q-Analogue Modified Laguerre Matrix Polynomials of Three Variables

       In this paper, the q-analogue modified Laguerre matrix polynomials of three variables are introduced as finite series and Some properties of these matrix polynomials are obtained.


Introduction
Matrix generalization of special functions has become important in the last two decades. The reason of importance have many motivations. For instance, using special matrix functions provides solutions for some physical problems. Also, special matrix functions are in connection with different matrix functions.
Throughout this paper, for a matrix in × , its spectrum ( ) denotes the set of all eigenvalues of .
(1.5) We conclude this section by recalling the Laguerre matrix polynomials. Let be a matrix in × such that − ∉ ( ) for every integer > 0 and be a complex number whose real part is positive. Then the Laguerre matrix polynomials ( , ) ( ) are defined by [11]: (1.6) The generating function of Laguerre matrix polynomials is given in [11] by , ∈ ℂ, | | < 1, ∈ ℂ, and Rodrigues formula is (1.9) In [3], it is shown that an appropriate combination of methods, relevant to operational calculus and to matrix polynomials, can be a very useful tool to establish and treat a new class of two variable Laguerre matrix polynomials in the following form: The generating relation for the matrix function , ( , ) ( , ) is given by the formula: , where { , , , } ∈ ℂ and | + | < 1.
Recently, q-calculus has served as a bridge between mathematics and physics. Therefore, there is a significant increase of activity in the area of the q-calculus due to its applications in mathematics, statistics and physics.
Let the q-analogues of Pochhammer symbol or q-shifted factorial be defined by [8] ( The q-analogue of the power (binomial) function ( ± ) ( [17]) is given by The formulas for the q-difference of a addition, a product and a quotient of functions are The q-exponential function is defined by [21]: Mohsen and Alsarahi [16] introduced q-analogue modified Laguerre polynomial of two variables by the following: where ϕ 1 1 is the basic hypergeometric or q-hypergeometric function. The generating relation for , ( , ) ( , ) is given by the formula: (1.23)

q-Analogue Modified Laguerre Matrix Polynomials of Three Variables
In this section, we introduce the q-analogue modified Laguerre matrix polynomial of three variables by the following generating function: where , , , , ∈ ℂ, | ( + )| < 1. Now, we get the series representation of the q-analogue modified Laguerre matrix polynomials in the form of the following theorem: Let us assume that is a matrix in × and , be a complex number whose real part is positive, then the series representation of the q-analogue modified Laguerre matrix polynomials   Thus, by same manner as above, we can obtain Hence, by continuing the above steps, we get the required relation (2.8).

Theorem 2.3
The q-analogue modified Laguerre matrix polynomials of three variables ,