Some relations on generalized Rice s matrix polynomials

Transcription

1 Available at Appl. Appl. Math. ISSN: Vol., Issue June 07, pp Applications and Applied Mathematics: An International Journal AAM Some relations on generalized Rice s matrix polynomials Ayman Shehata Department of Mathematics Faculty of Science Assiut University Assiut 756, Egypt Department of Mathematics College of Science and Arts Unaizah, Qassim University Qassim, Saudi Arabia drshehata006@yahoo.com Received: November 9, 06; Accepted: February 8, 07 Abstract The main aim of this paper is to obtain certain properties of generalized Rice s matrix polynomials such as their matrix differential equation, generating matrix functions, an expansion for them. We have also deduced the various families of bilinear and bilateral generating matrix functions for them with the help of the generating matrix functions developed in the paper and some of their applications have also been presented here. Keywords: Hypergeometric matrix functions; Rice s matrix polynomials; Matrix differential equations; Generating matrix functions MSC 00 No.: 5A60, 33C0, 33C60, 33C70. Introduction and Preliminaries An important generalization of special functions is special matrix functions. Special matrix 367

2 368 A. Shehata polynomials appear in connection with matrix analogues of Hermite, Chebyshev and Legendre matrix differential equations and the corresponding polynomial families, see Ataş et al. 03, Altin et al. 04, Çeim 03, Çeim et al. 0, Çeim et al. 03, Defez and Jódar 00, Erus-Duman and Çeim 04, Jódar et al. 995, Jódar and Sastre 998, Kargin and Kurt 04a, 04b, Kargin and Kurt 05 for a list of references. The author has earlier studied the Rice s matrix polynomials Shehata 04 and the present paper carries those studies ahead motivated by the importance of special matrix polynomials of several recent wors Ataş 04, Ataş et al. 0, Çeim and Ataş 05, Shehata 04 and Tasdelen et al. 0. We organize the present paper as follows: In Section, we give the matrix differential equation, generating matrix functions and an expansion of generalized Rice s matrix polynomials. The class of bilinear and bilateral generating matrix relations for generalized Rice s matrix polynomials have also been established in the Sections 3 and 4. In this section, we give some basic facts or properties, definitions, lemmas, theorems and some notations and terminology, which has been used in the next sections. Throughout this paper, the symbol C N N stands for the set of all square complex matrices of common order N and σa stands for the set of all the eigenvalues of A C N N. The matrices I and 0 will be denoted by the identity matrix and the null matrix zero matrix in C N N, respectively. Definition.. If A 0, A,..., A n are elements of C N N and A n 0, then by the matrix polynomial of degree n in x x is a real variable or complex variable, we mean an expression of the form Theorem.. Dunford and Schwartz 957 P x = A n x n + A n x n A x + A 0. If fz and gz are holomorphic functions of the complex variable z, which are defined in an open set Ω of the complex plane, and A, B are matrices in C N N with σa Ω and σb Ω, such that AB = BA, then fagb = gbfa, where fa and gb denote the images of functions fz and gz respectively, acting on the matrices A and B. Definition.. Jódar and Cortés 998a A matrix A in C N N is said be a positive stable matrix if Reµ > 0, for all eigenvalues µ σa; σa := spectrum of A. Fact.. Jódar and Cortés 998b For A C N N, let us denote the real numbers MA and ma as in the following MA = max{rez : z σa}; ma = min{rez : z σa}.

3 AAM: Intern. J., Vol., Issue June Definition.3. Jódar and Cortés 998a If P is a positive stable matrix in C N N, then the Gamma matrix function ΓP is defined by ΓP = 0 e t t P I dt; t P I = exp P I ln t. 3 Definition.4. For A C N N, the matrix form of the Pochhammer symbol or shifted factorial is defined by Jódar and Cortés 998 A n = AA + IA + I... A + n I = ΓA + niγ A ; n, A 0 = I. 4 Definition.5. Jódar and Cortés 998b The hypergeometric matrix function F A, B; C; z is defined in the form F A, B; C; z = z! A B C ], 5 where A, B, and C are commutative matrices in C N N such that C + ni is an invertible matrix for every integer n 0 and for z <. Lemma.. Lancaster 969 Let... denotes any matrix norm for which I =. If A < for a matrix A in C N N, then I A c exists and given by where c is a positive integer. Fact.. Jódar and Cortés 998b I A c = For any matrix A in C N N, we give the following relation c! A, 6 A x = F 0 A; ; x = n! A nx n ; x <. 7

