Pseudo Laguerre Matrix Polynomials , Operational Identities and Quasi-Monomiality

The main purpose of this paper is to introduce the matrix extension of the pseudo Laguerre matrix polynomials and to explore the formal properties of the operational rules and the principle of quasi-monomiality to derive a number of properties for pseudo Laguerre matrix polynomials.


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 [1]- [7].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 [5] [8] [9].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 [4] [9]- [19].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 [9] [18] [19] [20] [21].We say that a matrix A in A ≠ , then we call ( ) For any matrix A in N N × » , we have the following relation [22] ( ) ( ) Next, we recall that the Konhauser matrix polynomials are defined in [21] as In [23] being the matrix version of the Tricomi function defined in (see [4]).
For the purpose of this work we introduce the following matrix version of Kampé de Fériet double hypergeometric series [ ] x y and matrix version of the generalized hypergeometric function p q F [24] as follows: ; ; and ( ) ( ) ( ) ( ) In view of the definition (1.9) and the definition of the matrix version of the Gauss multiplication theorem ( ) where throughout this work ( ) denotes the array of m parameters ( )

Operational Identities and Quasi-Monomiality
First of all, we establish the following operational representations for pseudo Laguerre matrix polynomials Proof.In view of (1.10) and (1.11), we have ( )

ˆ1 , ks A A ks x A I D x
A ks I x 3) The desired result now follows by applying the identities (2.2) and (2.3) to the definition (1.5).Theorem 2.2.Let A be a matrix in Proof.The result follows directly from the formula and the derivatives operators According to the quasi-monomiality properties, we have ˆ, ; , , ; , ˆ, ; , , ; , . , , ; , 0.

P M L x y k A L x y k A P M L x y k A L x y k
From the lowering operators 1 P and 2 P in (2.6) and (2.7), we can define operators playing the role of the inverse operators  P − (see [ [8], Equation (15)]).Thus, we get ( ) and they satisfy , ; , ˆ, ; , , ; , . 1

L x y k A P L x y k A P L x y k A n
Clearly, we have ˆˆˆ, ; , , ; , , ; , .ˆˆˆ, ; , 0.
Moreover, from (2.5) in conjunction with (2.8), we get which yields the following recurrence relation (

Generating Functions and Expansions
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 [14], the following definition of Laguerre matrix polynomials has been in- is not an eigenvalue of A for every integer 0 α > and λ be a complex number whose real part is posi- tive.Such matrix polynomials have the following operational representation [14]: Let us consider 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 ( ) , , n H x y A (see [10]) and (2.4), we can easily derive the following bilateral generating function   Proof.According to the operational representation (2.4), we have

4 )
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 N N k A can be obtained by the method suggested in[23], thus getting Theorem 1.1.Let A be a matrix in N N × » satisfying the condition 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 con-Advances in Linear Algebra & Matrix Theory structed matrix polynomials in (1.5).

= 1 ˆ
assertion (2.3) and the definition(1.5).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[23] [25], a polynomial set » is quasi-monomial, if there exist two operators M and P , called multiplicative and derivative operators respectively, which when acting on the polynomialsThe operators M and P satisfy the commutation relation: a Weyl group structure.If M and P have differential reali- zation, then the differential equations satisfied by In this regard, the matrix polynomial set ( ) , ; , n L x y k A is quasi-monomial M. G. Bin-Saad, M. A. Pathan DOI: 10.4236/alamt.2018.8200891 Advances in Linear Algebra & Matrix Theory under the action of the multiplicative operator L x y k A P P L x y k A L x y k A − − = = Further, from (2.9)-(2.11),we can infer that ( ) , ; , n L x y k A are the natural solution of the following equation upon using (2.4) one obtains by routine calculations

3 . 1 .
Let A and B be a matrices inN N

F
is defined by (1.9).
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). Dattoli et al. introduced the two variable pseudo Laguerre polynomials n L x y k j in the form: