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Afrika Matematika ISSN 1012-9405 Afr. Mat.DOI 10.1007/s13370-013-0149-3

On Rice’s matrix polynomials

A. Shehata

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Afr. Mat.DOI 10.1007/s13370-013-0149-3

On Rice’s matrix polynomials

A. Shehata

Received: 3 August 2012 / Accepted: 16 March 2013© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2013

Abstract The main aim of this paper is to define and study of a new matrix polynomials,say, the Rice’s matrix polynomials. The convergence, radius of regularity, integral form,generating matrix functions and matrix recurrence relations satisfied by these Rice’s matrixpolynomials are derived. Furthermore, we study the operation of differential operators ofRice’s matrix polynomials and their applications are presented. The matrix differential equa-tion are obtained by them is presented. Finally, the study of the composition of Rice’s matrixpolynomials is investigated.

Keywords Hypergeometric matrix function · Rice’s matrix polynomials · Matrixdifferential equations · Integral representation · Recurrence relations · Differential operator

Mathematics Subject Classification (2000) 15A60 · 33C05 · 33C20 · 33C60 · 33C70 ·34A05

1 Introduction

Special matrix functions seen on statistics, Lie group theory and number theory are wellknown in [5,17,23]. Recently, an extension to the matrix framework of the classical familiesorthogonal polynomials of Hermite, Humbert, Jacobi, Gegenbauer, Laguerre, Bessel, Cheby-shev and pseudo Legendre matrix polynomials were introduced and studied in a number ofprevious papers, see for example, [1,6,7,10,11,14,19,22,30,32] for matrix in C

N×N . Fur-thermore, one can see more papers concerning orthogonal matrix polynomials in [4,9,13].Hermite matrix polynomials have been introduced and studied in [2,3,20,21,25] for matrices

A. ShehataDepartment of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt

A. Shehata (B)Department of Basic Applied Sciences, Unaizah Community College,Qassim University, Qassim 10363, Kingdom of Saudi Arabiae-mail: drshehata2006@yahoo.com

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in CN×N whose eigenvalues are all situated in the right open half-plane. In [7,33], the authors

introduced and studied Jacobi matrix polynomials. The hypergeometric matrix function hasbeen introduced as a matrix power series and an integral representation and the hypergeo-metric matrix differential equation in [15,18,27,28,31] and the explicit closed form generalsolution of it has been given in [16]. In [6], the authors introduced the Chebyshev matrixpolynomials and gave some results with Chebyshev matrix polynomials. In [13], the authorsstudied a new system of matrix polynomials, namely the Gegenbauer matrix polynomials in[26,29]. The reason of interest for this family of Rice’s matrix polynomials is due to theirintrinsic mathematical importance.

The primary goal of this paper is to consider a new system of matrix polynomials, namelythe Rice’s matrix polynomials. The structure of the paper is as follows: In Sect. 2 a definitionof Rice’s matrix polynomials is given and the convergence properties, radius of convergenceand an integral form are given. Some matrix differential recurrence relations, generatingmatrix functions and matrix recurrence relations are established, the effect of differentialoperator on these polynomials is investigated and the matrix differential equation satisfiedby them is presented in Sect. 3. Finally, we define the composite Rice’s matrix polynomialsand the convergence properties are investigated in Sect. 4.

Throughout this paper for a matrix A in CN×N , its spectrum σ(A) denotes the set of

all the eigenvalues of A. If A is a matrix in CN×N , its two-norm denoted by ||A||2 is

defined by [12]

||A||2 = supx �=0

||Ax ||2||x ||2

where for a vector y in CN , ||y||2 = (yT y)

12 is the Euclidean norm of y. We say that A in

CN×N is a positive stable matrix [14], if Re(z) > 0 for all λ ∈ σ(z), and denotes

M(A) = max{Re(z) : z ∈ σ(A)}; m(A) = min{Re(z) : z ∈ σ(A)}. (1.1)

If f (z) and g(z) are holomorphic functions of the complex variable z, which are defined inan open set � of the complex plane, and A is a matrix in C

N×N such that σ(A) ⊂ �, thenfrom the properties of the matrix functional calculus [8], it follows that

f (A)g(A) = g(A) f (A). (1.2)

Hence, if B in CN×N is a matrix for which σ(B) ⊂ � and also if AB = B A, then

f (A)g(B) = g(B) f (A). (1.3)

Let P and Q be two positive stable matrices in CN×N . The gamma matrix function �(P)

and the beta matrix function B(P, Q) have been defined in [14], as follows

�(P) =∞∫

0

e−t t P−I dt; t P−I = exp ((P − I ) ln t), (1.4)

and

B(P, Q) =1∫

0

t P−I (1 − t)Q−I dt, (1.5)

where I is the identity matrix in CN×N . Furthermore, if A is a matrix in C

N×N such that

A + nI is invertible for every non-negative integer n, (1.6)

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then �(A) is invertible, its inverse coincides with �−1(A) and one gets the formula [15]

(A)n = A(A+ I )(A+2I ), . . . , (A+(n−1)I )=�(A+nI )�−1(A); n ≥1; (A)0 = I.

(1.7)

From the relation (1.3) of [3], one obtains

(−1)k

(n − k)! I = (−n)k

n! I = (−nI )k

n! ; 0 ≤ k ≤ n. (1.8)

Jódar and Cortés have proved in [14,15] and n ≥ 1 is an integer, then

�(A) = limn→∞(n − 1)![(A)n]−1n A; n A = eA ln n, (1.9)

where (A)n is defined by (1.7). The hypergeometric matrix function 2 F1(A, B; C; z) hasbeen given in the form

2 F1(A, B; C; z) =∞∑

k=0

(A)k(B)k[(C)k]−1

n! zk, (1.10)

for matrices A, B and C in CN×N such that C + nI is invertible for all integer n ≥ 0 and for

|z| < 1. It has been seen by Jódar and Cortés [15] that the series is absolutely converges for|z| = 1 when

m(C) > M(A) + M(B),

where M(A), M(B) and m(C) are defined by (1.1).We will exploit the following relation due to [15]

(1 − z)−A = 1 F0(A;−; z) =∞∑

n=0

1

n! (A)nzn; |z| < 1. (1.11)

In the following, we introduce to define and study of a new matrix polynomial which repre-sents of the Rice’s matrix polynomials as given by the relation and the convergence properties,radius of convergence and an integral form are given.

2 On the Rice’s matrix polynomials

The Rice’s matrix polynomials Hn(A, B, z) is defined by means of the relation

Hn(A, B, z) = 3 F2(−nI, (n + 1)I, A; I, B; z)

=∞∑

k=0

zk

k! (−nI )k((1 + n)I )k(A)k[(I )k]−1[(B)k]−1, (2.1)

for matrices A and B in CN×N and commutative matrices in C

N×N such that B + k I isinvertible for all integer k ≥ 0.

For the sake of brevity we shall denote the expressions Hn(A, B, z), zk

k! (−nI )k((1 +n)I )k(A)k[(I )k]−1[(B)k]−1 and 1

k! (−nI )k((1 + n)I )k(A)k[(I )k]−1[(B)k]−1 by Hn, Uk(z)and Uk respectively.

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Now, we are going to study of the convergence properties of Rice’s matrix polynomials.Note that, If k is large enough so then for k > ‖B‖, then we will mention to the followingrelation which existed in Jódar and Cortés [14] in the form

‖(B + k I )−1‖ ≤ 1

k − ‖B‖; k > ‖B‖. (2.2)

Let us denote

α1(k) = ‖I −1‖‖(2I )−1‖, . . . , ‖(k I )−1‖; k > 1, (2.3)

α2(k) = ‖(B)−1‖‖(B + I )−1‖, . . . , ‖(B + (k − 1)I )−1‖; k > 1,

and note that

‖(A)k‖ ≤ (‖A‖)k, (2.4)

and taking into account the Pochhammer symbol or shifted factorial defined by

(a)k = a(a + 1)(a + 2), . . . , (a + k − 1) = �(a + k)

�(a); k ≥ 1; (a)0 = 1; a �= 0.

By (2.2), (2.3) and (2.4) for k > ‖B‖, we have∥∥∥∥ zk

k! (−nI )k((1 + n)I )k(A)k[(I )k ]−1[(B)k]−1∥∥∥∥

≤ |z|kk! ‖(−nI )k‖‖((1 + n)I )k‖‖(A)k‖α1(k)α2(k) (2.5)

≤ |z|kk! (‖ − nI‖)k(‖(1 + n)I‖)k(‖A‖)kα1(k)α2(k).

Now, we will investigate the convergence of the following series

∞∑k=0

|z|kk! (‖ − nI‖)k(‖(1 + n)I‖)k(‖A‖)kα1(k)α2(k).

By using the ratio test and the relation (2.3), it follows

limk−→∞

∣∣∣∣ (‖A‖)k+1(‖−nI‖)k+1(‖(1+n)I‖)k+1α1(k+1)α2(k+1)k!(‖A‖)k(‖−nI‖)k(‖(1+n)I‖)kα1(k)α2(k)(k+1)!

zk+1

zk

∣∣∣∣≤ lim

k→∞

∣∣∣∣ (‖A‖+k)(‖−nI‖+k)(‖(1+n)I‖+k)‖(I +k I )−1‖‖(B+k I )−1‖(k+1)

zk+1

zk

∣∣∣∣ (2.6)

≤ limk→∞

∣∣∣∣ (‖A‖+k)(‖ − nI‖+k)(‖(1+n)I‖+k)

(k−‖I‖)(k−‖B‖)(k+1)

∣∣∣∣ |z|=|z|,

where

‖Uk‖ ≤ (‖A‖)k(‖ − nI‖)k(‖(1 + n)I‖)kα1(k)α2(k)

k! .

The last limit shows that, thus, the power series (2.1) is absolutely convergent for |z| < 1 anddivergent for |z| > 1. The Rice’s matrix polynomials is absolutely convergent for |z| = 1when m(B) + m(I ) > M(−nI ) + M((1 + n)I ) + M(A) where M(−nI ), M((1 + n)I ),M(A), m(I ) and m(B) as defined in (1.1).

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Now, we begin the study of this function by calculating it radius of convergence R. Forthis purpose, we recall relation of [24], by (1.9), then

1

R= lim sup

k→∞(‖Uk‖) 1

k

= limk→∞ sup

(∥∥∥∥ (−nI )k((1 + n)I )k(A)k[(I )k]−1[(B)k]−1

k!∥∥∥∥) 1

k

= lim supk→∞

[∥∥∥∥k−A(A)k

(k − 1)! (k − 1)!k A

knI (−nI )k

(k − 1)! (k − 1)!k−nI k−(1+n)I ((1 + n)I )k

(k − 1)! (k − 1)!k(1+n)I (2.7)

k−I

(k − 1)! (k − 1)![(I )k ]−1k I k−B

(k − 1)! (k − 1)![(B)k]−1k B 1

k!∥∥∥∥] 1

k

= lim supk→∞

[∥∥∥∥(�−1(A)�−1(−nI )�−1((1 + n)I )�(I )�(B)

)k Ak−B (k − 1)!

k!∥∥∥∥] 1

k

≤ lim supk→∞

[∥∥∥∥k Ak−B 1

k

∥∥∥∥] 1

k ≤ lim supk→∞

[‖k A‖‖k−B‖k

] 1k

.

Using relation (8) of [15] for any square complex matrix A of size N , it follows that in theform

‖ et A ‖≤ et M(A)N−1∑j=0

(‖ A ‖ N12 t) j

j ! ; t ≥ 0, (2.8)

and considering that k A = eA ln k one gets

‖ k A ‖≤ k M(A)N−1∑j=0

(‖ A ‖ N12 ln k) j

j ! . (2.9)

Substitute from (2.8) and (2.9) into (2.7) one gets

1

R≤ lim sup

k→∞

⎧⎪⎨⎪⎩k M(A)

N−1∑j=0

(‖ A ‖ N12 ln k) j

j ! k−m(B)N−1∑j=0

(‖ B ‖ N

12 ln k

) j

j !1

k

⎫⎪⎬⎪⎭

1k

. (2.10)

Since

N−1∑j=0

(‖ A ‖ N12 ln k) j

j ! ≤ (N ln k)N−1N−1∑j=0

(‖ A ‖) j

j ! = (N ln k)N−1e‖A‖,

then

1

R≤ lim supk→∞

{k M(A)k−m(B)(N ln k)N−1e‖A‖(N ln k)N−1e‖B‖} 1

k = 1,

i.e. the radius of convergence of the Rice’s matrix polynomials Hn(A, B, z) is one and it isregular in a circle of radius r = 1.

