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On Hermite Matrix Polynomials of Two Variables



Ghazi S. Kahmmash
 
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ABSTRACT

This study deals with the two-variable Hermite matrix polynomials, some relevant matrix functions appear interims of the two-variable Hermite matrix polynomials the relationships with Hermite matrix polynomials of one variable, Chepyshev matrix polynomials of the second kind have been obtained and expansion of the. Gegenbauer matrix polynomials as series of Hermite matrix polynomials.

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  How to cite this article:

Ghazi S. Kahmmash , 2008. On Hermite Matrix Polynomials of Two Variables. Journal of Applied Sciences, 8: 1221-1227.

DOI: 10.3923/jas.2008.1221.1227

URL: https://scialert.net/abstract/?doi=jas.2008.1221.1227

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 CN×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 Hn (x, A) which appear as finite series solutions of second order matrix differential equations Y"-xAY+nAY = 0, for a matrix A in CN×N whose eigen values are all in right open half-plane. If A is a matrix in CN×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 Do is the complex plane cut along the negative real axis and log(z) denotes the principal logarithm of z, then z1/2 represents exp(1/2log(z)). If A is a matrix in CN×N with σ(A)⊂dDo then A½=Image for - On Hermite Matrix Polynomials of Two Variablesdenotes the image by z½ of the matrix functional calculus acting on the matrix A.

Let A is a matrix in CN×N such that,

Image for - On Hermite Matrix Polynomials of Two Variables
(1)

then nth Hermite matrix polynomial Hn (x, A) is defined by Jo’dar and Company (1996)

Image for - On Hermite Matrix Polynomials of Two Variables
(2)

and the following Rodrigues formula holds

Image for - On Hermite Matrix Polynomials of Two Variables
(3)

and satisfy the three terms recurrence relation ship.

Image for - On Hermite Matrix Polynomials of Two Variables
(4)

where, I is the identity matrix in CRxR.

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

Image for - On Hermite Matrix Polynomials of Two Variables
(5)

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

Image for - On Hermite Matrix Polynomials of Two Variables
(6)

and satisfy the recurrence relation ship.

Image for - On Hermite Matrix Polynomials of Two Variables
(7)

where, Hn (x, A) is defined in (2)
we shall use the relations (Defez and Jo’dar, 1998; Jo’dar and Company, 1996)

Image for - On Hermite Matrix Polynomials of Two Variables
(8)


Image for - On Hermite Matrix Polynomials of Two Variables
(9)

where, A (k, n) is a matrix on CN×N and the relation (Msayyed et al., 2004)

Image for - On Hermite Matrix Polynomials of Two Variables
(10)

Khan and Abukahmmash (1998) obtained the generating function for Hn (x, y) by

Image for - On Hermite Matrix Polynomials of Two Variables
(11)

for the Hermite polynomials of two variables

Image for - On Hermite Matrix Polynomials of Two Variables
(12)

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

Image for - On Hermite Matrix Polynomials of Two Variables
(13)

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

Image for - On Hermite Matrix Polynomials of Two Variables
(14)

where, Image for - On Hermite Matrix Polynomials of Two Variables(x,y) is a polynomial in two variables x and y of degree n in x and k in y thusImage for - On Hermite Matrix Polynomials of Two Variables(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 CN×N satisfying the condition (1). we define two variable Hermite matrix polynomials by

Image for - On Hermite Matrix Polynomials of Two Variables
(15)

we can write,

Image for - On Hermite Matrix Polynomials of Two Variables
(16)


Image for - On Hermite Matrix Polynomials of Two Variables
(17)

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

RECURRENCE RELATIONS

Now, since

Image for - On Hermite Matrix Polynomials of Two Variables
(18)

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

Image for - On Hermite Matrix Polynomials of Two Variables
(19)


Image for - On Hermite Matrix Polynomials of Two Variables
(20)

which with a shift of index on left yields,

Image for - On Hermite Matrix Polynomials of Two Variables
(21)

iteration of (21), gives

Image for - On Hermite Matrix Polynomials of Two Variables
(22)

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

Image for - On Hermite Matrix Polynomials of Two Variables
(23)

which with a shift of index on left yields,

Image for - On Hermite Matrix Polynomials of Two Variables

or

Image for - On Hermite Matrix Polynomials of Two Variables
(24)

iteration of (24), gives

Image for - On Hermite Matrix Polynomials of Two Variables
(25)

from (20), we get

Image for - On Hermite Matrix Polynomials of Two Variables
(26)

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

Image for - On Hermite Matrix Polynomials of Two Variables

(26), yields

Image for - On Hermite Matrix Polynomials of Two Variables
(27)

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

Image for - On Hermite Matrix Polynomials of Two Variables
(28)

multiplying (23), (19) and (28), by Image for - On Hermite Matrix Polynomials of Two Variables 2y and -t, respectively and adding, we get

Image for - On Hermite Matrix Polynomials of Two Variables

or

Image for - On Hermite Matrix Polynomials of Two Variables

equating the coefficients of tn, we get

Image for - On Hermite Matrix Polynomials of Two Variables

or

Image for - On Hermite Matrix Polynomials of Two Variables
(29)

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

Image for - On Hermite Matrix Polynomials of Two Variables
(30)

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

Image for - On Hermite Matrix Polynomials of Two Variables
(31)

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

Image for - On Hermite Matrix Polynomials of Two Variables

or

Image for - On Hermite Matrix Polynomials of Two Variables
(32)

RELATIONSHIPS BETWEEN
Hn (x, y, A) AND Hn (x, A)

