Acta Mathematica Academiae Paedagogicae Ny´ıregyh´aziensis

22 (2006), 247–264 www.emis.de/journals ISSN 1786-0091

RUSCHEWEYH DIFFERENTIAL OPERATOR SETS OF BASIC SETS OF POLYNOMIALS OF SEVERAL COMPLEX

VARIABLES IN HYPERELLIPTICAL REGIONS

G. F. HASSAN

Abstract. In this paper we study the effectiveness of Ruscheweyh differ- ential operator sum and product sets of basic sets of polynomials of several complex variables in hyperelliptical regions. These results extend and im- prove the existing relevant results of Ruscheweyh differential operator sets in hyperspherical regions.

1. Introduction

The idea of the basic sets of polynomials of one complex variable appeared in 1930’s by Whittaker [33,34,35] who laid down the definition of a basic sets and their effectiveness. The study of the basic sets of polynomials of sev- eral complex variables was initiated by Mursi and Makar [25,26], Nassif [27], Kishka and others [13,14,17,18,19], where the representation in polycylindrical and hyperspherical regions was considered. Also, there are studies on basic sets of polynomials such as in Clifford Analysis [1,2,3,4,5,6,7,8] and in Faber regions [11,28,31,32]. The problem of derived and integrated sets of basic sets of polynomials in one and two complex variables was studied by many authors [9,10,20,21,23,24], where they considered the unit disk ∆ = {z : |z| < 1} in the complex plane C, circles and hyperspherical regions. Recently, in [12], the author studied this problem in a new region which is called hyperelliptical re- gion. The purpose of this paper is to establish the effectiveness of Ruscheweyh differential operator sum and product sets of basic sets of polynomials of sev- eral complex variables in an open hyperellipse, in a closed hyperellipse and in the regions D(E_[r]) which means unspecified domain containing the closed hyperellipse E_[r]. These results extend my results concerning the effectiveness in hyperspherical regions found in [15].

2000Mathematics Subject Classification. 32A05, 32A15, 32A99.

Key words and phrases. Ruscheweyh differential operator, polynomials, hyperelliptical regions.

247

LetC represent the filed of complex variables and let z= (z₁, z₂, . . . , z_k) be an element of C^k, the space of several complex variables. To avoid lengthy scripts, the following notations are adopted throughout this paper:

m= (m₁, m₂, . . . , m_k), am= (am₁, am₂, . . . , am_k), hmi=m₁+m₂+· · ·+m_k, h= (h₁, h₂, . . . , h_k),

|z|={ Xk

s=1

|z_s|²}¹²; z^m = Yk

s=1

z_s^ms E_[r] =E_{[r1,r2,...,rk]}, D(E_[r]) =D(E_{[r1,r2,...,rk]}),

r=r₁, r₂, . . . , r_k, [r_i] = [r_i⁽¹⁾, r_i⁽²⁾, . . . , r_i^(k)], t^m=

Yk

s=1

t^m_s ^s,

α([r], [R]) = max{r₁ Yk

s=2

R_s, r_ν Yk

s=1,s6=ν

R_s, r_k

k−1Y

s=1

R_s},

where m₁, m₂, . . . , m_k, h₁, h₂, . . . , h_k are non-negative integers and ν ={2,3,4, . . . , k−1}.

In C^k, E[r] denotes an open hyperellipse region P^k

s=1

|zs|²

r²s < 1 and by E[r]

its closure, where rs, s ∈ I = {1,2, . . . , k} are positive numbers. Thus these regions satisfy the following inequalities [14]:

D(E_[r]) ={w⁰ :|w| ≤1}, (1.1)

E_[r] ={w:|w|<1}, (1.2)

E[r] ={w:|w| ≤1}, (1.3)

where w = (w₁, w₂, . . . , w_k); w_s = ^z_r^s_s and w⁰ = (w⁰₁, w₂⁰, . . . , w_k⁰); w⁰_s = _r^z+^s s ; s∈I.

Thus the function f(z), which is regular in E_[r] can be represented by the power series

(1.4) f(z) = X∞

m=0

a_mz^m =

X∞

(m1,m2,...,mk)=0

a_m1,m2,...,mkz₁^m1z^m₂ ². . . z_k^mk.