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2.1 Integral form of the Rice’s matrix polynomials

To get an integral form for the Rice’s matrix polynomials of complex variable. Suppose that−nI , (n + 1)I , A and I , B are commuting matrices in C

N×N , such that

AB = B A, (2.11)

and

A, B and B − A are positive stable matrices. (2.12)

By (1.7) and (2.12) one gets

(A)k[(B)k]−1 = �(A + k I )�−1(A)[�(B + k I )�−1(B)]−1 (2.13)

= �−1(A)�−1(B − A)�(B − A)�(A + k I )�−1(B + k I )�(B).

By Lemma 2 of [14] and (2.12), we see that

1∫

0

t A+(k−1)I (1 − t)B−A−I dt = B(A + k I, B − A)

= �(B − A)�(A + k I )�−1(B + k I ), (2.14)

and by (2.13) and (2.14) one get

(A)k[(B)k]−1 = �−1(A)�−1(B − A)

⎛⎝

1∫

0

t A+(k−1)I (1 − t)B−A−I dt

⎞⎠�(B), (2.15)

where Re(b−a) > 0 for all b−a ∈ σ(B − A) and Re(a +k) > 0 for all a +k ∈ σ(A+k I ).Hence, by (1.10) and (2.15), one can write

Hn(A, B, z) =∞∑

k=0

(−nI )k((n + 1)I )k(A)k[(I )k]−1[(B)k]−1

k! zk

=∞∑

k=0

(−nI )k((n + 1)I )k[(I )k ]−1�−1(A)�−1(B − A)�(B)

k!⎛⎝

1∫

0

t A−I (1 − t)B−A−I (t z)kdt

⎞⎠ (2.16)

=⎛⎝

1∫

0

2 F1(−nI, (n + 1)I ; I ; t z)t A−I (1 − t)B−A−I dt

⎞⎠

�−1(A)�−1(B − A)�(B).

This is the integral form of Rice’s matrix polynomials.In the following, we derive several matrix differential recurrence relations, the pure

matrix recurrence relations and Rice’s matrix differential equations from this Rice’s matrixpolynomials.

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3 The matrix contiguous function relations

Some matrix recurrence relation is carried out on the Rice’s matrix polynomials. In thisconnection the following contiguous functions relations follow, directly by increasing ordecreasing one in original relation A(A + I )k = (A + k I )(A)k together with the definitionsof the matrix contiguous functions relations follow, yield the formulas

Hn(A+, B, z) = 3 F2(−nI, (n + 1)I, A+; I, B; z)

=∞∑

k=0

zk

k! (−nI )k((n + 1)I )k(A + I )k [(I )k]−1[(B)k]−1

=∞∑

k=0

zk

k! (−nI )k((n + 1)I )k(A + k I )A−1(A)k[(I )k]−1[(B)k]−1 (3.1)

=∞∑

k=0

(A + k I )A−1Uk(z).

Similarly, we have

Hn(A−, B, z) =∞∑

k=0

(A − I )(A + (k − 1)I )−1Uk(z), (3.2)

Hn(A, B+, z) =∞∑

k=0

B(B + k I )−1Uk(z), (3.3)

and

Hn(A, B−, z) =∞∑

k=0

(B + (k − 1)I )(B − I )−1Uk(z). (3.4)

Using the differential operator θ = z ddz . Since θ zk = kzk , we see that

(θ I + A)Hn =∞∑

k=0

(A + k I )Uk(z). (3.5)

Hence, with the aid of (3.1)–(3.5), yield

(θ I + A)Hn = A Hn(A+, B, z). (3.6)

Similarly, it follows that

(θ I + B − I )Hn = (B − I ) Hn(A, B−, z). (3.7)

The result is the set of simple relations of four:

(A−B+ I ) Hn = A Hn(A+, B, z)−(B− I ) Hn(A, B−, z),

(A+nI ) Hn +(A−nI ) Hn−1(A, B, z) = A Hn(A+, B, z)+ A Hn−1(A+, B, z),

B[2A− I −(A+B− I )z] Hn = AB(1−z) Hn(A+, B, z)+B(A− I ) Hn(A−, B, z) (3.8)

+(B+nI )(nI −B+ I )z Hn(A, B+, z),

B Hn +B Hn−1(A, B, z) = (B+nI ) Hn(A, B+, z)+(B−nI ) Hn−1(A, B+, z).

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The �(θ) differential operator has been defined by Sayyed [24] in the form

�(θ) = 1 +N∑

k=1

θk; θk = θ θk−1. (3.9)

From (3.7), we obtain

θ Hn = (B − I ) [Hn(A, B−, z) − Hn(A, B, z)] , (3.10)

and

θ2 Hn = (B − I )(B − 2I ) Hn(A, B − 2I, z) (3.11)

− [(B − I )(B − 2I ) + (B − I )2] Hn(A, B − I, z)

+(B − I )2 Hn(A, B, z).

Thus by mathematical induction, we have the following general form

�(θ)Hn(A, B, z)=(

1+N∑

k=1

θk

)Hn(A, B, z)

= Hn(A, B, z)+N∑

k=1

⎧⎨⎩

k∏j=1

(B− j I ) Hn(A, B− j I, z)

−⎡⎣ k∏

j=1

(B− j I )+k−1∏j=1

(B− j I )N−1∑k=1

(B− j I )

⎤⎦ Hn(A, B−( j −1)I, z)

+⎡⎢⎣

k−1∏j=1

(B− j I )k−1∑j=1

(B− j I )+k−2∏j=1

(B− j I )

⎛⎝k−2∑

j=1

(B− j I )

⎞⎠

2

(3.12)

+k−3∑j=1

(B− j I )(B−( j +1)I )+k−4∑j=1

(B− j I )(B−( j +1)I ) . . .

