APMAdvances in Pure Mathematics2160-0368Scientific Research Publishing10.4236/apm.2012.26064APM-24670ArticlesPhysics&Mathematics On Humbert Matrix Polynomials of Two Variables haziS. Khammash1*A.Shehata2*Department of Mathematics, Al-Aqsa University, Gaza Strip, PalestineDepartment of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt* E-mail:ghazikhamash@yahoo.com(HSK);drshehata2006@yahoo.com(AS);091120120206423427July 13, 2012September 6, 2012 September 14, 2012© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

In this paper we introduce Humbert matrix polynomials of two variables. Some hypergeometric matrix representations of the Humbert matrix polynomials of two variables, the double generating matrix functions and expansions of the Humbert matrix polynomials of two variables in series of Hermite polynomials are given. Results of Gegenbauer ma-trix polynomials of two variables follow as particular cases of Humbert matrix polynomials of two variables.

Humbert Matrix Polynomials of Two Variables; Hypergeometric Matrix Function; Matrix Functional Calculus
1. Introduction

The special matrix functions appear in statistics, lie group theory and number theory [1-4] and the 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 [5-9].

If is the complex plane cut along the negative, real axis and log denotes the principal logarithm of z (Saks, S. and A. Zygmund, ), then represents if A is a matrix in the set of all the eigenvalues of is denoted by the set of all the eigenvalues of A is denoted by . If and are holomorphic functions of the complex variable z, which are defined in an open set of the complex plane, and A is a matrix in such that . Then from the properties of the matrix functional calculus, (Dunford N. and J. Schwartz J. ), it follows that . If A is a matrix with , then denotes the a image by of the matrix functional calculus acting on the matrix A. we say that A is a positive stable matrix if for all .

For any matrix P in we will exploit the following relation due to 

Khammash , define the Gegenbauer matrix polynomials of two variables by

From which it follows that is a matrix polynomial in two variables x and y of degree precisely n in x and k in y.

Also we recall that if are matrix in for and that it follows that (Defez and Jódar )

and, for m is a positive integer such that , then

We define Humbert matrix polynomials of two variables and discuss its special cases. Some hypergeometric matrix representations of the Humbert matrix polynomials of two variables, the double generating matrix functions and expansions of the Humbert matrix polynomials of two variables in series of Hermite polynomials are given. Some particular cases are also discussed.

2. Definition of Humbert Matrix Polynomials of Two Variables

Let A be a positive stable matrix in for a positive integer m, we define Humbert matrix polynomials by

Now (7) it can be written in the form and by (3) and (6) respectively, one gets By equating the coefficients of in (7) and (8), we obtain an explicit representation of the Humbert matrix polynomials of two variables. In the form

from which it follows that is a matrix polynomial in two variables x and y of degree precisely n in x and k in y. In (9) setting m = 2, we get the Gegenbauer matrix polynomials of two variables  as particular case of the Humbert matrix polynomials of two variables.

3. Hypergeometric Matrix Representation for <img src="12-5300219\d564118e-53f8-4807-bba8-ec5e4de8e557.jpg" />

We study here the representation of the hypergeometric matrix representation for the Humbert matrix polynomials of two variables. There are some facts and notations used throughout the development in Sections 3 - 5, which are listed here.

Fact 1.  The reciprocal scalar Gamma Function , is an entire functions of the complex variable z. Thus, for , the Riesz-Dunford functional calculus  shows that is well defined and is indeed, the inverse of , Hence: if is such that is invertible for every integer . Then .

Fact 2.  If A, B and C are members of for which is invertible for every integer . The hypergeometric matrix function is defined by it converges for .

Notation 1.  For , the matrix version of the pochhammer symbol (the shifted factorial) is

with .

Note that , where j is a positive integer, then when ever . Also, the product in (10) is commutative, and then it is easy to see that  and where m is a positive integer.

Notation 2.  Now, in view of Notation 2, the explicit representation (9) for , becomes Thus we get the following hypergeometric representation of Humbert matrix polynomials of two variables.

For m = 2 (11), we gives hypergeometric representation of Gegenbauer matrix polynomials of two variables .

The above facts and notations will be used throughout the next two sections.

4. Additional Double Generating Matrix Functions

Now, since

By using (5), one gets

By using Notation 1, the following generating matrix functions for Humbert matrix polynomials of two variables follows  have thus discovered the family of double generating function of the Humbert polynomials of two variables

If B is a positive stable matrix in , then let us now return to (12) and consider the double sum. Then in similar manner, we get

5. Expansions of <img src="12-5300219\7ed81cab-da22-453b-8b23-df7ddeeb1f32.jpg" /> in Series of Hermite <img src="12-5300219\1e91a1b1-6049-4eb0-8410-88c3d7b18fd3.jpg" />

Now, we derive expansions of in series of Hermite According to , the expansion of in a series of Hermite matrix polynomials was given in the form:

which with the aid of (5) and (9), one gets From (16), we get Now replacing by and equating the coefficients of  , we get

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