For the functionf(z), we have from [13,14] that

(1.5) M(f, E_[r]) = sup

E_[r]

|f(z)|,

then it follows that

hmi→∞lim sup









|am| σ_m Q^k

s=1

{r_s}^hmi−ms









1 hmi

≤1/

Yk

s=1

r_s,

where

σ_m = inf

|t|=1

1

t^m = {hmi}^(1/2)hmi Qk

s=1

m^(1/2)ms ^s

, (see [14] and [27])

1≤σ_m ≤

³√ k

´_hmi

on the assumption m^(1/2)ms ^s = 1, whenever m_s = 0, s ∈I.

Definition 1.1 ([25,26]). A set of polynomials

{P_m[z]}={P₀[z], P₁[z], . . . , P_n[z], . . .}

is said to be basic when every polynomial in the complex variables z_s, s ∈I, can be uniquely expressed as a finite linear combination of the elements of the set{P_m[z]}.

Thus according to [25, Th.5] the set {P_m[z]} will be basic if and only if there exists a unique row-finite matrixP such that

(1.6) P P =P P =I,

where P ={P_m,h} is the matrix of coefficients, P = {P_m,h} is the matrix of operators of the set {P_m[z]}and I is the infinite unit matrix.

For the basic set {Pm[z]} and its inverse {Pm[z]}, we have Pm[z]=X

h

Pm,h z^h, (1.7)

z^m =X

h

P_m,hP_h[z]=X

h

P_m,hP_h[z], (1.8)

Pm[z]=X

h

Pm,h z^h. (1.9)

Thus, (1.4) becomes

f(z)=X

m

Π_mP_m[z], where

Πm =X

h

Ph,m ah =X

h

Ph,mf^(h)(0) h_s! , whereh! = h(h−1)(h−2)· · ·3·2·1. The series P

m

Π_mP_m[z] is the associated basic series of f(z).

Definition 1.2 ([13,14]). The associated basic series P

m

Π_m P_m[z] is said to represent f(z) in E_[r] (resp. E_[r], D(E_[r])) when P

m

Π_mP_m[z] converges uni- formly to f(z)in E[r] (resp. E[r], some hyperellipse surrounding the hyperel- lipse E_[r], not necessarily the former hyperellipse).

Definition 1.3 ([13,14]). The basic set{P_m[z]} is said to be effective inE_[r]

(resp. E_[r], D(E_[r])) when the associated basic series represents in E_[r] (resp.

E_[r], some hyperellipse surrounding the hyperellipse E_[r], not necessarily the former hyperellipse) every function which is regular there.

To study the convergence properties of such basic sets of polynomials in hyperelliptical regions we consider the following notations,

M(P_m, E_[r]) = sup

E[r]

|P_m[z]|, (1.10)

G(P_m, E_[r]) =X

h

¯¯P_m,h¯

¯M(P_h, E_[r]), (1.11)

Ω(P_m, E_[r]) = σ_m Yk

s=1

{r_s}^hmi−ms G(P_h, E_[r]), (1.12)

where Ω(P_m, E_[r]) is called the Cannon sum of the basic set {P_m[z]} for the hyperellipseE_[r] (see [13,14]).

Also, the Cannon function for the basic sets of polynomials in hyperelliptical regions [13,14] were defined as follows:

(1.13) Ω(P, E_[r]) = lim

hmi→∞sup {Ω(P_m, E_[r])}^hmi1 .

Let N_m = N_m1,m2,...,mk be the number of non-zero coefficients P_m,h in the representation (1.8). A basic set satisfying the condition

(1.14) lim

hmi→∞{N_m}^hmi1 =a, a >1,

is called general basic set and ifa= 1, then the basic set is called Cannon set [25,26]

Now, let D_m = D_m1,m2,...,mk be the degree of the polynomial of the highest degree in the representation (1.8). That is to say, if D_h = D_h1,h2,...,hk is the degree of the polynomial P_h[z], then D_h ≤ D_m ∀ h_s ≤ m_s;s ∈ I, and since the elements of the basic set are linearly independent, then

Nm ≤1 + 2 +· · ·+ (Dm+ 1) ≤λ1D²_m, where λ₁ be a constant.

