Certain pl ( m , n )-Kummer Matrix Function of Two Complex Variables under Differential Operator

The main aim of this paper is to define and study of a new matrix functions, say, the pl(m, n)-Kummer matrix function of two complex variables. The radius of regularity, recurrence relation and several new results on this function are established when the positive integers p is greater than one. Finally, we obtain a higher order partial differential equation satisfied by the pl(m, n)-Kummer matrix function and some special properties.


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][4][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][7][8][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 2 H matrix function of two complex variables under differential operators [11][12][13].In [14][15][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.
, then from the properties of the matrix functional calculus [18], it follows that The reciprocal gamma function denoted by z is an entire function of the complex variable .Then for any matrix where and one gets [6]  Jódar and Cortés have proved in [6], that  [19] and keeping in mind ( ) Summarizing, the following result has been established.
Theorem 2.1.Let A and be matrices in are invertible for all integer .Then, the pl(m, n)-Kummer matrix function is an entire function.
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 , , .
By the same way, we have Now, we consider the following differential operators where , Putting in this relation 1 m  and instead of and n respectively, then and so that we can be written the relation Therefore, the power series , the pl(m, n)-Kummer matrix function is a solution of the matrix differential equation ; , (2.7) In this paper, we affect by differential operator D the pl(m, n)-Kummer matrix function, successively, then we have ; ; , ; , ; ; , i.e. the (m, n)-Kummer matrix function is a solution to this matrix differential equation Therefore, the following result has been established.
Thus by mathematical induction, we have the following general form , 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.

p 2 A
Throughout this paper for a matrixA in N N C  , its spectrum   A  denotes the set of all the eigenvalues of A .If A is a matrix in N N C  , its two-norm denoted by is norm of .y If   f z and   g z zare holomorphic functions of the complex variable , defined in an open set  of the complex plane, and if A and are a matrix in B

Theorem 2 . 2 .
Let A and be matrices in B N N C  .Then the pl(m, n)-Kummer matrix function is solution of this matrix differential equation Special cases: we can be written the matrix function N