⎤⎦

Hn(A, B−( j −2)I, z)+· · ·+(−1)k(B − I )k Hn(A, B, z)

⎫⎬⎭ ,

where N is a finite positive integer.Consider the differential operator θ = z d

dz , θ zk = kzk , yields that

θ I (θ I + I − I )(θ I +B− I )Hn

=∞∑

k=1

k zk

k! (k I + I − I )(k I +B− I )(−nI )k((n+1)I )k

(A)k[(I )k]−1[(B)k]−1

=∞∑

k=1

zk

(k − 1)! (−nI )k((n + 1)I )k(A)k[(I )k−1]−1[(B)k−1]−1. (3.13)

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Now, we replace k by k + 1 and we have

θ I (θ I )(θ I +B− I )Hn =∞∑

k=0

zk+1

k! (−nI )k+1((n+1)I )k+1(A)k+1[(I )k]−1[(B)k]−1

=∞∑

k=0

zk+1

k! (−nI +k I )((n+k+1)I )(A+k I )(−nI )k((n+1)I )k

(A)k[(I )k ]−1[(B)k]−1 (3.14)

= z(θ I −nI )(θ I +(n+1)I )(θ I + A)Hn .

Thus, we have shown that Hn(A, B, z) is a solution of the following matrix differentialequation

[θ I (θ I )(θ I + B − I ) − z(θ I − n I )(θ I + (n + 1)I )(θ I + A)] Hn = 0, (3.15)

an equation which may also be written

(1−z)z2 H ′′′n +(B+2I −(A+4I )z I )z H ′′

n +[B−(2A+(n+2)(1−n)I )z]

H ′n +n(n+1)A Hn =0.

These results are summarized below.

Theorem 3.1 For each natural number n > 0, the Rice’s matrix polynomials Hn(A, B, z)satisfies the matrix differential equation

(1 − z)z2 H ′′′n + (B + 2I − (A + 4I )z I )z H ′′

n + [B − (2A + (n + 2)(1 − n)I )z]

H ′n + n(n + 1)A Hn = 0. (3.16)

Since the series Hn(A, B, z) is absolutely convergent we can differentiate it term by term.We can find some properties related with differentiation of the Rice’s matrix polynomials ofcomplex variables with respect to z in the following forms

d

dzHn

=∞∑

k=1

(−nI )k((n+1)I )k(A)k [(I )k ]−1[(B)k ]−1

(k−1)! zk−1

=∞∑

k=0

(−nI )k+1((n+1)I )k+1(A)k+1[(I )k+1]−1[(B)k+1]−1

k! zk

=∞∑

k=0

(−nI )((n+1)I )A(I )−1(B)−1(−nI + I )k((n+1)I + I )k(A+ I )k [(I + I )k ]−1[(B+ I )k ]−1

k! zk

=(−nI )((n+1)I )A(I )−1(B)−1 Hn+1(A+ I, B+ I, z), (3.17)

and in general

dk

dzkHn(A, B, z) = (−nI )k((n + 1)I )k(A)k[(I )k ]−1[(B)k]−1 Hn+k(A + k I, B + k I, z),

(3.18)

for k = 1, 2, 3, . . .,

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dn

dznHn(A, B, z) = (−nI )n((n + 1)I )n(A)n[(I )n]−1[(B)n]−1 H2n(A + nI, B + nI, z).

(3.19)

In the next theorem, we obtain another representation for the Rice’s matrix polynomials.

Theorem 3.2 Let A and B be matrices in CN×N . The Rice’s matrix polynomials defined by

(2.1) have the following properties

Hn(A, B, z) =∞∑

k=0

(−1)k zk

k!(n − k)!22k(

1

2I

)k(I )n+k(A)k[(I )2k]−1[(B)k]−1. (3.20)

Proof By (2.1), we can write

Hn(A, B, z) =∞∑

k=0

(−nI )k((1 + n)I )k zk

k! (A)k[(I )k]−1[(B)k]−1; 0 ≤ k ≤ n. (3.21)

From the relation (1.8), we get

Hn(A, B, z) =∞∑

k=0

(−1)kn!zk

k!(n − k)! ((1 + n)I )k(A)k[(I )k ]−1[(B)k]−1

=∞∑

k=0

(−1)k zk

k!(n − k)! (I )n((1 + n)I )k(A)k[(I )k ]−1[(B)k]−1.

Clearly

(I )n = I (I + I )(I + 2I ), . . . , (I + (n − 1)I ) = (1, 2, 3, . . . , n)I = n!I.According to (1.7), we find that

(I )n+k = (I )n((1 + n)I )k, (3.22)

and using (3.22), we get

Hn(A, B, z) =∞∑

k=0

(−1)k zk

k!(n − k)! (I )n+k(A)k[(I )k ]−1[(B)k]−1.

From (1.7), we see that

(I )2k = 22k(

1

2I

)k(I )k, (3.23)

this gives another representation for Rice’s matrix polynomials in the form

Hn(A, B, z) =∞∑

k=0

(−1)k zk

k!(n − k)!22k(

1

2I

)k(I )n+k(A)k[(I )2k]−1[(B)k]−1,

and the proof of Theorem 3.2 is completed. ��We now give representations a generating matrix function in a series of the Rice’s matrix

polynomials follow readily from (3.20).

Theorem 3.3 Let A and B be matrices in CN×N . Then a generating matrix function for

Rice’s matrix polynomials has the following representation

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∞∑n=0

Hn(A, B, z)tn = (1 − t)−I2 F1

(1

2I, A; B;− 4zt

(1 − t)2

), (3.24)

where the hypergeometric matrix function 2 F1(. . . , . . . ; . . . ; . . .) is given as

2 F1

(1

2I, A; B;− 4zt

(1 − t)2

)=

∞∑k=0

(−1)k

k!(

4zt

(1 − t)2

)k (1

2I

)k(A)k[(B)k]−1,

such that B + k I is invertible for all integer k ≥ 0 and for | 4zt(1−t)2 | < 1.

Proof We consider the series

∞∑n=0

Hn(A, B, z)tn =∞∑

n=0

∞∑k=0

(−1)k(4z)k tn

k!(n − k)!(

1

2I

)k(I )n+k(A)k[(I )2k]−1[(B)k]−1

=∞∑

n=0

∞∑k=0

(−1)k(4z)k tn+k

k!n!(

1

2I

)k(I )n+2k(A)k[(I )2k ]−1[(B)k]−1.

From (1.7) and (1.11), we can write

∞∑n=0

Hn(A, B, z)tn =∞∑

n=0

∞∑k=0

(−1)k(4z)k tn+k

k!n! ((2k + 1)I )n

(1

2I

)k(A)k[(B)k]−1

=∞∑

k=0

(−1)k(4z)k tk

k!(

1

2I

)k(A)k[(B)k]−1

∞∑n=0

tn

n! ((2k + 1)I )n

=∞∑

k=0

(−1)k(4z)k tk

k!(

1

2I

)k(A)k[(B)k]−1(1 − t)−(2k+1)I

= (1 − t)−I2 F1

(1

2I, A; B;− 4zt

(1 − t)2

).

We obtain an generating matrix function for the Rice’s matrix polynomials in the form

∞∑n=0

Hn(A, B, z)tn = (1 − t)−I2 F1

(1

2I, A; B;− 4zt

(1 − t)2

).

Hence the equation (3.24) is established and the proof of Theorem 3.3 is completed. ��Now, we can use the expansion of Rice’s matrix polynomials together with their properties

to prove the following result.

Theorem 3.4 Let A and B be matrices in CN×N . For non-negative integral n and expansion

of Rice’s matrix polynomials

zn I =(B)n

(1

2I

)n]−1[(A)n]−1

n∑k=0

(−1)kn!22n(n−k)! ((1+2k)I )(I )2n[(I )n+k+1]−1 Hk(A, B, z).

(3.25)

Proof We put that

ν = − 4t

(1 − t)2 .

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Then

t = 1 − 2

1 + √1 − ν

= − ν

(1 + √1 − ν)2

,

and (3.24) becomes

2 F1

(1

2I, A; B; zν

)=(

2

1 + √1 − ν

)I ∞∑k=0

(−1)k Hk(A, B, z)νk

(1 + √1 − ν)2k

or

2 F1

(1

2I, A; B; zν

)=

∞∑k=0

(−1)k Hk(A, B, z)νk

22k

(2

1 + √1 − ν

)(2k+1)I

,

we found that

(2

1 + √1 − ν

)(2α−1)I

= 2 F1

(α I,

(α − 1

2

)I ; 2α I ; ν

). (3.26)

The use of (3.26) with α = k + 1 leads to

2 F1

(1

2I, A; B; zν

)=

∞∑k=0

2 F1

((k+1)I,

(k+ 1

2

)I ; (2k+2)I ; ν

)(−1)kνk

22kHk(A, B, z)

=∞∑

n=0

∞∑k=0

(−1)kνn+k

n!22k((1+k)I )n

((k+ 1

2

)I

)n[((2k+2)I )n]−1

Hk(A, B, z),

and using (3.22) and (3.23), we get

((2k + 1)I )2n = 22n((

k + 1

2

)I

)n((k + 1)I )n,

(I )2k+1 = ((2k + 1)I )(I )2k ,

(I )2k+n+1 = (I )2k+1((2k + 2)I )n = ((2k + 1)I )(I )2k((2k + 2)I )n,

and hence

2 F1

(1

2I, A; B; zν

)=

∞∑n=0

∞∑k=0

(−1)kνn+k

22nn!22k((1 + 2k)I )2n[(2k + 2)n]−1 Hk(A, B, z)

=∞∑

n=0

∞∑k=0

(−1)kνn+k

22nn!22k((1 + 2k)I )(I )2k((1 + 2k)I )2n[(I )n+2k+1]−1

Hk(A, B, z)

=∞∑

n=0

∞∑k=0

(−1)kνn+k

22n+2kn! ((1 + 2k)I )(I )2n+2k [(I )n+2k+1]−1 Hk(A, B, z).

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Therefore

2 F1

(1

2I, A; B; zν

)=

∞∑n=0

νnzn

n!(

1

2I

)n(A)n[(B)n]−1

=∞∑

n=0

∞∑k=0

(−1)kνn+k

22n+2kn! ((1 + 2k)I )(I )2n+2k[(I )n+2k+1]−1 Hk(A, B, z)

=∞∑

n=0

n∑k=0

(−1)kνn

22n(n − k)! ((1 + 2k)I )(I )2n[(I )n+k+1]−1 Hk(A, B, z),

(3.27)

this yield

zn

n!(

1

2I

)n(A)n[(B)n]−1 = ∑n

k=0(−1)k

22n(n−k)! ((1 + 2k)I )(I )2n[(I )n+k+1]−1 Hk(A, B, z).

Expanding the left-hand side of (3.27) into powers of ν and identifying the coefficients ofνn on both sides gives (3.25). Therefore, the expression (3.25) is established and the proofof Theorem 3.4 is completed. ��

In the next corollary, we obtain another representation an expansion of the Rice’s matrixpolynomials as follows.

Corollary 3.1 For non-negative integral n and another representation expansion of Rice’smatrix polynomials the following holds

zn I = (B)n[(A)n]−1n∑

k=0

(−1)kn!(n − k)! ((1 + 2k)I )(I )n[(I )n+k+1]−1 Hk(A, B, z). (3.28)

Proof Using the Theorem 3.4 and (3.23), we get directly the equation (3.28). The proof ofCorollary 3.1 is completed. ��

In the following theorem, we obtain the properties of Rice’s matrix polynomials as follows.

Theorem 3.5 Let A and B be matrices in CN×N and B + k I is invertible for every integer

k ≥ 0. The Rice’s matrix polynomials satisfy the following differential recurrence relations

zH ′n(A, B, z) + zH ′

n−1(A, B, z) = n[Hn(A, B, z) − Hn−1(A, B, z)

] ; n ≥ 1, (3.29)

zH ′n(A, B, z) − nHn(A, B, z) = −

n−1∑k=0

[Hk(A, B, z) + 2zH ′

k(A, B, z)] ; n ≥ 1, (3.30)

and

zH ′n(A, B, z) − nHn(A, B, z) =

n−1∑k=0

(−1)n−k(1 + 2k)Hk(A, B, z); n ≥ 1. (3.31)

Proof In order to derive (3.29)–(3.31), we put

W = W (I, A, B, z, t) =∞∑

n=0

Hn(A, B, z)tn = (1 − t)−I2 F1

(1

2I, A; B;− 4zt

(1 − t)2

)

= (1 − t)−I

(− 4zt

(1 − t)2

), (3.32)

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where

=

(− 4zt

(1 − t)2

)= 2 F1

(1

2I, A; B;− 4zt

(1 − t)2

).