Therefore, the condition (1.14) for a basic set to be Cannon set implies the following condition:

(1.15) lim

hmi→∞{D_m}^hmi1 = 1 (see[30])

Results on the effectiveness of the basic set{P_m[z]}in hyperelliptical regions are:

Theorem 1.1 ([13,14]). The necessary and sufficient condition for a Cannon basic set {P_m[z]} of polynomials of several complex variables to be effective in the closed hyperellipse E[r] is that Ω(P, E[r]) = Q^k

s=1

rs.

Theorem 1.2 ([13,14]). The necessary and sufficient condition for a Cannon basic set {P_m[z]} of polynomials of several complex variables to be effective in the open hyperellipse E_[r] is that Ω(P, E_[R])< α([r],[R]).

Theorem 1.3([13,14]). The Cannon basic set {P_m[z]} of polynomials of sev- eral complex variables will be effective in D(E_[r]), if and only if,

Ω(P, D(E_[r])) = Yk

s=1

r_s.

For more information about the study of basic sets of polynomials, we refer to [3,4,7,13,15,16,19,22,31,32].

2. Ruscheweyh differential operator sum sets of polynomials of several complex variables

Ruscheweyh differential operator Dⁿ [15] acting on the monomial z^m is defined by:

Dⁿz^m =



 [P^k

s=1

D_zⁿ_s^s]z^m; m6= 0,

1; m= 0

where

(2.1) Dⁿ_zs^s z_s^ms = z_s

n_s! (z_s^ns+ms−1)^(ns), (see [29]) the derivatives are repeatedns-times;s∈I. Thus,

(2.2) Dⁿz^m =





 Pk s=1

Ãn_s+m_s−1 n_s

!

z^m; m6= 0

1; m= 0.

Applying Dⁿ into (1.8) we get







 Pk s=1

Ã

n_s+m_s−1 n_s

!

z^m =P

h

P_m,hP_h⁽ⁿ⁾[z]; m6= 0 1 =P

h

P_0,hP_h⁽ⁿ⁾[z]; m= 0

where

P_m⁽ⁿ⁾[z]=DⁿP_m[z]

=Pm,0+X

h≥1

Pm,h

Xk

s=1

µn_s+h_s−1 ns

¶ z^h

=X

h

α_n,hP_m,hz^h and

α_n,h =





 Pk s=1

Ã

ns+hs−1 n_s

!

; h6= 0

1; h= 0

The set {Pm⁽ⁿ⁾[z]} is said to be Ruscheweyh differential operator sum set of polynomials of several complex variables.

Now, consider the next question: If we apply the operator Dⁿ on a basic set of polynomials of several complex variables {P_m[z]} then can we say that {Pm⁽ⁿ⁾[z]} is still basic? In fact the aim of the following section is to give an answer to this question.

To construct the basic property of the set{Pm⁽ⁿ⁾[z]} we write DⁿP_m[z]=X

h

α_n,hP_m,hz^h =X

h

P_m,h⁽ⁿ⁾z^h. The matrix of coefficients P⁽ⁿ⁾ of this set are

P⁽ⁿ⁾ = (α_n,h P_m,h).

Also the matrix of operatorsP⁽ⁿ⁾ follows from the representation z^m = 1

α_n,m X

h

P_m,h P_h⁽ⁿ⁾[z]=X

h

P⁽ⁿ⁾_m,hP_h⁽ⁿ⁾[z], that is to say

P⁽ⁿ⁾ = µ 1

αn,m

P_m,h

¶ . Therefore

P⁽ⁿ⁾ P⁽ⁿ⁾ = ÃX

h

P_m,h⁽ⁿ⁾P⁽ⁿ⁾_h,k

!

= ÃX

h

αn,h Pm,h

1

α_n,h Ph,k

!

=P P =I.

where I is the infinite unit matrix.

Also,

P⁽ⁿ⁾ P⁽ⁿ⁾ = ÃX

h

P⁽ⁿ⁾_m,h P_h,k⁽ⁿ⁾

!

=

Ãα_n,k αn,m

X

h

P_m,h P_h,k

!

=

µα_n,k α_n,m δ^m_k

¶

=I.

