rl}& (\mathcal{Q}{)}_{m+n,\kappa }(\mathcal{Q}+\left(m+n\right)\kappa I{)}_{r,\kappa }=(\mathcal{Q}{)}_{r,\kappa }(\mathcal{Q}+r\kappa I{)}_{m+n,\kappa },\end{array}$ (59) $\begin{array}{rl}& \begin{array}{l}(1-\kappa t{)}^{\frac{-\mathcal{Q}}{\kappa }}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};\frac{z}{1-\kappa \text{t}},\frac{w}{1-\kappa \text{t}}\right)\\ =\sum _{m,n,r\ge 0}(\mathcal{Q}{)}_{r,\kappa }\frac{t^r}{r!}\frac{{\left(\mathcal{Q}+r\kappa I\right)}_{m+n,\kappa }{\left(\mathcal{N}\right)}_{n,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{n,\kappa }(\mathfrak{D}{)}_{m+n,\kappa }^{-1}}{m!n!}{z}^m{w}^n\\ =\sum _{r\ge 0}(\mathcal{Q}{)}_{r,\kappa }\frac{t^r}{r!}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q}+r\kappa I,\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right).\end{array}\end{array}$(59) The proof of Equation (56) is completed.

For Equation (57), we substitute Equation (15) in Equation (57), we get: (60) $\begin{array}{r}\begin{array}{l}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w+\kappa \text{t}\right)\\ =\sum _{m,n\ge 0}\frac{{\left(\mathcal{Q}\right)}_{m+n,\kappa }{\left(\mathcal{N}\right)}_{m,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{n,\kappa }(\mathfrak{D}{)}_{m+n,\kappa }^{-1}}{m!n!}(z{)}^m(w+\kappa \text{t}{)}^n.\end{array}\end{array}$(60) Using the series expansion $(w+\kappa t{)}^n=\sum _{r=0}^n\frac{{\left(\kappa t\right)}^r{w}^{n-r}n!}{\left(n-r\right)!r!}$ in Equation (60), we get $\begin{array}{l}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w+\kappa \text{t}\right)\\ =\sum _{m,n,r\ge 0}\frac{{\left(\kappa t\right)}^r}{r!}\frac{{\left(\mathcal{Q}\right)}_{m+n+r,\kappa }{\left(\mathcal{N}\right)}_{m,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{n+r,\kappa }(\mathfrak{D}{)}_{m+n+r,\kappa }^{-1}}{m!n!}(z{)}^m(w{)}^n\\ =\sum _{m,n,r\ge 0}\frac{{\left(\kappa t\right)}^r}{r!}\frac{{\left(\mathcal{Q}\right)}_{r,\kappa }{\left(\mathcal{Q}+r\kappa I\right)}_{m+n,\kappa }{\left(\mathcal{N}\right)}_{m,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{r,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}+r\kappa I\right)}_{n,\kappa }\left(\mathfrak{D}+r\kappa I{)}_{m+n,\kappa }^{-1}\right(\mathfrak{D}{)}_{r,\kappa }^{-1}}{m!n!}(z{)}^m(w{)}^n\\ =\sum _{r\ge 0}\frac{{\left(\mathcal{Q}\right)}_{r,\kappa }{\left(\kappa t\right)}^r}{r!}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q}+r\kappa I,\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }}+r\kappa I;\mathfrak{D}+r\kappa I;z,w\right)({\mathcal{N}}^{\mathrm{\prime }}{)}_{r,\kappa }(\mathfrak{D}{)}_{r,\kappa }^{-1}.\end{array}$

Remark 5.1:

When $\kappa \to 1$ in Equation (56), then we get infinite matrix summation formulas of ${\mathcal{F}}_1\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)$ in [20,24]. Moreover, when $\kappa \to 1$ in Equation (57), then we get infinite matrix summation formulas of ${\mathcal{F}}_1\left(\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)$ in [20].

The list of infinite matrix summation formulas associated with the $\kappa$-Appell matrix function ${\mathcal{F}}_{2,\kappa }$ are presented in the following theorem. Since the proofs are similar to Theorem 5.1, we omit them.

Theorem 5.2:

Let $\kappa \in {\mathbb{R}}^{+},z,w\in \mathbb{C},\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }},\mathfrak{D}$, and ${\mathfrak{D}}^{\mathrm{\prime }}$ be commutative matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, with $\mathfrak{D}+n\kappa I$ and ${\mathfrak{D}}^{\mathrm{\prime }}+n\kappa I$ are invertible matrixes for all integrals $n\ge 0$. Then the following infinite matrix summation formulas hold for the $\kappa$-Appell matrix function ${\mathcal{F}}_{2,\kappa }:$ (61) $\begin{array}{r}\begin{array}{l}\sum _{r\ge 0}(\mathcal{Q}{)}_{r,\kappa }\frac{t^r}{r!}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q}+r\kappa I,\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)\\ =(1-\kappa t{)}^{\frac{-\mathcal{Q}}{\kappa }}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};\frac{z}{1-\kappa \text{t}},\frac{w}{1-\kappa \text{t}}\right),\end{array}\end{array}$(61) where $\left|\frac{z}{1-\kappa \text{t}}\right|+\left|\frac{w}{1-\kappa \text{t}}\right|<\frac{1}{\kappa }$, and $|t|<\frac{1}{\kappa }$. (62) $\begin{array}{r}\begin{array}{l}\sum _{r\ge 0}(\mathcal{N}{)}_{r,\kappa }\frac{t^r}{r!}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N}+r\kappa I,{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)\\ =(1-\kappa t{)}^{\frac{-\mathcal{N}}{\kappa }}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};\frac{z}{1-\kappa \text{t}},w\right),\end{array}\end{array}$(62) where $\left|\frac{z}{1-\kappa \text{t}}\right|+|w|<\frac{1}{\kappa }$, $|t|<\frac{1}{\kappa }$ and $\mathcal{Q}\mathcal{N}=\mathcal{N}\mathcal{Q}$. (63) $\begin{array}{r}\begin{array}{l}\sum _{r\ge 0}(\mathcal{Q}{)}_{r,\kappa }\frac{{\left(\kappa t\right)}^r}{r!}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q}+r\kappa I,\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }}+r\kappa I;\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }}+r\kappa I;z,w\right)({\mathcal{N}}^{\mathrm{\prime }}{)}_{r,\kappa }({\mathfrak{D}}^{\mathrm{\prime }}{)}_{r,\kappa }^{-1}\\ ={\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w+\kappa \text{t}\right),\end{array}\end{array}$(63) where $|z|+|w+\kappa \text{t}|<\frac{1}{\kappa }$, $|t|<\frac{1}{\kappa }$ and ${\mathcal{N}}^{\mathrm{\prime }}\mathfrak{D}=\mathfrak{D}{\mathcal{N}}^{\mathrm{\prime }},{\mathcal{N}}^{\mathrm{\prime }}{\mathfrak{D}}^{\mathrm{\prime }}={\mathfrak{D}}^{\mathrm{\prime }}\mathcal{N}{}^{\mathrm{\prime }}$.

Remark 5.2:

The proofs are similar to Theorem 5.1, for proof Equation (62) using the matrix identities $\sum _{r\ge 0}(\mathcal{N}+m\kappa I{)}_{r,\kappa }\frac{t^r}{r!}=(1-\kappa t{)}^{\frac{-\left(\mathcal{N}+m\kappa I\right)}{k}},|t|<\frac{1}{\kappa },$ $(\mathcal{N}{)}_{m,\kappa }(\mathcal{N}+m\kappa I{)}_{r,\kappa }=(\mathcal{N}{)}_{r,\kappa }(\mathcal{N}+r\kappa I{)}_{m,\kappa }.$When $\kappa \to 1$, then we get infinite matrix summation formulas of ${\mathcal{F}}_2(\mathcal{A},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w)$ in [20].

Theorem 5.3:

Let $\kappa \in {\mathbb{R}}^{+},z,w\in \mathbb{C},\mathcal{Q},\mathcal{Q}{}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }},\text{and }\mathfrak{D}$ be matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, with $\mathfrak{D}+n\kappa I$ is invertible matrix for all integrals $n\ge 0,\text{ and }\mathcal{Q}{\mathcal{Q}}^{\mathrm{\prime }}={\mathcal{Q}}^{\mathrm{\prime }}\mathcal{Q}$. Then the following infinite matrix summation formulas hold for $\kappa$-Appell matrix function ${\mathcal{F}}_{3,\kappa }:$ (64) $\begin{array}{r}\begin{array}{l}\sum _{r\ge 0}({\mathcal{Q}}^{\mathrm{\prime }}{)}_{r,\kappa }\frac{{\left(-t\right)}^r}{r!}{\mathcal{F}}_{3,\kappa }\left(\mathcal{Q},-r\kappa I,\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,\frac{\left(1+\kappa t\right)w}{\kappa t}\right)\\ =(1+\kappa t{)}^{\frac{-\mathcal{Q}}{\kappa }}{\mathcal{F}}_{3,\kappa }\left(\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right),\end{array}\end{array}$(64) where $|z|<\frac{1}{\kappa },\left|\frac{\left(1+\kappa \text{t}\right)w}{\kappa \text{t}}\right|<\frac{1}{\kappa }$ and $|t|<\frac{1}{\kappa }$. (65) $\begin{array}{r}\begin{array}{l}\sum _{r\ge 0}(\mathcal{Q}{)}_{r,\kappa }\frac{{\left(\kappa t\right)}^r}{r!}{\mathcal{F}}_{3,\kappa }\left(\mathcal{Q}+r\kappa I,{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N}+r\kappa I,{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D}+r\kappa I;z,w\right)(\mathfrak{D}{)}_{r,\kappa }^{-1}(\mathcal{N}{)}_{r,\kappa }\\ ={\mathcal{F}}_{3,\kappa }\left(\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z+\kappa \text{t},w\right),\end{array}\end{array}$(65) where $|z+\kappa \text{t}|<\frac{1}{\kappa },|w|<\frac{1}{\kappa }$, $|t|<\frac{1}{\kappa }and\mathcal{Q}\mathcal{N}=\mathcal{N}\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }}\mathcal{N}=\mathcal{N}{\mathcal{Q}}^{\mathrm{\prime }}.$

Proof.

By using Equation (17) and the identity $(-r\kappa I{)}_{n,\kappa }=\frac{{\left(-\kappa \right)}^{n}r!}{\left(r-n\right)!},$ we get $\begin{array}{rl}& \begin{array}{l}\sum _{r\ge 0}({\mathcal{Q}}^{\mathrm{\prime }}{)}_{r,\kappa }\frac{{\left(-t\right)}^r}{r!}{\mathcal{F}}_{3,\kappa }\left(\mathcal{Q},-r\kappa I,\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,\frac{\left(1+\kappa t\right)w}{\kappa t}\right)\\ =\sum _{m,\kappa ,r=0}^{\mathrm{\infty }}\sum _{n=0}^r({\mathcal{Q}}^{\mathrm{\prime }}{)}_{r,\kappa }\frac{{\left(-t\right)}^{r-n}}{\left(r-n\right)!}\frac{{\left(\mathcal{Q}\right)}_{m,\kappa }{\left(\mathcal{N}\right)}_{m,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{n,\kappa }(\mathfrak{D}{)}_{m+n,\kappa }^{-1}}{m!n!}(z{)}^m(1+\kappa \text{t})w{)}^n\end{array}\end{array}$ $\begin{array}{rl}& =\sum _{m,n,\kappa ,r\ge 0}({\mathcal{Q}}^{\mathrm{\prime }}{)}_{n+r,\kappa }\frac{{\left(-t\right)}^r}{\left(r\right)!}\frac{{\left(\mathcal{Q}\right)}_{m,\kappa }{\left(\mathcal{N}\right)}_{m,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{n,\kappa }(\mathfrak{D}{)}_{m+n,\kappa }^{-1}}{m!n!}{z}^m{w}^n(1+\kappa t{)}^n\end{array}$ $\begin{array}{rl}& =\sum _{m,n,\kappa ,r\ge 0}({\mathcal{Q}}^{\mathrm{\prime }}+n\kappa I{)}_{r,\kappa }\frac{{\left(-t\right)}^r}{\left(r\right)!}\frac{{\left(\mathcal{Q}\right)}_{m,\kappa }{\left({\mathcal{Q}}^{\mathrm{\prime }}\right)}_{n,\kappa }{\left(\mathcal{N}\right)}_{n,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{n,\kappa }(\mathfrak{D}{)}_{m+n,\kappa }^{-1}}{m!n!}{z}^m{w}^n(1+\kappa t{)}^n\end{array}$ $\begin{array}{rl}& =\sum _{m,n,\kappa \ge 0}(1+\kappa t{)}^{\frac{-{\mathcal{Q}}^{\mathrm{\prime }}}{\kappa }}\frac{{\left(\mathcal{Q}\right)}_{m,\kappa }{\left({\mathcal{Q}}^{\mathrm{\prime }}\right)}_{n,\kappa }{\left(\mathcal{N}\right)}_{m,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{n,\kappa }(\mathfrak{D}{)}_{m+n,\kappa }^{-1}}{m!n!}{z}^m{w}^n\end{array}$ $\begin{array}{rl}& =(1+\kappa t{)}^{\frac{-\mathcal{Q}}{\kappa }}{\mathcal{F}}_{3,\kappa }\left(\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right).\end{array}$This completes the proof of Equation (64).

We notice that the proof of the formula in Equation (65) is similar to proof formula in Equation (57).

Remark 5.3:

When $\kappa \to 1,$ then we get infinite matrix summation formulas of ${\mathcal{F}}_3\left(\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)$ in [20].

Theorem 5.4:

Let $\kappa \in {\mathbb{R}}^{+},z,w\in \mathbb{C},\mathcal{Q},\mathcal{N},\mathfrak{D},$ and ${\mathfrak{D}}^{\mathrm{\prime }}$be matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$ with $\mathfrak{D}+n\kappa I$ and ${\mathfrak{D}}^{\mathrm{\prime }}+n\kappa I$ are invertible matrixes for all integrals $n\ge 0$. Then the following infinite matrix summation formulas hold for the $\kappa$-Appell matrix function ${\mathcal{F}}_{4,\kappa }:$ (66) $\begin{array}{r}\begin{array}{l}\sum _{r\ge 0}(\mathcal{Q}{)}_{r,\kappa }(\mathcal{N}{)}_{r,\kappa }\frac{{\left(\kappa t\right)}^r}{r!}{\mathcal{F}}_{4,\kappa }\left(\mathcal{Q}+r\kappa I,\mathcal{N}+r\kappa I;\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }}+r\kappa I;z,w\right)({\mathfrak{D}}^{\mathrm{\prime }}{)}_{r,\kappa }^{-1}\\ ={\mathcal{F}}_{4,\kappa }\left(\mathcal{Q},\mathcal{N};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w+\kappa \text{t}\right),\end{array}\end{array}$(66) where $\mathcal{Q}\mathcal{N}=\mathcal{N}\mathcal{Q},\sqrt{|z|}+\sqrt{|w+\kappa \text{t}|}<\frac{1}{\sqrt{\kappa }}$.

Proof.

The proof is similar to Equation (57).

Remark 5.4:

When $\kappa \to 1,$ then we get infinite matrix summation formulas of ${\mathcal{F}}_4\left(\mathcal{Q},{\mathcal{Q}}^{\mathrm{\prime }},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)$ in [20].

6. Differentiation formulas

In this section, we establish some differentiation formulas for the $\kappa$-Appell matrix functions ${\mathcal{F}}_{1,\kappa }and{\mathcal{F}}_{2,\kappa }$. In the sequel, the notation $D_z^n\mathcal{F}={\left(\frac{d\mathcal{F}}{dz}\right)}^n$ will be used.

Theorem 6.1:

Let $\kappa \in {\mathbb{R}}^{+},z,w\in \mathbb{C}$, $\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }}\text{and }\mathfrak{D}$ be positive stable matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, such that $\mathfrak{D}+n\kappa I$ and ${\mathfrak{D}}^{\mathrm{\prime }}+n\kappa I$ are invertible matrices for all integrals $n\ge 0$. Then, for $|z|<\frac{1}{\kappa },|w|<\frac{1}{\kappa }$ the differentiation formulas of ${\mathcal{F}}_{1,\kappa }$ are defined as follows: (67) $\begin{array}{rl}& \begin{array}{l}{D}_z^n\left[{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)\right]=(\mathcal{Q}{)}_{n,\kappa }(\mathcal{N}{)}_{n,\kappa }[{\left(\mathfrak{D}\right)}_{n,\kappa }{]}^{-1}\\ {\mathcal{F}}_{1,\kappa }\left(\mathcal{Q}+n\kappa I,\mathcal{N}+n\kappa I,{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D}+n\kappa I;z,w\right),\end{array}\end{array}$(67) (68) $\begin{array}{rl}& \begin{array}{l}{D}_z^n\left[z^{\frac{\mathcal{Q}}{\kappa }+\left(n-1\right)I}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,zw\right)\right]=\frac{1}{{\kappa }^n}{z}^{\frac{\mathcal{Q}}{\kappa }-I}(\mathcal{Q}{)}_{n,\kappa }\\ {\mathcal{F}}_{1,\kappa }\left(\mathcal{Q}+n\kappa I,\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,zw\right),\end{array}\end{array}$(68) (69) $\begin{array}{rl}& \begin{array}{l}{D}_w^n\left[w^{\frac{{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }+\left(n-1\right)I}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)\right]=\frac{1}{{\kappa }^n}{w}^{\frac{{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }-I}({\mathcal{N}}^{\mathrm{\prime }}{)}_{n,\kappa }\\ {\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }}+n\kappa I;\mathfrak{D};z,w\right).\end{array}\end{array}$(69)