4 370 A. Shehata Notation.. For A is an arbitrary matrix in C N N and using 4, we have the following relations A n+ = A n A + ni = A A + I n, A = A A + I ni = A n = { n! n! I, 0 n; 0, > n, { A n I A ni ], 0 n; 0, > n., 8 Definition.6. Defez et al. 004 Let A and B be matrices in C N N satisfying the conditions Rez >, z σa and Rew >, w σb. 9 For n 0, the Jacobi matrix polynomials P A,B n x is defined by the hypergeometric matrix function Lemma.. P A,B n x = B + I n n! F A + B + n + I, ni; B + I; x. 0 In 00, Defez and Jódar have shown that for matrices A, n and B, n in C N N n 0, 0, then the following relations are satisfied: n B, n = B, n, A, n = n m ] A, n m ; m N. when Furthermore, we have the following relations n B, n = B, n +, n m ] A, n = A, n + m ; m N, where x] denotes the greatest integer in x.

5 AAM: Intern. J., Vol., Issue June Generalized Rice s matrix polynomials Definition.. Let P and Q be matrices in C N N satisfying the condition 9, A and B are matrices in C N N, B satisfying the condition B + ni is an invertible matrix for all integers n 0. 3 Let us consider the generalized Rice s matrix polynomials H n P,Q A, B, z by hypergeometric matrix function H n P,Q A, B, z = P + I n n! A; P + I, B; z = P + I n n! 3F ni, P + Q + n + I, z! ni P + Q + n + I A P + I ] B ] = P + I n n P + Q + n + I A P + I ] B ], z!n! 4 where 0 n and all matrices are commutative. Remar.. For the scalar case N =, taing P = Q = 0, A = ξ, B = p and z = ν in 4 gives the scalar Rice s polynomials H n ξ, p, ν see Rainville 945 and Rice 940 Corollary.. H n ξ, p, ν = 3 F n, n +, ξ;, p; ν. For the purpose of this wor, with the help of Theorem see Shehata 06b, setting p = 3 and q =, we give the convergence properties for the generalized Rice s matrix polynomials: The power series 4 is convergent for with z < and diverges for z >. The power series 4 is absolutely convergent for z = when mb + mp + I > MA + MP + Q + n + I + M ni. 3 If the power series 4 is conditionally convergent for z = when MA + MP + Q + n + I + M ni < mb + mp + I 4 The power series 4 is diverges for z = when MA + MP + Q + n + I + M ni. mb + mp + I MA + MP + Q + n + I + M ni, where MA and mb are defined in.

6 37 A. Shehata Remar.. We mention that for special case P = Q = 0, this reduces to Rice s matrix polynomials in Shehata 04. Remar.. We note that by taing A = B in 4, we obtain the result H P,Q n A, A, z = P Q,P n z = P + I n n! F ni, P + Q + n + I; P + I; z, 5 where P Q,P n x is the Jacobi matrix polynomials defined in 0. Theorem.. For n > 0, the generalized Rice s matrix polynomials satisfies the following matrix differential equation zz D 3 H n P,Q A, B, z + zb + P + I ] z A + P + Q + 4I D H n P,Q A, B, z + BP + I + nn + zi ] za + IP + Q + I + nzp + Q D H n P,Q A, B, z 6 + nap + Q + n + I H P,Q n A, B, z = 0 ; D = d dz, where all matrices are commutative. Proof: Consider the differential operator θ = z d dz, θz = z, yields that θ θ I + P + I Iθ I + B IH P,Q n A, B, z = = z! I + P + I II + B I ni P + Q + n + I A P + I ] B ] z =! ni P + Q + n + I = A P + I ] B ]. Now, we replacing by +, we have