By differentiating (3.32) with respect to z and t yields respectively

∂W

∂z= −4t (1 − t)−3I ′,

∂W

∂t= (1 − t)−2I − 4z(1 + t)(1 − t)−4I ′. (3.33)

Therefore W satisfies the matrix partial differential equation

z(1 + t)∂W

∂z− t (1 − t)

∂W

∂t= −tW. (3.34)

Equation (3.34) can be rewritten in the forms

z∂W

∂z− t

∂W

∂t= −tW − t2 ∂W

∂t− zt

∂W

∂z, (3.35)

z∂W

∂z− t

∂W

∂t= − t

1 − tW − 2zt

1 − t

∂W

∂z, (3.36)

and

z∂W

∂z− t

∂W

∂t= − t

1 + tW − 2t2

1 + t

∂W

∂t. (3.37)

Since

W =∞∑

n=0

Hn(A, B, z)tn,

equation (3.35) yields that

∞∑n=0

[zH ′n(A, B, z) − nHn(A, B, z)]tn

= −∞∑

n=0

Hn(A, B, z)tn+1 −∞∑

n=0

nHn(A, B, z)tn+1 −∞∑

n=0

zH ′n(A, B, z)tn+1

= −∞∑

n=0

nHn−1(A, B, z)tn −∞∑

n=0

zH ′n−1(A, B, z)tn,

this leads to (3.29). Equation (3.36) implies that

∞∑n=0

[zH ′n(A, B, z) − nHn(A, B, z)]tn

= −∞∑

n=0

tn+1∞∑

k=0

Hk(A, B, z)tk − 2z∞∑

n=0

tn+1∞∑

k=0

H ′k(A, B, z)tk

= −∞∑

n=0

∞∑k=0

Hk(A, B, z)tn+k+1 − 2z∞∑

n=0

∞∑k=0

H ′n(A, B, z)tn+k+1

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= −∞∑

n=0

n∑k=0

Hk(A, B, z)tn+1 − 2z∞∑

n=0

n∑k=0

H ′k(A, B, z)tn+1

= −∞∑

n=1

n−1∑k=0

Hk(A, B, z)tn − 2z∞∑

n=1

n−1∑k=0

H ′k(A, B, z)tn,

which leads to (3.30). From (3.37), we obtain

∞∑n=0

[zH ′n(A, B, z) − nHn(A, B, z)]tn

= −∞∑

n=0

(−1)ntn+1∞∑

k=0

Hk(A, B, z)tk − 2∞∑

n=0

(−1)ntn+1∞∑

k=0

k Hk(A, B, z)tk

= −∞∑

n=0

∞∑k=0

(−1)n Hk(A, B, z)tn+k+1 − 2∞∑

n=0

∞∑k=0

k(−1)n Hk(A, B, z)tn+k+1

= −∞∑

n=0

n∑k=0

(−1)n−k Hk(A, B, z)tn+1 − 2∞∑

n=0

n∑k=0

k(−1)n−k Hk(A, B, z)tn+1

= −∞∑

n=1

n−1∑k=0

(−1)n−k−1 Hk(A, B, z)tn − 2∞∑

n=0

n−1∑k=0

k(−1)n−k−1 Hk(A, B, z)tn

=∞∑

n=1

n−1∑k=0

(−1)n−k Hk(A, B, z)tn + 2∞∑

n=0

n−1∑k=0

k(−1)n−k Hk(A, B, z)tn

=∞∑

n=1

n−1∑k=0

(−1)n−k(1 + 2k)Hk(A, B, z)tn,

this gives (3.31). Formulas (3.29)–(3.31) are called the recurrence formulas for Rice’s matrixpolynomials. Thus the proof of Theorem 3.5 is completed. ��

Now, we can state and prove the following theorem:

Theorem 3.6 The Rice’s matrix polynomials Hn(A, B, z) defined in (2.1), satisfy the purematrix recurrence relations

n(2n − 3)(B + (n − 1)I )Hn − (2n − 1) [(n − 2)(B − (n − 1)I )

+2(n − 1)(2n − 3)I − 2(2n − 3)(A + (n − 1)I )z] Hn−1 + (2n − 3)[2(n − 1)2 I − n(B − (n − 1)I ) + 2(2n − 1)(A − (n − 1)I )z

]Hn−2 (3.38)

+(n − 2)(2n − 1)(B − (n − 1)I )Hn−3 = 0.

Proof To simplify the exposition, we let

Hn(A, B, z) =∞∑

k=0

zk

k! (−nI )k((1 + n)I )k(A)k[(I )k ]−1[(B)k]−1 =∞∑

k=0

Uk(z). (3.39)

We list a sequence of the Hk(A, B, z) and zHk(A, B, z) (k = n, n − 1, . . .) in which forclarity, we exhibit each in both its explicit form and also in a form involving Uk(z) as definedabove. Thus

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Hn−1(A, B, z)=∞∑

k=0

zk

k! ((−n+1)I )k(nI )k(A)k[(I )k]−1[(B)k]−1

=∞∑

k=0

(n−k)I [(n+k)I ]−1Uk(z), (3.40)

zHn−1(A, B, z)=∞∑

k=0

zk+1

k! ((−n+1)I )k(nI )k(A)k[(I )k]−1[(B)k]−1

=∞∑

k=1

zk

(k−1)! ((−n+1)I )k−1(nI )k−1(A)k−1[(I )k−1]−1[(B)k−1]−1

=∞∑

k=0

−k2(B+(k−1)I )[(n+k)I ]−1[(n+k−1)I ]−1[A+(k−1)I ]−1Uk(z),

(3.41)

Hn−2(A, B, z)=∞∑

k=0

zk

k! ((−n+2)I )k((n−1)I )k(A)k[(I )k]−1[(B)k]−1

=∞∑

k=0

(n−k)I (n−k−1)I [(n+k)I ]−1[(n+k−1)I ]−1Uk(z), (3.42)

zHn−2(A, B, z)=∞∑

k=0

zk+1

k! ((−n+2)I )k((n − 1)I )k(A)k[(I )k]−1[(B)k]−1

=∞∑

k=1

zk

(k−1)! ((−n+2)I )k−1((n−1)I )k−1(A)k−1[(I )k−1]−1[(B)k−1]−1

=∞∑

k=0

−k2(B+(k−1)I )(n−k)(n−k−2)I [(n+k)I ]−1[(n+k−1)I ]−1

[(n+k−2)I ]−1[A+(k−1)I ]−1Uk(z), (3.43)

and

Hn−3(A, B, z) =∞∑

k=0

zk

k! ((−n + 3)I )k((n − 2)I )k(A)k[(I )k ]−1[(B)k]−1

=∞∑

k=0

(n − k)I (n − k − 1)I (n − k − 2)I [(n + k)I ]−1[(n + k − 1)I ]−1

[(n + k − 2)I ]−1Uk(z). (3.44)