Hence the basic property of Ruscheweyh differential operator sum set {Pm⁽ⁿ⁾[z]} follows directly from Theorem 5 of [25].

3. Effectiveness of Ruscheweyh differential operator sum set of polynomials in closed hyperellipse

Let {P_m[z]} is a basic set of polynomials of several complex variables and {Pm⁽ⁿ⁾[z]} is the Ruscheweyh differential operator sum set. Consider the next question: If the set {P_m[z]} is effective in closed hyperellipse E_[r], do the set {Pm⁽ⁿ⁾[z]} still effective in E[r]? In this section we will give the answer of this question.

Let Ω(Pm⁽ⁿ⁾, E[r]) be the Cannon sum of the set{Pm⁽ⁿ⁾[z]}for the hyperellipse E_[r] , then

Ω(P_m⁽ⁿ⁾, E_[r]) = σ_m Yk

s=1

{r_s}^hmi−msX

h

¯¯

¯P⁽ⁿ⁾_m,h

¯¯

¯M(P_h⁽ⁿ⁾, E_[r])

= σ_m αn,m

Yk

s=1

{r_s}^hmi−msX

h

¯¯P_m,h¯

¯ M(P_h⁽ⁿ⁾, E_[r]) (3.1)

where,

M(P_h⁽ⁿ⁾, E[r]) = max

E_[r]

¯¯

¯P_h⁽ⁿ⁾[z]

¯¯

¯.

Now, we let, D_m be the degree of the polynomial of the highest degree in the representation (1.8). Hence by Cauchy’s inequality we see that

M(P_m⁽ⁿ⁾, E[r]) = max

E[r]

¯¯P_m⁽ⁿ⁾[z]¯

¯≤X

h

¯¯

¯P_m,h⁽ⁿ⁾

¯¯

¯ Qk s=1

{rs}^hs σ_h

=X

h

α_n,h|P_m,h| Qk s=1

{r_s}^hs

σ_h ≤ M(P_m, E_[r])X

h

α_n,h

=M(P_m, E_[r])

"

1 +X

h≥1

α_n,h

#

=M(Pm, E_[r])

"

1 +X

h≥1

Xk

s=1

µn_s+h_s−1 n_s

¶#

≤KN_mDⁿ_mM(P_m, E_[r])

≤K Dⁿ⁺²_m M(P_m, E_[r]) (3.2)

where K be a constant and the powernhere because we differentiatedns-times.

Thus the relation between the Cannon sums of the two sets {P_m[z]}and {Pm⁽ⁿ⁾[z]} can be obtained from the relations (3.1) and (3.2) as follows (3.3) Ω(P_m⁽ⁿ⁾, E_[r])≤K Dⁿ⁺²_m

α_n,m Ω(P_m, E_[r]).

Consider condition (1.15), we obtain that Ω(P⁽ⁿ⁾, E_[r]) = lim

hmi→∞sup{Ω(P_m⁽ⁿ⁾, E_[r])}^hmi1

≤ lim

hmi→∞sup{K Dⁿ⁺²_m

α_n,m Ω(P_m, E_[r])}^hmi1

≤Ω(P, E_[r]) = Yk

s=1

r_s. (3.4)

But

Ω(P⁽ⁿ⁾, E_[r])≥ Yk

s=1

r_s (see [14]).

Then,

(3.5) Ω(P⁽ⁿ⁾, E[r]) =

Yk

s=1

rs.

Therefore, according to (3.5) and using Theorem 1.1, we deduce that the effectiveness of the original set {Pm[z]} in E[r] implies the effectiveness of Ruscheweyh differential operator sum set{Pm⁽ⁿ⁾[z]} inE_[r].

Hence, we obtain the following theorem

Theorem 3.1. If the Cannon basic set {P_m[z]} of polynomials of the sev- eral complex variables z_s, s∈ I for which the condition (1.15) is satisfied, is effective in the closed hyperellipse E_[r], then the Ruscheweyh differential oper- ator sum set {Pm⁽ⁿ⁾[z]} of polynomials associated with the set {P_m[z]} will be effective in E_[r].

If, condition (1.15) is not satisfied then the set{Pm⁽ⁿ⁾[z]}can not be effective in E_[r], where the set {P_m[z]} is effective in E_[r]. To ensure this, we give the following example.

