**INTRODUCTION**

Laguerre, Hermite and Gegenbauer matrix polynomials were introduced and studied (Defez and Jo’dar, 1998; Msayyed* et al*., 2004; Jo’dar *et al*., 1994), for matrix in C^{N×N}. Moreover, some properties of the Hermite matrix polynomials are given (Defez and Jo’dar, 1998; Defez *et al*., 2002) and a generalized form of the Hermite matrix polynomials has been introduced and studied in (Msayyed *et al*., 2003).

Jo’dar and Company (1996) introduced the class of Hermite matrix polynomials H_{n} (x, A) which appear as finite series solutions of second order matrix differential equations Y"-xAY+nAY = 0, for a matrix A in C^{N×N} whose eigen values are all in right open half-plane. If A is a matrix in C^{N×N}, it spectrum σ(A) denotes the set of all eigenvalues of A. If f(z), g(z) are holomorphic functions in an open set Ω of the complex plane and if σ(A)⊂dΩ we denote by f(A), g(A), respectively, the image by the Riesez-Dunford functional calculus of the functions f(z), g(z), respectively, acting on the matrix A and

f(A) g(A) = g(A) f(A)

If D_{o} is the complex plane cut along the negative real axis and
log(z) denotes the principal logarithm of z, then z^{1/2} represents
exp(1/2log(z)). If A is a matrix in C^{N×N} with σ(A)⊂dD_{o}
then A^{½}=denotes
the image by z^{½} of the matrix functional calculus acting on the
matrix A.

Let A is a matrix in C^{N×N} such that,

then n^{th} Hermite matrix polynomial H_{n} (x, A) is defined
by Jo’dar and Company (1996)

and the following Rodrigues formula holds

and satisfy the three terms recurrence relation ship.

where, I is the identity matrix in C^{RxR}.

by Jo’dar and Company (1996) we also have

Batahan (2006) define the two-variable Hermite matrix polynomials by

and satisfy the recurrence relation ship.

where, H_{n} (x, A) is defined in (2)

we shall use the relations (Defez and Jo’dar, 1998; Jo’dar and Company,
1996)

where, A (k, n) is a matrix on C^{N×N} and the relation (Msayyed *et
al*., 2004)

Khan and Abukahmmash (1998) obtained the generating function for H_{n}
(x, y) by

for the Hermite polynomials of two variables

where, H_{n} (x) is the well known Hermite polynomial of one-variable
and it’s equivalent to the following explicit representation of H_{n}
(x, y) by

Kahmmash (2007) define Gegenbauer matrix polynomials of two variables by

where, (x,y)
is a polynomial in two variables x and y of degree n in x and k in y thus(x,y)
is a polynomial in two variables x and y of degree n+k.

The aim of this study is to establish the two variable extension of the Hermite matrix polynomials and the generating function for these matrix polynomials and Rodrigues formula, expansion series of Hermite matrix polynomials. Chepyshev matrix polynomial of the second kind and expansion of the Gegenbauer matrix polynomials as series of Hermite matrix polynomials.

**HERMITE MATRIX POLYNOMIALS OF TWO VARIABLES**

Let A be a matrix in C^{N×N} satisfying the condition (1). we define
two variable Hermite matrix polynomials by

we can write,

for y = 0, (17) reduces to Hermite matrix polynomial H_{n} (x, A) of one variable (Jo’dar and Company, 1996)

**RECURRENCE RELATIONS**

Now, since

differentiating (18) partially w.r.t. y, we get

which with a shift of index on left yields,

iteration of (21), gives

differentiating (22) partially w.r.t. x, we get.

which with a shift of index on left yields,

or

iteration of (24), gives

from (20), we get

let k = 2, in (25), we get

(26), yields

Equation 22, 25 and 27
are similar to the results given by Batahan (2006) differentiating (18) partially
w.r.t. wt, we get.

multiplying (23), (19) and (28), by
2y and -t, respectively and adding, we get

or

equating the coefficients of t^{n}, we get

or

combination of (24), (26) and (29) yields

from (29) and (30) we obtain the pure recurrence relation

from (24) and (29) the partial differential Equation given by

or

**RELATIONSHIPS BETWEEN **

H_{n} (x, y, A) AND H_{n} (x, A)

Since,

where, H_{n} (x, A) is well known Hermite matrix polynomial of one
variable (Jo’dar and Company, 1996) replacing x by and
t by
in (33), we get

in view of (15), we get

equating coefficients of t", we get

now

for y = 0, (35) reduces

and for x = 0, we get

for x = y = 0, we get

**THE RODRIGUES FORMULA**

Examination of the defining relation

in the light of maclaurin’s theorem gives us at once.

the function is independent of t. So we may write

or

or

a formula of the same nature as Rodrigue’s formula for Hermite matrix polynomial of one variablenote, setting y = 0 in (42) it gives the Rodrigue’s formula (Jo’dar and Company, 1996) of H_{n} = (x, A),

**EXPANSION OF TWO-VARIABLES HERMITE MATRIX POLYNOMIALS**

Since

it follows that

equating coefficients of t^{n}, we get

for y = 0, (44), gives the expansion of one variable Hermite matrix polynomials (Defez and Jo’dar, 1998).

**THE CHEPYSHEV MATRIX POLYNOMIALS **

The two-variable Hermite matrix polynomials will be exploited here to define
a matrix Version of Chepyshev polynomials. The Chepyshev polynomials of the
second kind (Davis, 1975) are defined by

suppose that A is a matrix in C^{N×N} satisfying the condition (1), by (16) it follows that

since the summation in the right-hand side of the above equality is finite, then the series and the integral can be permuted. also, in view of

we can write

hence, the chepyshev matrix polynomials of the second kind can be defined by

or by using the two-variable Hermite matrix polynomials in the form

in similar way, we define the generalized Chepyshev matrix polynomials of the second kind as follows

and

it’s evident that

**EXPAND THE GEGENBAUER MATRIX POLYNOMIALS OF TWO VARIABLES IN ****SERIES OF H**_{n} (x, y, A)

Let us now employ (14), (9) and (44) and taking into account that each matrix
commutes with it self.

from (14), one gets

which on applying (9) becomes.

From (44), we get

by (10) it follows that

we may write as

again from (9), one gets

equating the coefficient of t^{n+k} we obtain an expansion of the two-variable
Gegenbauer matrix Polynomials as series of two-variable Hermite matrix polynomials
in the form