Since,

Image for - On Hermite Matrix Polynomials of Two Variables
(33)

where, Hn (x, A) is well known Hermite matrix polynomial of one variable (Jo’dar and Company, 1996) replacing x by Image for - On Hermite Matrix Polynomials of Two Variablesand t by Image for - On Hermite Matrix Polynomials of Two Variables in (33), we get

Image for - On Hermite Matrix Polynomials of Two Variables
(34)

in view of (15), we get

Image for - On Hermite Matrix Polynomials of Two Variables

equating coefficients of t", we get

Image for - On Hermite Matrix Polynomials of Two Variables
(35)

now

Image for - On Hermite Matrix Polynomials of Two Variables
(36)

for y = 0, (35) reduces

Image for - On Hermite Matrix Polynomials of Two Variables
(37)

and for x = 0, we get

Image for - On Hermite Matrix Polynomials of Two Variables
(38)

for x = y = 0, we get

Image for - On Hermite Matrix Polynomials of Two Variables
(39)

THE RODRIGUES FORMULA

Examination of the defining relation

Image for - On Hermite Matrix Polynomials of Two Variables
(40)

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

Image for - On Hermite Matrix Polynomials of Two Variables
(41)

the function Image for - On Hermite Matrix Polynomials of Two Variables is independent of t. So we may write

Image for - On Hermite Matrix Polynomials of Two Variables

Image for - On Hermite Matrix Polynomials of Two Variables  

Image for - On Hermite Matrix Polynomials of Two Variables

or

Image for - On Hermite Matrix Polynomials of Two Variables

or

Image for - On Hermite Matrix Polynomials of Two Variables
(42)

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 Hn = (x, A),

Image for - On Hermite Matrix Polynomials of Two Variables

EXPANSION OF TWO-VARIABLES HERMITE MATRIX POLYNOMIALS

Since

Image for - On Hermite Matrix Polynomials of Two Variables
(43)

it follows that

Image for - On Hermite Matrix Polynomials of Two Variables

equating coefficients of tn, we get

Image for - On Hermite Matrix Polynomials of Two Variables
(44)

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

Image for - On Hermite Matrix Polynomials of Two Variables
(45)

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

Image for - On Hermite Matrix Polynomials of Two Variables

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

Image for - On Hermite Matrix Polynomials of Two Variables

we can write

Image for - On Hermite Matrix Polynomials of Two Variables
(46)

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

Image for - On Hermite Matrix Polynomials of Two Variables

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

Image for - On Hermite Matrix Polynomials of Two Variables

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

Image for - On Hermite Matrix Polynomials of Two Variables

and

Image for - On Hermite Matrix Polynomials of Two Variables

it’s evident that

Image for - On Hermite Matrix Polynomials of Two Variables
(47)

EXPAND THE GEGENBAUER MATRIX POLYNOMIALS OF TWO VARIABLES IN SERIES OF Hn (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

Image for - On Hermite Matrix Polynomials of Two Variables
(48)

which on applying (9) becomes.

Image for - On Hermite Matrix Polynomials of Two Variables

From (44), we get

Image for - On Hermite Matrix Polynomials of Two Variables

Image for - On Hermite Matrix Polynomials of Two Variables

by (10) it follows that

Image for - On Hermite Matrix Polynomials of Two Variables

we may write as

Image for - On Hermite Matrix Polynomials of Two Variables

again from (9), one gets

Image for - On Hermite Matrix Polynomials of Two Variables

equating the coefficient of tn+k we obtain an expansion of the two-variable Gegenbauer matrix Polynomials as series of two-variable Hermite matrix polynomials in the form

Image for - On Hermite Matrix Polynomials of Two Variables
(49)
REFERENCES
1:  Batahan, R.S., 2006. Anew extension of Hermite matrix polynomials and its applications. Linear Algebra Appl., 419: 82-92.
CrossRef  |  Direct Link  |  

2:  Davis, P.J., 1975. Interpolation and Approximation. Dover Publications, Inc., Mineola, New York.

3:  Defez, E. and L. Jodar, 1998. Some application of the hermite matrix polynomials series expansions. J. Comput. Applied Math., 99: 105-117.
CrossRef  |  

4:  Defez, E., M. Garcia-Honrubia and R.J. Villanueva, 2002. A procedure for computing the exponential of a matrix using Hermite matrix polynomials. Far East J. Applied Math., 6: 217-231.
Direct Link  |  

5:  Jodar, L., R. Company and E. Navarro, 1994. Laguerre matrix polynomials and systems of second order differential equations. Applied Numer. Math., 15: 53-63.
CrossRef  |  

6:  Jodar, L. and R. Company, 1996. Hermite matrix polynomials and second order matrix differential equations. J. Anal. Theory Appl., 12: 20-30.
Direct Link  |  

7:  Kahmmash, G.S., 2007. A study of a two variables gegenbauer matrix polynomials and second order matrix partial differential equations. Int. J. Math. Anal., 2: 807-821.
Direct Link  |  

8:  Khan, M.A. and G.S. Abukahmmash, 1998. On hermite polynomials of two variables suggested by S.F. Rajab’s laguerre polynomials of two variables. Ball. Coll. Math. Soc., 90: 61-76.

9:  Sayyed, K.A.M., M.S. Metwally and R.S. Batahan, 2003. On generalized Hermite matrix polynomials. Elect. J. Linear. Algebra, 10: 272-279.
Direct Link  |  

10:  Msayyed, K.A., M.S. Metwally and R.S. Bataha, 2004. Gegenbauer matrix polynomials and order matrix differential equations. Divulgaciones Matematics, 12: 101-115.
Direct Link  |  

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