If the coefficients of Uk(z) in the above series are written with a least common denominator,and the numerators are polynomials of degree at most four in k, then a linear combinationof the six series would have as numerator of its coefficient of Uk(z) a polynomial of degreefour in k. Such a linear combination would leave five undetermined constants with whichto make the five coefficients in the fourth degree polynomial vanish. This suggests that forn ≥ 3 there exists a linear recurrence relation of the form

Hn(A, B, z) + (C1 + C2 z)Hn−1(A, B, z) + (C3 + C4 z)Hn−2(A, B, z)

+C5 Hn−3(A, B, z) = 0 (3.45)

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in which C1, C2, C3, C4 and C5 are rational functions in n and are independent of z. Todetermine the coefficients in (3.45) we replace the several Hk(A, B, z) and zHk(A, B, z) bytheir respective forms where the factor Uk(z) occurs explicitly under the summation sign.Equating coefficients of Uk(z), we have for k ≥ 0

I + C1(n − k)I [(n + k)I ]−1 + C2[−k2(B + (k − 1)I )[(n + k)I ]−1[(n + k − 1)I ]−1

[A + (k − 1)I ]−1]+C3

[(n − k)I (n − k − 1)I [(n + k)I ]−1[(n + k − 1)I ]−1]

+C4[−k2(B + (k − 1)I )(n − k)I (n − k − 2)I [(n + k)I ]−1[(n + k − 1)I ]−1

[(n + k − 2)I ]−1[A + (k − 1)I ]−1]+C5

[(n − k)I (n − k − 1)I (n − k − 2)I [(n + k)I ]−1[(n + k − 1)I ]−1

[(n + k − 2)I ]−1] = 0. (3.46)

Clearing the above expression of fractions gives the following identity in k:

(n + k)I (n + k − 1)I (n + k − 2)I (A + (k − 1)I )

+C1(n − k)I (n + k − 1)I (n + k − 2)I (A + (k − 1)I )

+C2[−k2(B + (k − 1)I )(n + k − 2)I

]+C3 [(n − k)I (n − k − 1)I (n + k − 2)I (A + (k − 1)I )] (3.47)

+C4[−k2(B + (k − 1)I )(n − k)I (n − k − 2)I

]+C5 [(n − k)I (n − k − 1)I (n − k − 2)I (A + (k − 1)I )] = 0.

From this identity we can readily determine the coefficients in (3.45). Substituting the valuesthus obtained and clearing the result of fractions, we get, for n ≥ 3, the four-term recurrencerelation:

nI (2n − 3)I (B + (n − 1)I )Hn − (2n − 1)I [(n − 2)I (B − (n − 1)I )

+2(n − 1)I (2n − 3)I − 2(2n − 3)I (A + (n − 1)I )z] Hn−1 + (2n − 3)I (3.48)[2(n − 1)2 I − n(B − (n − 1)I ) + 2(2n − 1)I (A − (n − 1)I )z

]Hn−2

+(n − 2)I (2n − 1)I (B − (n − 1)I )Hn−3 = 0.

In the following, we introduce to define of composite Rice’s matrix polynomials and theradius of convergence is obtained. ��

4 Composite Rice’s matrix polynomials

Let us introduce the following notation [27]

k = (k1, k2, . . . , ki ),

(k) = k1 + k2 + · · · + ki ,

(k)! = k1!k2! . . . ki !,zk = zk1

1 zk22 . . . zki

i ,

A = (A1, A2, . . . , Ai ),

B = (B1, B2, . . . , Bi ),

(A)k = (A1)k1(A2)k2 . . . (Ai )ki ,

(B)k = (B1)k1(B2)k2 . . . (Bi )ki ,

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and

H = (H1, H2, . . . , Hi ).

Suppose that

Hl(Al , Bl , zl)=∑kl≥0

zkll

(kl)! (−nl I )kl ((nl +1)I )kl (Al)kl [(Il)kl ]−1[(Bl)kl ]−1, l =1, 2, . . . , i,

(4.1)

are i Rice’s matrix polynomials with square complex matrices A1, A2, . . . , Ai and B1, B2

,. . . , Bi of the same order N .Construct the Rice’s matrix polynomials Hn(A, B, z) defined by

Hn(A, B, z) = 3 F2(−nI, (n + 1)I, A; I , B; z)

=∑k≥0

zk

(k)! (−nI )k((1 + n)I )k(A)k[(I )k]−1[(B)k]−1. (4.2)

This function, will be called the composite Rice’s matrix polynomials of several complexvariables z1, z2, . . . , zi .

We begin the study of this function by calculating its radius of convergence R. For thispurpose, we recall relation (1.3.10) of [24] and keeping in mind that σk ≥ 1. Hence

1

R= lim sup

(k)→∞

( ‖ Uk ‖σk

) 1(k)

= lim sup(n)→∞

( ‖ (−nI )k((1 + n)I )k(A)k [(I )k ]−1[(B)k ]−1 ‖(k)!

) 1(k) ( 1

σk

) 1(k)

≤ lim sup(k)→∞

( ‖ ((−nI )1)k1 , . . . , ((−nI )i )ki (((n+1)I )1)k1 , . . . , (((n+1)I )i )ki (A1)k1 , . . . , (Ai )ki ‖k1!k2!, . . . , ki !

) 1(k)

([(I1)k1 ]−1, . . . , [(Ii )ki ]−1[(B1)k1 ]−1, . . . , [(Bi )ki ]−1) 1(k)

≤ lim sup(k)→∞

(∥∥∥∥∥(

k−(−nI )11 ((−nI )1)k1

(k1 − 1)!

)(k1 − 1)!k(−nI )1

1 , . . . ,

(k−(−nI )i

i ((−nI )i )ki

(ki − 1)!

)(ki − 1)!k(−nI )i

i

(k−((n+1)I )1

1 (((n+1)I )1)k1

(k1 − 1)!

)(k1−1)!k((n+1)I )1

1 , . . . ,

(k−((n+1)I )i

i (((n + 1)I )i )ki

(ki − 1)!

)(ki −1)!k((n+1)I )i

i

(k−A1

1 (A1)k1

(k1 − 1)!

)(k1 − 1)!k A1

1 , . . . ,

(k−Ai

i (Ai )ki

(ki − 1)!

