1. Introduction
Many Special matrix functions appear in connection with statistics [1], mathematical physics, theoretical physics, group representation theory, Lie groups theory [2] and orthogonal matrix polynomials are closely related [3-5]. The hypergeometric matrix function has been introduced as a matrix power series and an integral representation and the hypergeometric matrix differential equation in [6-9] and the explicit closed form general solution of it has been given in [10]. The author has earlier studied the Kummer’s and Horn’s
matrix function of two complex variables under differential operators [11-13]. In [14-16], extension to the matrix function framework of the classical families of p-Kummer’s matrix function,
and q-Appell matrix function and Humbert matrix function have been proposed.
Throughout this paper for a matrix
in
, its spectrum
denotes the set of all the eigenvalues of
. If
is a matrix in
, its two-norm denoted by
is defined by [17]
![](https://www.scirp.org/html/16-7401156\6813fb45-122c-4e9a-af87-84b9851adb7a.jpg)
where for a vector
in
,
is the Euclidean norm of
.
If
and
are holomorphic functions of the complex variable
, defined in an open set
of the complex plane, and if
and
are a matrix in
with
and
also and if
, then from the properties of the matrix functional calculus [18], it follows that
(1.1)
The reciprocal gamma function denoted by
is an entire function of the complex variable
. Then for any matrix
in
, the image of
acting on
denoted by
is a well defined matrix. Furthermore, if
(1.2)
where
is the identity matrix in
, then
is invertible, its inverse coincides with
and one gets [6]
(1.3)
Jódar and Cortés have proved in [6], that
(1.1)
2. On pl(m, n)-Kummer Matrix Function
We We define the pl(m, n)-Kummer matrix function
of two complex variables in the form
(2.1)
where
,
![](https://www.scirp.org/html/16-7401156\909dfd1b-044d-4f27-abe4-88421928c31a.jpg)
[19] and
,
and
are non-negative integer numbers. Notice that
is a non-negative integer number.
For simplicity, we can write the
in the form
,
in the form
and
in the form
.
We begin the study of this function by calculating its radius of regularity
of such function for this purpose we recall relation (1.3.10) of [19] and keeping in mind that
. Hence
![](https://www.scirp.org/html/16-7401156\a8aaa677-e880-4985-ab7e-17968b5c23f3.jpg)
where
![](https://www.scirp.org/html/16-7401156\73b0e067-d0f3-4740-9d81-8d7a630791e0.jpg)
Summarizing, the following result has been established.
Theorem 2.1. Let
and
be matrices in
such that
are invertible for all integer
. Then, the pl(m, n)-Kummer matrix function is an entire function.
For
, we have
![](https://www.scirp.org/html/16-7401156\c6ac93bd-d36c-4713-9040-c52179c7810c.jpg)
i.e., the l(m, n)-Kummer matrix function is an entire function.
Some matrix recurrence relations are carried out on the pl(m, n)-Kummer matrix function. In this connection the following matrix contiguous functions relations follow, directly by increasing or decreasing one in original relation
![](https://www.scirp.org/html/16-7401156\058d9df3-4394-4ff3-9a5b-0080ce5d488b.jpg)
(2.2)
Similarly
(2.3)
By the same way, we have
(2.4)
Now, we consider the following differential operators
![](https://www.scirp.org/html/16-7401156\fc1474ae-5803-44b1-bc7b-7d07117d84ca.jpg)
where
,
and
.
It is clear that
![](https://www.scirp.org/html/16-7401156\e179dbe2-a5e1-44f7-aba5-4dfc39184ce2.jpg)
(2.5)
So that
(2.6)
Putting in this relation
and
instead of
and
respectively, then
![](https://www.scirp.org/html/16-7401156\04e0d793-e3c8-44e1-93f1-ec1adf21b4da.jpg)
and so that we can be written the relation
and
instead of
and
yields
![](https://www.scirp.org/html/16-7401156\fae50e3a-ab0b-4f61-b46d-7e5b1c24e5f4.jpg)
Therefore, the power series
, as follows
![](https://www.scirp.org/html/16-7401156\8bcf2b75-aeec-4c56-94ff-0dcdb8fe771d.jpg)
i.e., the pl(m, n)-Kummer matrix function is a solution of the matrix differential equation
![](https://www.scirp.org/html/16-7401156\88e0c6d1-5eef-4c23-a890-eba534b38c14.jpg)
(2.7)
In this paper, we affect by differential operator D the pl(m, n)-Kummer matrix function, successively, then we have
![](https://www.scirp.org/html/16-7401156\d7d26ba9-dbe0-4c06-91e8-4150be826b23.jpg)
i.e. the (m, n)-Kummer matrix function is a solution to this matrix differential equation
(2.8)
Then
![](https://www.scirp.org/html/16-7401156\fd6fb552-c33c-416e-a239-214fd5bc679d.jpg)
Therefore, the following result has been established.
Theorem 2.2. Let
and
be matrices in
. Then the pl(m, n)-Kummer matrix function is solution of this matrix differential equation
(2.9)
The
differential operator has been defined by Sayyed [19] in the form
![](https://www.scirp.org/html/16-7401156\355e9364-e0b1-4169-a02a-88e76e758a64.jpg)
From (2.1), (2.3) and (2.5), we obtain
![](https://www.scirp.org/html/16-7401156\e72bfde7-8110-4cfa-b9cd-c512c5c6d4d6.jpg)
(2.10)
hence
(2.11)
and
(2.12)
Thus by mathematical induction, we have the following general form
![](https://www.scirp.org/html/16-7401156\5e8873c6-f3ae-4051-95fc-226656130308.jpg)
(2.13)
where
is a finite positive integer.
Special cases: we can be written the matrix function
in the form
(2.14)
we see that
![](https://www.scirp.org/html/16-7401156\08424991-6c34-40a3-a0bd-96d128baafd7.jpg)
i.e., the
is a solution to this matrix differential equation
(2.15)
Also
![](https://www.scirp.org/html/16-7401156\dfc72e72-09df-49c7-8daf-806a750128dd.jpg)
i.e., the
is a solution for the matrix partial differential equations
![](https://www.scirp.org/html/16-7401156\0fac512e-5968-4e14-aba8-56a7fc026809.jpg)
The results of this paper are variant, significant and so it is interesting and capable to develop its study in the future. One can use the same class of differential operators for some other function of several complex variables. Hence, new results and further applications can be obtained.
3. Acknowledgements
The Author expresses his sincere appreciation to Dr. M. S. Metwally, (Department of Mathematics, Faculty of Science (Suez), Suez Canal University, Egypt) for his kind interest, encouragements, help, suggestions, comments and the investigations for this series of papers.
The author would like to thank the referees for his comments and suggestions on the manuscript.