Example 1. Consider the set {P_m[z]} of polynomials of the several complex variables z_s∈I given by

(3.6)









P_m[z] =σ_m Q^k

s=1

z_s^ms +σ_am Q^k

s=1

z_s^ams, m6= 0, P_m[z] =σ_m Q^k

s=1

z_s^ms, otherwise, where a=b^hmi, b >1, then

Yk

s=1

z_s^ms =z^m = 1

σ_m [Pm[z]−Pam[z]], and the Cannon sum Ω(P_m, E_[r]) will given by

Ω(P_m, E_[r]) = Yk

s=1

[r^hmi_s + 2 r_s^{hmi+(a−1)ms}], which leads to

Ω(P, E_[1])≤ lim

hmi→∞sup{Ω(Pm, E_[1])}^hmi1 = 1;

i.e. the set {P_m[z]} is effective in E_[r] for r_s= 1, s ∈I.

Now, construct Ruscheweyh differential operator sum set {Pm⁽ⁿ⁾[z]} as fol-

lows; 





Pm⁽ⁿ⁾[z]=σ_mα_n,m z^m+σ_am α_n,am Q^k

s=1

z_s^ams; (m)6= 0 Pm⁽ⁿ⁾[z]=σ_mα_n,mz^m otherwise.

Thus it follows that,

z^m = 1

σ_mα_n,m [P_m⁽ⁿ⁾[z] −P_am⁽ⁿ⁾[z]]

and the Cannon sum Ω(Pm⁽ⁿ⁾, E_[r]) will given by Ω(P_m⁽ⁿ⁾, E_[r]) = σ_m

Yk

s=1

{r_s}^hmi−msX

h

¯¯

¯P⁽ⁿ⁾_m,h

¯¯

¯ M(P_h⁽ⁿ⁾, E_[r])

= 1

αn,m

[α_n,m Yk

s=1

r_s^hmi+ 2α_n,am Yk

s=1

r_s^{hmi+(a−1)ms}]

= Yk

s=1

r_s^hmi+ζ(a) Yk

s=1

r_s^{hmi+(a−1)ms}, where ζ(a)>1 is a constant depending only on a and

Ω(P⁽ⁿ⁾, E[r]) = 1 +ζ(a)>1,

that is to say that the Ruscheweyh differential operator sum set {Pm⁽ⁿ⁾[z]} is not effective in E_[r] for r_s = 1, s ∈ I, although the original set {P_m[z]} is effective in E_[r]. The reason for this, obviously, that condition (1.15) is not satisfied by the set{P_m[z]} as required.

4. Effectiveness of Ruscheweyh differential operator sum set of polynomials in open hyperellipse and the region D(E_[r]) Now, we establish the effectiveness property for Ruscheweyh differential op- erator sum set {Pm⁽ⁿ⁾[z]} in open hyperellipse E_[r] and the region D(E_[r]).

Suppose that the Cannon set {Pm[z]} is effective in E[r]. Then from the properties of Cannon functions, it follows from Theorem 1.1 of [14], that (4.1) Ω(P, E_[R])< α([r], [R]) for all 0< R_s < r_s, s∈I.

Construct the sets of numbers {r^(s)_i , s ∈ I} , (cf. [14]), in such a way that 0< r₀^(s) < r_s, s∈I and

r^(s)₀ r^(j)₀ = r_s

rj

; s, j ∈I, (4.2)

r_i+1^(s) = 1

2(r_s+r_i^(s)); s∈I; i≥0.

(4.3)

It follows, easily, from (4.2) and (4.3) that

(4.4) r^(s)_i

r^(j)_i = r_s

r_j; s, j ∈I; i≥0.

Therefore it follows that

R_s < r_i^(s)< r_s; s ∈I; i≥0.

