On p and q-Horn ’ s Matrix Function of Two Complex Variables

The main aim of this paper is to define and study of a new Horn’s matrix function, say, the p and q-Horn’s matrix function of two complex variables. The radius of regularity on this function is given when the positive integers p and q are greater than one, an integral representation of 2( , , , ; ; , ) Hq p A A B B C z w   is obtained, recurrence relations are established. Finally, we obtain a higher order partial differential equation satisfied by the p and q-Horn’s matrix function.


Introduction
Many special functions encountered in mathematical physics, theoretical physics, engineering and probability theory are special cases of hypergeometric functions [1].Hypergeometric series in one and more variables occur naturally in a wide variety of problems in applied mathematics, statistics [2][3][4], and operations research and so on [5].In [6,7], the hypergeometric matrix function has been introduced as a matrix power series and an integral representation.Moreover, Jódar and Cortés introduced, studied the hypergeometric matrix function ( , ; ; )  F A B C z , the hypergeometric matrix differential equation in [8] and the explicit closed form general solution of it has been given in [9].Upadhyaya and Dhami have earlier studied the generalized Horn's functions of matrix arguments with real positive definite matrices as arguments [10] and this function 7 H also [11], while the author has earlier studied the Horn's matrix function H 2 of two complex variables under differential operators [7].In [12,13], extension to the matrix function framework of the classical families of p-Kummer's matrix functions and p and q-Appell matrix functions have been proposed.
Our purpose here is to introduce and study an extension of the matrix functions of two variables.This paper is organized as follows: Section 2 contains the definition of the p and q-Horn's matrix function of two variables, its radius of regularity and integral relation of the p and q-Horn's matrix function is given.Some matrix recu-rrence relations are established in Section 3. Finally, the effect of differential operator on this function is investigated and p and q-Horn's matrix partial differential equation are obtained in Section 4.
Throughout this paper 0 will denote the complex plane cut along the negative real axis.The spectrum of a matrix , then from the properties of the matrix functional calculus [15], it follows that The reciprocal gamma function denoted by is an entire function of the complex variable .Then for any matrix , the image of acting on A denoted by is invertible for every non negative integer where I is the identity matrix in N N C  , then ( ) A  is invertible, its inverse coincides with 1 ( ) A   and one gets [8] 1 0 Jódar and Cortés have proved in [16], that .
Let P and Q be commuting matrices in N N C  such that the matrices and are invertible for every integer .Then according to [8], we have

Definition of p and q-Horn's Matrix Function
Suppose that p and q are positive integers.The p and q-Horn's matrix function 2 ( , , , ; ; , ) of two complex variables is written in the form where and , , ( , ) For simplicity, we can write the 2 ( , , , ; ; , )  .We begin the study of this function by calculating its radius of regularity R of such function for this purpose we recall relation (1.3.10) of [17,18] and keeping in mind that 2 , 1 2 . We define the radius of regularity of the function 2 ( , , , ; ; , ) , Using Stirling formula and take m n ( 1) Summarizing, the following result has been established.As a conclusion, we get the following result.
Theorem 2.1.Let A , A , , and be matrices in . Then, the p and q-Horn's matrix function is an entire function in the case that, at least, one of the integers p and q are greater than one.
Integral form of the p and q-Horn Matrix Function Suppose that A and are matrices in the space C N N C  of the square complex matrices, such that A C CA   , A , and are positive stable matrices.

C C A 
By (1.3), (1.4) and (1.7) one gets Substituting from (2.1) and (2.2), we see that Therefore, the following result has been established.Theorem 2.2.Let A , A , , and be matrices in Then the p and q-Horn's matrix function of two complex variables satisfies the following integral form

Matrix Recurrence Relations
Some recurrence relation are carried out on the p and q-Horn's matrix function.In this connection the following contiguous functions relations follow, directly by increasing or decreasing one in original relation

The p and q-Horn's Matrix Function under the Differential Operator
Consider the differential operator D on the p and q-Horn's matrix function of two complex variables, defined in [7,17] as 1, otherwise . This operator has the property .( ) Dz w m n z w   p q For the and -Horn's matrix function the following relations hold By the way, we have From (4.1), (4.2) and ( 4.3), we get H Also from (4.2), (4.3) and (4.4), we see that H Now, we append this section by introducing the dif- to the entire functions in successive manner as follows; , , ; ; , )  ( , , ,  ; ; ,  ( )   H H We can written the 2 ( , , , ; ; , )