Proof.

These formulas can be established using the series expansions of ${\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w\right)$ in Equation (15). Here we prove Formula (68), the other differentiation formulas can be proven in a similar manner. $\begin{array}{l}{D}_z^n\left[z^{\frac{\mathcal{Q}}{\kappa }+\left(n-1\right)I}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,zw\right)\right]\\ =\sum _{s,r\ge 0}\frac{{\left(\mathcal{Q}\right)}_{s+r,\kappa }{\left(\mathcal{N}\right)}_{s,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{r,\kappa }(\mathfrak{D}{)}_{s+r,\kappa }^{-1}}{s!r!}{w}^r{D}_z^n\left[z^{\frac{\mathcal{Q}+\left(n-1+s+r\right)\kappa I}{k}}\right]\\ =\frac{1}{{\kappa }^n}\sum _{s,r\ge 0}\frac{{\left(\mathcal{Q}\right)}_{s+r,\kappa }{\left(\mathcal{N}\right)}_{s,\kappa }{\left(\mathcal{N}{}^{\mathrm{\prime }}\right)}_{r,\kappa }(\mathfrak{D}{)}_{s+r,\kappa }^{-1}}{s!r!}{w}^r\times (\mathcal{Q}+\left(s+r\right)\kappa I{)}_{n,\kappa }{z}^{\frac{\mathcal{Q}+\left(s+r-1\right)\kappa I}{k}}.\end{array}$Using the $\kappa$-Pochhammer matrix identity $(\mathcal{Q}{)}_{s+r,\kappa }(\mathcal{Q}+\left(s+r\right)\kappa I{)}_{n,\kappa }=(\mathcal{Q}{)}_{n,\kappa }(\mathcal{Q}+n\kappa I{)}_{s+r,\kappa },$ we get $\begin{array}{l}{D}_z^n\left[z^{\frac{\mathcal{Q}}{\kappa }+\left(n-1\right)I}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,zw\right)\right]\\ =\frac{1}{{\kappa }^n}(\mathcal{Q}{)}_{n,\kappa }\sum _{s,r\ge 0}\frac{{\left(\mathcal{Q}+n\kappa I\right)}_{s+r,\kappa }{\left(\mathcal{N}\right)}_{s,\kappa }{\left({\mathcal{N}}^{\mathrm{\prime }}\right)}_{r,\kappa }(\mathfrak{D}{)}_{s+r,\kappa }^{-1}}{s!r!}{w}^r{z}^{\frac{\mathcal{Q}+\left(s+r-1\right)\kappa I}{\kappa }}\\ =\frac{1}{{\kappa }^n}(\mathcal{Q}{)}_{n,\kappa }{z}^{\frac{\mathcal{Q}}{\kappa }-I}{\mathcal{F}}_{1,\kappa }\left(\mathcal{Q}+n\kappa I,\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,zw\right).\end{array}$

Remark 6.1:

When $\kappa \to 1,$ then we obtain differentiation formulas of ${\mathcal{F}}_1$ in [24].

Theorem 6.2:

Let $\kappa \in {\mathbb{R}}^{+},z,w\in \mathbb{C}\text{ and }\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }},\mathfrak{D}$ and ${\mathfrak{D}}^{\mathrm{\prime }}$ be positive stable matrices in ${\mathbb{C}}^{\mathcal{M}\times \mathcal{M}}$, such that $\mathfrak{D}+n\kappa I$ and ${\mathfrak{D}}^{\mathrm{\prime }}+n\kappa I$are invertible matrices for all integrals $n\ge 0$. Then, for $\mathcal{Q}\mathcal{N}=\mathcal{N}\mathcal{Q}\text{ and }\mathfrak{D}{\mathfrak{D}}^{\mathrm{\prime }}={\mathfrak{D}}^{\mathrm{\prime }}\mathfrak{D}$ the differentiation formulas of ${\mathcal{F}}_{2,\kappa }$ are defined as the following form: (70) $\begin{array}{rl}& \begin{array}{l}{D}_z^n[{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)]=(\mathcal{Q}{)}_{n,\kappa }(\mathcal{N}{)}_{n,\kappa }[{\left(\mathfrak{D}\right)}_{n,\kappa }{]}^{-1}\\ [{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q}+n\kappa I,\mathcal{N}+n\kappa I,{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D}+n\kappa I,\mathcal{C}{}^{\mathrm{\prime }};z,w\right)],\end{array}\end{array}$(70) (71) $\begin{array}{rl}& \begin{array}{l}{D}_w^n[{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)]=(\mathcal{Q}{)}_{n,\kappa }(\mathcal{N}{}^{\mathrm{\prime }}{)}_{n,\kappa }[{\left({\mathfrak{D}}^{\mathrm{\prime }}\right)}_{n,\kappa }{]}^{-1}\\ [{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q}+n\kappa I,\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }}+n\kappa I;\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }}+n\kappa I;z,w\right)],\end{array}\end{array}$(71) (72) $\begin{array}{rl}& \begin{array}{l}{D}_w^n\left[w^{\frac{\mathcal{Q}}{\kappa }+\left(n-1\right)I}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)\right]=\frac{1}{{\kappa }^n}{w}^{\frac{\mathcal{Q}}{\kappa }-I}(\mathcal{Q}{)}_{n,\kappa }\\ [{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q}+n\kappa I,\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)],\end{array}\end{array}$(72) (73) $\begin{array}{rl}& \begin{array}{l}{D}_w^n\left[w^{\frac{{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }+\left(n-1\right)I}{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)\right]=\frac{1}{{\kappa }^n}{w}^{\frac{{\mathcal{N}}^{\mathrm{\prime }}}{\kappa }-I}({\mathcal{N}}^{\mathrm{\prime }}{)}_{n,\kappa }\\ [{\mathcal{F}}_{2,\kappa }\left(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }}+n\kappa I;\mathfrak{D},{\mathfrak{D}}^{\mathrm{\prime }};z,w\right)].\end{array}\end{array}$(73)

Proof.

These formulas can be established using the series expansions of ${\mathcal{F}}_{2,\kappa }(\mathcal{Q},\mathcal{N},{\mathcal{N}}^{\mathrm{\prime }};\mathfrak{D};z,w)$ in Equation (16). Proof of the differentiation formulas are similar that of Theorem 6.1.

Remark 6.2:

When $\kappa \to 1,$ then we get differentiation formulas of ${\mathcal{F}}_2$ in [22].

7. Conclusion

The $\kappa$-Appell hypergeometric functions are a generalization of the standard hypergeometric functions and are used in a variety of fields, such as physics, engineering, and mathematics [6–11]. Their applications include solving differential equations, computing integrals, modelling various phenomena, such as wave propagation, and particularly obtaining exact solutions for various problems in quantum mechanics [12,13]. Additionally, they are related to other special functions, such as Meijer $G$-function and Fox $H$-function, which also have important applications in various areas of science and engineering. A recent surge in research has occurred on extend the classical Appell hypergeometric functions to the matrix setting, with many authors contributing to this field of study.

This work is devoted to the study of $\kappa$-Appell hypergeometric matrix functions within the analysis. This study is assumed to be the improvement and generalization of the scalar case [30,33,37], in which the matrix case has never been investigated. If we take $\kappa \to 1$, we can obtain some results to reduce the classical Appell hypergeometric functions ${\mathcal{F}}_1,{\mathcal{F}}_2,{\mathcal{F}}_3,\text{ and }{\mathcal{F}}_4$ in [19–24,36]. In this work, we introduced the $\kappa$-Appell hypergeometric matrix functions and further investigated their mathematical properties, including integral representation, differential formulas, transformation formulas and infinite matrix summation formulas. The results are novel and will be valuable in future research.

Interesting problems arise from this work. Future research in this area must focus on exploring a $\kappa$-generalization of other special functions, similar to what was performed for the scalar case in references [38–41]. Such problems offer researchers a wealth of opportunities to expand this new area.

Acknowledgements

All authors are also indebted to the anonymous reviewers for their helpful, valuable comments and suggestions to improve this manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).