Now, since the set {P_m[z]} accord to (4.1), in view of (1.12) and (1.13), then corresponding to the numbers r^(s)_i , s∈ I, there exists a constant K ≥1 such that

σm

Yk

s=1

{r_i^(s)}^hmi−ms G(Pm, E_[ri])< K{r⁽¹⁾_i+1 Yk

s=2

r^(s)_i }^hmi,

from which we get, in view of (4.4), the following inequality

G(P_m, E_[ri])< K σ_m{r⁽¹⁾_i+1

r⁽¹⁾_i }^hmi Yk

s=1

{r_i^(s)}^ms

= K

σ_m Yk

s=1

{r⁽¹⁾_i+1

r⁽¹⁾_i r_i^(s)}^ms

= K

σ_m Yk

s=1

{r^(s)_i+1

r^(s)_i r_i^(s)}^ms

= K

σ_m Yk

s=1

{r^(s)_i+1}^ms; (m_s ≥0; s∈I).

(4.5)

Now, for the numbers rs, Rs;s ∈ I, we have at least one of the following cases:

(i) ^R_R¹_s ≤ ^r_r¹_s;s∈I or (ii) ^R_R^ν

s ≤ ^r_r^ν

s;s ∈I,ν = 2 or 3 or . . . ork−1 or (iii) ^R_R^k

s ≤ ^r_r^k

s;s ∈I.

Suppose now, that relation (i) is satisfied, then from the construction of the sets {r_i^(s), s∈I}, we see that

(4.6) R₁

R_s ≤ r₁

r_s = r⁽¹⁾_i+1

r^(s)_i+1; s∈I.

Thus the Cannon sum of the set{Pm⁽ⁿ⁾[z]}for the hyperellipseE_[R],in view of (4.5) and (4.6) lead to

Ω(P_m⁽ⁿ⁾, E_[R]) = σ_m Yk

s=1

{R_s}^hmi−msX

h

¯¯

¯P⁽ⁿ⁾_m,h

¯¯

¯ M(P_h⁽ⁿ⁾, E_[R])

=

σ_m Q^k

s=1

{R_s}^hmi−ms α_n,m

X

h

¯¯P_m,h¯¯ M(P_h⁽ⁿ⁾, E_[R])

< L σm

Qk s=1

{Rs}^hmi−ms α_n,m

X

h

¯¯P_m,h¯

¯ M(P_h, E_[ri])

=L

σ_m Q^k

s=1

{R_s}^hmi−ms

α_n,m G(P_m, E_[ri])

< KL α_n,m

Yk

s=1

{R_s}^hmi−ms {r_i+1^(s)}^ms

= KL α_n,m

Yk

s=1

{r_i+1^(s)}^ms{R1

R_s}^ms Yk

s=2

{Rs}^hmi

≤ KL αn,m

Yk

s=1

{r^(s)_i+1}^ms{r₁ rs

}^ms Yk

s=2

{R_s}^hmi

= KL α_n,m

Yk

s=1

{r_i+1^(s)}^ms{r_i+1⁽¹⁾ r_i+1^(s) }^ms

Yk

s=2

{R_s}^hmi

= KL α_n,m {r⁽¹⁾_i+1

Yk

s=2

Rs}^hmi; which implies that

Ω(P⁽ⁿ⁾, E_[R]) = lim

hmi→∞sup {Ω(P_m, E_[R])}^hmi1

≤r_i+1⁽¹⁾ Yk

s=2

R_s< r₁ Yk

s=2

R_s, (4.7)

where

L= 1 + X

(h)≥1

Xk

s=1

µns+hs−1 ns

¶Y_k

s=1

{R^(s)_i

r^(s)_i }^hs ∀0< R_s < r_s; s∈I.

Also, if relation (ii) is satisfied for ν = 2 or 3 or . . . or k−1, then we shall have

(4.8) Rν

R_s ≤ rν

r_s = r^(ν)_i+1

r^(s)_i+1; s ∈I.

Thus (4.5) and (4.8) leads Ω(P_m⁽ⁿ⁾, E_[R])< KL

α_n,m Yk

s=1

{Rs}^hmi−ms {r_i+1^(s)}^ms

= KL αn,m

Yk

s=1

{r_i+1^(s)}^ms{R_ν Rs

}^ms { Yk

s=1,s6=ν

R_s}^hmi

≤ KL α_n,m

Yk

s=1

{r^(s)_i+1}^ms{rν

r_s}^ms { Yk

s=1,s6=ν

R_s}^hmi

= KL αn,m

Yk

s=1

{r_i+1^(s)}^ms{r_i+1^(ν) r_i+1^(s) }^ms {

Yk

s=1,s6=ν

R_s}^hmi

= KL α_n,m {r_i+1^(ν)

