1. Preliminaries and Definitions
In the last two decade, matrix polynomials have become more important and some results in the theory of classical orthogonal polynomials have been extended to orthogonal matrix polynomials see for instance  -  . Orthogonal matrix polynomials are important from both the theoretical and practical points of view, they appear in connection with representation theory, matrix expansion problems, prediction theory and in the matrix quadrature integration problems, see for example    . Numerous problems of chemistry, physics and mechanics are related to second order matrix differential equation. Moreover, some properties of the Hermite and Laguerre matrix polynomials and a generalized form of the Hermite matrix polynomials have been introduced and studied in   -  . Other classical orthogonal polynomials as Gegenbauer, Chebyshev, Jacobi and Konhauser polynomials have been extended to orthogonal matrix polynomials, and some results have been investigated, see for example      . We say that a matrix A in is a positive stable if for all , where is the set of the eigenvalues of A. If are elements of and , then we call
a matrix polynomial of degree n in x. If is invertible for every integer then
Thus we have
For any matrix A in , we have the following relation 
Next, we recall that the Konhauser matrix polynomials are defined in  as
In  Dattoli et al. introduced the two variable pseudo Laguerre polynomials in the form:
In this work, we construct a matrix version of the pseudo Laguerre matrix polynomials given by (1.4) as follows:
Definition 1.1. Let A be a matrix in satisfying the condition for every , and . We define the pseudo-Laguerre matrix polynomials by the series
The relevant generating function for the polynomials can be obtained by the method suggested in  , thus getting
Theorem 1.1. Let A be a matrix in satisfying the condition for every , ,and . Then
being the matrix version of the Tricomi function defined in (see  ).
Proof. If we use the series (1.7) in right-hand side of (1.6), we get
Now, by letting , we obtain the left-hand side of the assertion (1.6).W
We must emphasize that the matrix polynomials in (1.6) are a generalized form of Konhauser matrix polynomials defined by (1.3) and indeed we have
For the purpose of this work we introduce the following matrix version of Kampé de Fériet double hypergeometric series and matrix version of the generalized hypergeometric function  as follows:
In view of the definition (1.9) and the definition of the matrix version of the Gauss multiplication theorem
it is not difficult to show that
where throughout this work denotes the array of m parameters For an arbitrary matrix the following two formulas are well-known consequences of the derivative operator and the integral 
where and .
Note that, in this work we apply the concept of the right-Riemann-Liouville fractional calculus to obtain operational identities and relations. Motivated by the works mentioned above, we aim in this work to present systematic investigation of the matrix version of the pseudo Laguerre polynomials given by (1.5) and exploit methods of operational nature and the monomiality principle to derive a number of operational representations, operators and generating functions constructed matrix polynomials in (1.5).
2. Operational Identities and Quasi-Monomiality
First of all, we establish the following operational representations for pseudo Laguerre matrix polynomials .
Theorem 1.1. Let A be a matrix in satisfying the condition for every and . Then
Proof. In view of (1.10) and (1.11), we have
The desired result now follows by applying the identities (2.2) and (2.3) to the definition (1.5).W
Theorem 2.2. Let A be a matrix in satisfying the condition for every and . Then
Proof. The result follows directly from the formula
the assertion (2.3) and the definition (1.5).W
The use of the monomiality principle has offered a powerful tool for studying the properties of families of special functions and polynomials. We know that according to the monomiality principle   , a polynomial set is quasi-monomial, if there exist two operators and , called multiplicative and derivative operators respectively, which when acting on the polynomials yield 
The operators and satisfy the commutation relation:
and thus display a Weyl group structure. If and have differential realization, then the differential equations satisfied by are
In this regard, the matrix polynomial set is quasi-monomial under the action of the multiplicative operator
and the derivatives operators
According to the quasi-monomiality properties, we have
Therefore, the identities
in differential forms give us
Moreover, regarding the Lie bracket defined by , we led to
From the lowering operators and in (2.6) and (2.7), we can define operators playing the role of the inverse operators and (see [  , Equation (15)]). Thus, we get
and they satisfy
Clearly, we have
Further, from (2.9)-(2.11), we can infer that are the natural solution of the following equation
Moreover, from (2.5) in conjunction with (2.8), we get
which yields the following recurrence relation
then upon using (2.4) one obtains by routine calculations
3. Generating Functions and Expansions
First, in the identity (2.1) multiply throughout by , sum and then employ the formulae (1.10) and (1.11) and the result
Next, let us consider the generating relation
which according to operational identity (2.4)and the formulae (1.10) and (1.11) yields the following bilinear generating function
In  , the following definition of Laguerre matrix polynomials has been introduced:
where A be a matrix in , and is not an eigenvalue of A for every integer and be a complex number whose real part is positive. Such matrix polynomials have the following operational representation  :
Let us consider the generating relation
Now, directly from (2.4) and (3.1) by employing the previously outlined method leading to the bilinear generating function, we obtain from (3.2) the following bilateral generating function
Similarly, from the operational representation of the two variable Hermite matrix polynomials (see  )
and (2.4), we can easily derive the following bilateral generating function
Theorem 3.1. Let A and B be a matrices in satisfying the conditions for every or , , and . Then
where is defined by (1.9).
where is defined by (1.8).
Proof. According to the operational representation (2.4), we have
which in view of (1.2), the operator in (1.11) and the definition of Pochhammer symbol (1.2), yields the right-hand side of Equation (3.4). Similarly, one can prove the result (3.3).W
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