)(ki − 1)!k Ai

i

k I11

(k1 − 1)! (k1 − 1)![(I1)k1]−1k−I1

1 , . . . ,k Ii

i

(ki − 1)! (ki − 1)![(Ii )ki ]−1k−Iii

k B11

(k1 − 1)! (k1 − 1)![(B1)k1]−1k−B1

1 , . . . ,k Bi

i

(ki − 1)! (ki − 1)![(Bi )ki ]−1k−Bii

∥∥∥∥∥1

k1!k2!, . . . , ki !) 1

(k)

≤ lim sup(k)→∞

(‖ �−1((−nI )1) ‖ · · · ‖ �−1((−nI )i ) ‖‖ �−1(((n + 1)I )1) ‖ · · · ‖ �−1(((n + 1)I )i ) ‖)

(‖ �−1(A1) ‖ · · · ‖ �−1(Ai ) ‖‖ �(I1) ‖ · · · ‖ �(Ii ) ‖‖ �(B1) ‖ · · · ‖ �(Bi ) ‖) 1(k)

123

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On Rice’s matrix polynomials

lim sup(k)→∞

(‖ k(−nI )1

1 ‖ · · · ‖ k(−nI )ii ‖‖ k((n+1)I )1

1 ‖ · · · ‖ k((n+1)I )ii ‖‖ k A1

1 ‖ · · · ‖ k Aii ‖

k1k2, . . . , ki

) 1(k)

(‖ k−I1

1 ‖ · · · ‖ k−Iii ‖‖ k−B1

1 ‖ · · · ‖ k−Bii ‖

) 1(k)

, (4.3)

where

σk =⎧⎨⎩(

k1+···+kik1

) k12(

k1+···+kik2

) k22

, . . . ,(

k1+···+kiki

) ki2

, k �= 0

1, k = 0.

For positive numbers μl and positive integer k, we can write

kl = μl k, l = 1, 2, . . . , i.

Substitute from (2.8) and (2.9) into (4.3) one gets

1

R≤ lim sup

k(μ1+···+μi )→∞

×⎧⎨⎩(μ1k)M((−nI )1)

N−1∑j=0

(‖ (−nI )1 ‖ N12 ln μ1k) j

j ! , . . . , (μi k)M((−nI )i )

N−1∑j=0

(‖ (−nI )i ‖ N12 ln μi k) j

j !

(μ1k)M(((n+1)I )1)N−1∑j=0

(‖ ((n + 1)I )1 ‖ N12 ln μ1k) j

j ! , . . . , (μi k)M(((n+1)I )i )N−1∑j=0

(‖ ((n + 1)I )i ‖ N12 ln μi k) j

j !

(μ1k)M(A1)N−1∑j=0

(‖ A1 ‖ N12 ln μ1k) j

j ! , . . . , (μi k)M(Ai )N−1∑j=0

(‖ Ai ‖ N12 ln μi k) j

j !

(μ1k)−m(I1)

N−1∑j=0

(‖ I1 ‖ N12 ln μ1k) j

j ! , . . . , (μi k)−m(Ii )

N−1∑j=0

(‖ Ii ‖ N12 ln μi k) j

j !

(μ1k)−m(B1)

N−1∑j=0

(‖ B1 ‖ N12 ln μ1k) j

j ! , . . . , (μi k)−m(Bi )

N−1∑j=0

(‖ Bi ‖ N12 ln μi k) j

j !

1

(μ1k)!, . . . , (μi k)!} 1

k(μ1+···+μi )

. (4.4)

Since

N−1∑j=0

(‖ Ai ‖ N12 ln μi k) j

j ! ≤ (N ln μi k)N−1N−1∑j=0

(‖ Ai ‖) j

j ! = (N ln μi k)N−1e‖Ai ‖,

then

1

R≤ lim sup

k(μ1+···+μi )→∞{kM((−nI )1)+···+M((−nI )i )kM(((n+1)I )1)+···+M(((n+1)I )i )kM(A1)+···+M(Ai )} 1

k(μ1+···+μi )

lim supk(μ1+···+μi )→∞

{k−m(I1)−···−m(Ii )k−m(B1)−···−m(Bi )} 1k(μ1+···+μi )

lim supk(μ1+···+μi )→∞

((N ln μ1k)N−1e‖(−nI )1‖ . . . (N ln μi k)N−1e‖(−nI )i ‖

) 1k(μ1+···+μi )

lim supk(μ1+···+μi )→∞

((N ln μ1k)N−1e‖((n+1)I )1‖ . . . (N ln μi k)N−1e‖((n+1)I )i ‖

) 1k(μ1+···+μi )

lim supk(μ1+···+μi )→∞

((N ln μ1k)N−1e‖A1‖ . . . (N ln μi k)N−1e‖Ai ‖

) 1k(μ1+···+μi )

123

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A. Shehata

lim supk(μ1+···+μi )→∞

((N ln μ1k)N−1e‖I1‖ . . . (N ln μi k)N−1e‖Ii ‖

) 1k(μ1+···+μi )

lim supk(μ1+···+μi )→∞

((N ln μ1k)N−1e‖B1‖ . . . (N ln μi k)N−1e‖Bi ‖

) 1k(μ1+···+μi ) = 1,

i.e. the radius of convergence of the composite Rice’s matrix polynomials is one and it isregular in the sphere SR ; R = 1 (c.f. [24]). There are many way of investigating of Rice’smatrix polynomials and give some properties of the results. Starting from the modified formsof the definition of Rice’s and composite Rice’s matrix polynomials is one of these directmethods and clearly some directions to develop more researches and studies in that area. Theresults of this paper are original, variant, significant and so it is interesting and capable todevelop its study in the future.

5 Open problem

One can use the same class of new integral representation, operational methods, familiesof multilinear and multilateral generating functions and orthogonality property for the mul-tivariable extension of the generalized Rice’s matrix polynomials. Hence, new results andfurther applications can be obtained. Further results and applications will be discussed in aforthcoming paper.

Acknowledgments (1) The Author expresses his sincere appreciation to Dr. Mahmoud Tawfik Mohamed,(Department of Mathematics and Science, Faculty of Education(New Valley), Assiut University, New Valley,EL-Kharga 72111, Egypt) for his kind interest, encouragements, help, suggestions, comments and the inves-tigations for this series of papers. (2) The author would like to thank the referees for their valuable commentsand suggestions which have led to the better presentation of the paper.

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