7 AAM: Intern. J., Vol., Issue June θ θ I + P θ I + B IH n P,Q A, B, z z + = ni + P + Q + n + I + A + P + I ] B ]! = zθ I niθ I + P + Q + n + Iθ I + AH P,Q n A, B, z. Thus, we show that H n P,Q A, B, z is a solution of the following matrix differential equation θ θ I + P θi + B I zθ I n I ] θ I + P + Q + n + IθI + A H n P,Q A, B, z = 0. 7 Thus, H P,Q n A, B, z, given by 4, satisfies 6 in z <. Theorem.. Let A, B, P and Q be matrices in C N N satisfying the conditions 9 and 3. Then, a generating matrix function for generalized Rice s matrix polynomials has the following form for t <, Proof: P + Q + I n P + I n ] H n P,Q A, B, zt n P Q I = t 3 F P + Q + I, 4zt P + Q + I, A; P + I, B;, t <. t 4zt 8 Starting from 4 and using the result, we have = = P + Q + I n P + I n ] H P,Q n n From 8 and 7, we can write A, B, zt n z t n!n! P + Q + I n+a P + I ] B ] z t n+ P + Q + I n+ A P + I ] B ].!n!

8 374 A. Shehata P + Q + I n P + I n ] H P,Q n A, B, zt n = z t n+!n! P + Q + I P + Q + + I n A P + I ] B ] t n = n! P + Q + + I 4z t n P + Q + I! P + Q + I A P + I ] B ] = t 4z t! P + Q + I P + I ] B ] = t P + Q + I P Q I A 3 F P + Q + I, 4zt P + Q + I, A; P + I, B; t Hence the proof of Theorem. is completed. P Q +I Now, we can use the series of generalized Rice s matrix polynomials together with their properties to prove the following result. Theorem.3. For a non-negative integer n, an expansion of generalized Rice s matrix polynomials is given as follows. n z n I =P + I n B n A n ] n! P + Q + + I n! ] ] P + Q + I P + Q + I n++ P + I H P,Q A, B, z. 9 Proof: Equation 8 can be written in the form 3F P + Q + I, P + Q + I, A; P + I, B; 4zt t = t P +Q+I P + Q + I P + I ] H P,Q A, B, zt. 0 In 0, we put that ν = 4t t.

9 AAM: Intern. J., Vol., Issue June Then, 0 can be written t = + ν = ν + ν, 3F P + Q + I, P + Q + I, A; P + I, B; ν z = n ν P +Q++I + P + Q + I ν ] P + I H P,Q A, B, z. Now, replacing we have + ν P +Q++I = F P + Q + + I,, P + Q + + I; P + Q + + I; ν 3F P + Q + I, P + Q + I, A; P + I, B; ν z ν = F P + Q + + I, P + Q + + I; P + Q + + I; ν P + Q + I P + I ] H P,Q A, B, z ν n+ = P + Q + + I n! n ] P + Q + + I P + Q + + I n P + Q + I P + I ] H P,Q A, B, z ν n+ = n! P + Q + + I n+ n ] P + Q + + I n P + Q + I P + I ] H P,Q A, B, z. n

10 376 A. Shehata Using 8, we get P + Q + + I n = P + Q + I n+ P + Q + I ]. Thus, we obtain Therefore, ν n+ P + Q + I n+ n+ P + Q + + I n! ] P + Q + I n++ P + Q + I P + I ] H P,Q A, B, z. P + Q + I P + Q + I n! n ] ] n P + I n B n ν n z n = A n n n n! ] P + Q + I n P + Q + + I P + Q + I n++ P + Q + I P + I ] H P,Q A, B, zν, 3 which yields equation 9 on equating coefficients of ν n and using 8. Therefore, the proof of Theorem.3 is completed. Now, we mention some interesting special cases of our results of this section. Corollary.. Substituting P = Q = 0 in 4, we obtain a nown result in Shehata 04 n z n I =B n A n ] n! n! + II n ] I n++ H A, B, z. 4 Corollary.3. Taing A = B and replacing z by x in 4, we have a nown result in Defez et al. 004

11 AAM: Intern. J., Vol., Issue June x n I = n P + I n n n! P + Q + + I n! P + Q + I P + Q + I n++ ] P + I ] P Q,P x. 5 We now give a generating matrix function for the generalized Rice s matrix polynomials. Theorem.4. A generating matrix function for the generalized Rice s matrix polynomials is derived as n! C nh m P,Q ni, B, zt n = t C C, B, zt 6 ; t <, zt t t <, H P,Q m where P C = CP and QC = CQ. Proof: By using 4, 7 and, we have n! C nh m P,Q ni, B, zt n = m! P + I m m mi C n P + Q + m + I P + I ] B ] = m m! P + I m! mi n! C + I nt n C z t n+ n!! P + Q + m + I P + I ] B ] zt = m! P + I m m! mi t C I C P + Q + m + I B ] P + I ] zt = m! t C P + I m m C P + Q + m + I P + I ] B ] mi! zt t = m! t C P + I m 3 F mi, P + Q + m + I, C; P + I, B; zt = t C H m P,Q C, B, zt, t t which completes the proof of formula 6.

12 378 A. Shehata Corollary.4. For C = B, Equation 6 further reduces to following generating matrix function representation: n! C nh m P,Q ni, C, zt n = t C + zt 7 ; t <, zt t t <. P Q,P m 3. Bilinear and bilateral generating matrix functionsfor the generalized Rice s matrix polynomials The present section is a further attempt to establish a general theorem on a novel class of bilinear and bilateral generating matrix functions of various generalized Rice s matrix polynomials by using the similar method considered in Ataş 04, Ataş et al. 03, Ataş et al. 0, Altin et al. 04, Çeim and Ataş 05, Tasdelen et al. 0. In fact, we obtain the main results in the following theorem. Theorem 3.. Corresponding to a non-vanishing matrix function Ω µ y, y,..., y s of s complex variables y, y,..., y s, s N and involving a complex parameter µ, give a order, let us consider the following Λ µ,ν y, y,..., y s ; z = a Ω µ+ν y, y,..., y s z ; 8 a 0, µ, ν C, where the coefficients a are assumed to non-vanishing in order for the matrix function on the left-hand side to be non-null. Suppose also that the matrix polynomials Ψ n,m,µ,ν x; y, y,..., y s ; η = m n] a P + Q + I n m P + I n m ] H P,Q n m A, B, xω µ+νy, y,..., y s η ; n, m N, 9 where A, B, P and Q are matrices in C N N satisfying the conditions 9 and 3, and as usual the notation n] means the greatest integer less than or equal to n n N p p 0, p N. Then, we have Ψ n,m,µ,ν x; y, y,..., y s ; η t n P Q I = t t m 3 F P + Q + I, 30 P + Q + I, A; P + I, B; t Λ µ,ν y, y,..., y s ; η,

13 AAM: Intern. J., Vol., Issue June provided that each member of 30 exists. Proof: For the sae of convenience, let S denote the member of the assertion 30 of Theorem 3.. Then, by substituting for the matrix polynomials Ψ n,m,µ,ν x; y, y,..., y s ; η from the definition 9 into the left hand side of 30, we obtain t m the familiar generating matrix function = Ψ n,m,µ,ν x; y, y,..., y s ; η t n t m m n] a P + Q + I n m P + I n m ] H P,Q n m A, B, x Ω µ+ν y, y,..., y s η t n m. Upon changing the order of summation in 3, if we replace n by n + m, we can write Ψ n,m,µ,ν x; y, y,..., y s ; η t n = t m a P + Q + I n P + I n ] H n P,Q A, B, xω µ+ν y, y,..., y s η t n ] = P + Q + I n P + I n ] H n P,Q A, B, xt n ] a Ω µ+ν y, y,..., y s η P Q I = t 3 F P + Q + I, P + Q + I, A; P + I, B; t Λ µ,ν y, y,..., y s ; η, which completes the proof of Theorem 3.. By expressing the multivariable matrix function Ω µ+ν y, y,..., y s, N 0 and s N in terms of simpler matrix function of one and more variables, we can give further applications of Theorem 3.. In the following, we obtain the results which provide a class of bilinear generating matrix functions for the generalized Rice s matrix polynomials. Corollary 3.. Let Λ µ,ν y; z = a P + Q + I µ+ν P + I µ+ν ] H P,Q µ+ν A, B, yz ; a 0, µ, ν N 0 3

14 380 A. Shehata and Ψ n,m,µ,ν x; y; η = m n] a P + Q + I n m P + I n m ] H P,Q n m A, B, x P + Q + I µ+ν P + I µ+ν ] H P,Q µ+ν A, B, yη ; n, m N, where P, P, Q and Q are matrices in C N N satisfying the condition 9, and A, A, B and B are matrices in C N N, B and B satisfying the condition 3. Then, we have Ψ n,m,µ,ν x; y; η t n = t P Q I 3F t m P + Q + I, P + Q + I, A; P + I, B; t provided that each member of 3 exists. Proof: Λ µ,ν y; η, Equation 3 can be proved using the same method as in proof of Theorem 3.. Remar 3.. For the generalized Rice s matrix polynomials, by the generating matrix functions in 8 and taing a =, µ = 0 and ν =, we have m n] P + Q + I n m P + I n m ] H P,Q n m A, B, x P + Q + I P + I ] H P,Q A, B, yη t n m = t P Q I 3F P + Q + I, P + Q + I, A; P + I, B; η P Q I t 3 F P + Q + I, P + Q + I, A ; P + I, B ; 4yη. η In the next section, we proceed to discuss the various applications of the Theorem 3. for the case of certain special matrix functions including the generalized Rice s matrix polynomials Further Remars and Applications As we remared above that the Theorem 3. provides a very generalization of certain classes of bilateral generating matrix functions for the Chebyshev, Gegenbauer, Jacobi, Legendre, Laguerre, modified Laguerre, Hermite and the generalized Rice s matrix polynomials, the same are being deduced now as nown or new consequences of the Theorem 3..

15 AAM: Intern. J., Vol., Issue June Definition 4.. Let D be a matrix in C N N satisfying the condition. Then, the Chebyshev matrix polynomials of the second ind are defined by means of the series see Metwally et al. 05 U n x, D = n] n! x D n,!n! and the generating matrix function I xt D + t I = U n x, Dt n ; t <, x, 33 where I xt D + t I and xt D t I are invertible matrices in C N N. Corollary 4.. If Λ µ,ν y; z = a U µ+ν y, Dz ; a 0, µ, ν N 0, and Ψ n,m,µ,ν x; y; η = m n] a P + Q + I n m P + I n m ] H P,Q n m A, B, xu µ+νy, Dη ; n, m N, where D is a matrix in C N N satisfying the condition, then we have Ψ n,m,µ,ν x; y; η t n = t P Q I 3F t m P + Q + I, P + Q + I, A; P + I, B; t Λ µ,ν y; η, 34 provided that each member of 34 exists. Remar 4.. Using the generating matrix function 33 for the U y, D and by taing a =, µ = 0 and

16 38 A. Shehata ν =, we have m n] P + Q + I n m P + I n m ] H P,Q n m A, B, xu y, D η t n m = P + Q + I n P + I n ] H P,Q n A, B, xt n ] ] U y, Dη = t P Q I 3F P + Q + I, P + Q + I, A; P + I, B; t for t <, η <, x <, y and <. t Definition 4.. Jódar et al. 995 Let D be a matrix in C N N satisfying the condition Gegenbauer matrix polynomials are defined by I yη D + η I x / σd for all x Z+ {0}. 35 C D n x = n] x n D n.!n! Notice that the Gegenbauer matrix polynomials are generated as follows see Jódar et al. 995 and Sayyed et al. 004: F x, t, D = xt + t D = Cn D xt n. 36 If r and r are the roots of the quadratic equation xt + yt = 0 and r is the minimum of the set {r, r }, then the matrix function F x, t, D regarded as a matrix function of t, is analytic in the dis t < r for every real number in x. Corollary 4.. Let Λ µ,ν y; z = a Cµ+νyz D ; a 0, µ, ν N 0 and Ψ n,m,µ,ν x; y; η = m n] a P + Q + I n m P + I n m ] H P,Q n m A, B, xcd µ+νyη ; n, m N,

17 AAM: Intern. J., Vol., Issue June where D is a matrix in C N N satisfying the condition 35. Then, we have Ψ n,m,µ,ν x; y; η t n = t P Q I 3F t m P + Q + I, P + Q + I, A; P + I, B; t Λ µ,ν y; η, 37 provided that each member of 37 exists. Remar 4.. Using the generating matrix function 36 for the C D y and taing a =, µ = 0 and ν =, we have m n] P + Q + I n m P + I n m ] H P,Q n m A, B, xcd yη t n m = = P + Q + I n P + I n ] H P,Q n P + Q + I n P + I n ] H P,Q n A, B, xc D yη t n ] ] A, B, xt n C D yη = t P Q I 3F P + Q + I, P + Q + I, A; P + I, B; yη + η D. t The Jacobi matrix polynomials are generated by see Altin et al. 04 D + E + I n P D,E n xd + I n ] t n = t D E I F D + E + I, tx D + E + I; D + I;, t 38 where D and E are matrices in C N N all of whose eigenvalues, z, satisfy the condition Rez > with t <, x < and <. t Corollary 4.3. Let tx Λ µ,ν y; z = a P D,E µ+ν yz ; a 0, µ, ν N 0,

18 384 A. Shehata and m n] Ψ n,m,µ,ν x; y; η = P + Q + I n m P + I n m ] H P,Q n m A, B, xa P D,E µ+ν yη ; n, m N, where D and E are matrices in C N N satisfying the condition 9. Then, we have Ψ n,m,µ,ν x; y; η t n = t P Q I 3F t m P + Q + I, P + Q + I, A; P + I, B; t Λ µ,ν y; η, 39 provided that each member of 39 exists, where DE = ED. Remar 4.3. Using the generating matrix function 38 for the P D,E y and taing a = D + E + I D + I ], µ = 0 and ν =, we have Definition 4.3. m n] P + Q + I n m P + I n m ] H P,Q n m A, B, xx D + E + I P D,E yd + I ] η t n m = P + Q + I n P + I n ] H n P,Q A, B, xd + E + I P D,E yd + I ] η t n ] = P + Q + I n P + I n ] H n P,Q A, B, xt n ] D + E + I P D,E yd + I ] η P Q I = t 3 F P + Q + I, P + Q + I, A; P + I, B; t η D E I F D + E + I, D + E + I; ηy D + I;. η Let us consider the Legendre matrix polynomials P n x, D defined in Shehata 06a as follows: n n +! x P n x, D = Γ D + IΓD, n 0!n!

19 AAM: Intern. J., Vol., Issue June where D is a matrix in C N N satisfying the condition 0 < Reλ <, for all λ σd. 40 and D + I is an invertible matrix for all integers 0 and x <. The Legendre matrix polynomials in 40 are generated by see Shehata 06a: P n x, Dt n = t tx F I; D; ; t 4 t <, tx t <. Now using the Theorem 3., we get the following result which provides a class of bilateral generating matrix relation for the Legendre matrix polynomials and the generalized Rice s matrix polynomials. Corollary 4.4. If Λ µ,ν y; z = a P µ+ν y, Dz ; a 0, µ, ν N 0, and Ψ n,m,µ,ν x; y; η = m n] a P + Q + I n P + I n ] H P,Q n A, B, xp µ+ν y, Dη ; n, m N, where D is a matrix in C N N satisfying the condition 40, then we have Ψ n,m,µ,ν x; y; η t n = t P Q I 3F t m P + Q + I, P + Q + I, A; P + I, B; t Λ µ,ν y; η, 4 provided that each member of 4 exists. Remar 4.4. For the case of Legendre matrix polynomials P y, D, by the generating matrix function 4

20 386 A. Shehata and taing a =, µ = 0 and ν =, we have m n] P + Q + I n m P + I n m ] H P,Q n m A, B, xp y, D η t n m = P + Q + I n P + I n ] H P,Q n A, B, xt n ] ] P y, Dη = t P Q I 3F P + Q + I, P + Q + I, A; P + I, B; t ηy F I; D; ; η <, η η ηy η <. Next, if we set s = and Λ µ,ν y, w; z = a L D,λ µ+ν y, wz, a 0, µ, ν N 0 in Theorem 3., then we get the following result which provides a class of bilateral generating matrix functions for Laguerre matrix polynomials of two variables and generalized Rice s matrix polynomials. Definition 4.4. Let D be a matrix in C N N such that / σd for every integer > 0, 43 and λ is a complex number with Re λ > 0. Then, the Laguerre matrix polynomials are defined by see Jódar and Sastre 998, Shehata 05a, 05b, 05c L D,λ n x, y = n D + I n D + I ] y n λx.!n! According to Khan and Hassan 00, Laguerre matrix polynomials of two variables are generated by where t, x, y C and yt <. Corollary 4.5. λxt L D,λ n x, yt n = yt D+I exp, 44 yt Let Λ µ,ν y, w; z = a L D,λ µ+ν y, wz ; a 0, µ, ν N 0

21 AAM: Intern. J., Vol., Issue June and Ψ n,m,µ,ν x; y; w; η = m n] a P + Q + I n m P + I n m ] H P,Q n m A, B, xld,λ µ+ν y, wη ; n, m N, where D is a matrix in C N N satisfying the condition 43, then we have Ψ n,m,µ,ν x; y; w; η t n = t P Q I 3F t m P + Q + I, P + Q + I, A; P + I, B; t Λ µ,ν y, w; η, 45 provided that each member of 45 exists. Remar 4.5. For the Laguerre matrix polynomials L D,λ y, w, by the generating matrix function 44 and taing a =, µ = 0 and ν =, we have m n] P + Q + I n m P + I n m ] H P,Q n m A, B, xld,λ y, w η t n m = L D,λ P + Q + I n P + I n ] H P,Q n A, B, xt n ] ] y, wη = t P Q I 3F P + Q + I, wη D+I exp P + Q + I, A; P + I, B; t λ yη. wη Further, for the application of our Theorem 3., we get the following result on bilateral generating matrix relation involving modified Laguerre matrix polynomials. Definition 4.5. Let D be a matrix in C N N satisfying the condition 43 and λ is a complex parameter with Reλ > 0, then the n th modified Laguerre matrix polynomials f n D,λ x, y is defined see Khan and Hassan 00 and Shehata 05b n f n D,λ D n λ x y n x, y =.!n! We recall that the modified Laguerre matrix polynomials of two variables are defined by the generating matrix function in Khan and Hassan 00 f n D,λ x, yt n = yt D e λxt, t, x, y C, yt <. 46

22 388 A. Shehata Corollary 4.6. If we put s = in the above theorem, we get the following result. Let and Λ µ,ν y, w; z = Ψ n,m,µ,ν x; y, w; η = a f D,λ µ+ν y, wz ; a 0, µ, ν N 0 m n] a P + Q + I n m P + I n m ] H P,Q D,λ n m A, B, xf µ+ν y, wη ; n, m N where D is a matrix in C N N satisfying the condition 43, then we have Ψ n,m,µ,ν x; y, w; η t n = t P Q I 3F t m P + Q + I, P + Q + I, A; P + I, B; t Λ µ,ν y, w; η, 47 provided that each member of 47 exists. Remar 4.6. Using the generating matrix function 46 for the modified Laguerre matrix polynomials f D,λ y, w and taing a =, µ = 0 and ν =, we have for m n] P + Q + I n m P + I n m ] H P,Q n m η t n m = f D,λ D,λ A, B, xf µ+ν y, w P + Q + I n P + I n ] H P,Q n A, B, xt n ] ] µ+ν y, wη = t P Q I 3F P + Q + I, wη D e λyη, P + Q + I, A; P + I, B; t <, η <. η 4yη We now consider the Hermite matrix polynomials H n y, w, D of two variables satisfying the following generating matrix function in Shehata 05c t n n! H ny, w, D = exp yt D wt I ; t <, 48

23 AAM: Intern. J., Vol., Issue June where D is a positive stable matrix in C N N satisfying the condition, then we obtain the following result which provides a class of bilateral generating matrix functions for the matrix version of the Hermite matrix polynomials of two variables and generalized Rice s matrix polynomials. Corollary 4.7. Let Λ µ,ν y, w; z = a H µ+ν y, w, Dz ; a 0, µ, ν N 0 and Ψ n,m,µ,ν x; y, w; η = m n] a P + Q + I n m P + I n m ] H P,Q n m A, B, xh µ+νy, w, Dη ; n, m N, where D is a matrix in C N N satisfying the condition. Then, we have Ψ n,m,µ,ν x; y, w; η t n = t P Q I 3F t m P + Q + I, P + Q + I, A; P + I, B; t Λ µ,ν y, w; η, 49 provided that each member of 49 exists. Remar 4.7. Using the generating matrix function 48 for the Hermite matrix polynomials H y, w, D and taing a =, µ = 0 and ν =, we have! m n] P + Q + I n m P + I n m ] H P,Q n m A, B, x! H y, w, D η t n m = ]! H y, w, Dη P + Q + I n P + I n ] H P,Q n A, B, xt n ] P + Q + I, A; P + I, B; t = t P Q I 3F P + Q + I, exp yη D wη I. Acnowledgments The author is thanful to anonymous referees and the Editor-in-Chief Professor Aliabar Montazer Haghighi for useful comments and suggestions towards the improvement of this paper.

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