Yk

s=1,s6=ν

R_s}^hmi Therefore

(4.9) Ω(Pⁿ, E_[R])≤r^(ν)_i+1 Yk

s=1,s6=ν

R_s < r_ν Yk

s=1,s6=ν

R_s

where ν = 2 or 3 or . . . or k −1. Similarly if relation (iii) is satisfied, we proceed similarly as above to show,

(4.10) Ω(P⁽ⁿ⁾, E_[R])< r_k

k−1Y

s=1

R_s. Thus, it follows in view of (4.7), (4.9) and (4.10) that (4.11) Ω(P⁽ⁿ⁾, E_[R])< α([r],[R]).

Therefore, according to (4.11) and using Theorem 1.2 the Ruscheweyh differ- ential operator sum set{Pm⁽ⁿ⁾[z]}is effective in the open hyperellipseE_[r]when the original set {P_m[z]} is effective in E_[r].

Hence, we obtain the following theorem:

Theorem 4.1. If the Cannon basic set {P_m[z]} of polynomials of the several complex variables zs, s ∈ I is effective in the open hyperellipse E_[r], then the Ruscheweyh differential operator sum set {Pm⁽ⁿ⁾[z]} of polynomials associated with the set {P_m[z]} will be effective in E_[r].

Now, using a similar proof as done to Theorem 4.1, the following relation follows

Ω[P⁽ⁿ⁾, D(E_[r]] = Yk

s=1

r_s when Ω[P, D(E_[r]] = Yk

s=1

r_s Therefore, by using Theorem 1.3, we obtain the following theorem

Theorem 4.2. If the Cannon basic set {Pm[z]} of polynomials of the sev- eral complex variables z_s, s ∈ I is effective in the region D(E_[r]), then the Ruscheweyh differential operator sum set {Pm⁽ⁿ⁾[z]} of polynomials associated with the set {Pm[z]} will be effective in D(E[r]).

Now, we define the Ruscheweyh differential operator D^∗n acting on the monomial z^m, such that

D^∗nz^m =





·Qk

s=1

Dⁿ_zs^s

¸

z^m; m6= 0,

1; m= 0,

where D_zⁿ_s^s is defined as in (2.1). Thus,

(4.12) D^∗nz^m =





 Qk s=1

Ãn_s+m_s−1 n_s

!

z^m; m6= 0

1; m= 0.

Thus inserting the operator D^∗n into (1.8) we get





 Qk s=1

Ã

n_s+m_s−1 n_s

!

z^m =P

hP_m,h P_h⁽ⁿ⁾[z]; m6= 0 1 =P

hP_0,h P_h⁽ⁿ⁾[z]; m= 0,

where

P_m^∗(n)[z]=D^∗n Pm[z]

=P_m,0+X

h≥1

P_m,h Yk

s=1

µn_s+h_s−1 n_s

¶ z^h

=X

h

γ_n,h P_m,h z^h where,

γn,h =





 Qk s=1

Ãn_s+h_s−1 n_s

!

; h6= 0

1; h= 0

The set {Pm^∗(n)[z]} is said be Ruscheweyh differential operator product set of polynomials of several complex variables.

Similarly we may proceed as in Section 2 to prove the basic property of this set{Pm^∗(n)[z]}, such that

P^∗(n) P^∗(n) =P^∗(n) P^∗(n) =I.

For {Pm^∗(n)[z]}, we can proceed very similar as in Theorem 3.1,Theorem 4.1 and Theorem 4.2 to prove the following theorems:

Theorem 4.3. If the Cannon basic set {Pm[z]} of polynomials of the several complex variables z_s, s∈I for which the condition (1.15) is satisfied, is effec- tive in the closed hyperellipse E_[r], then the Ruscheweyh differential operator product set {Pm^∗(n)[z]} of polynomials associated with the set {Pm[z]} will be effective in E_[r].

Theorem 4.4. If the Cannon basic set {P_m[z]} of polynomials of the several complex variables z_s, s ∈ I is effective in the open hyperellipse E_[r], then the Ruscheweyh differential operator product sum set {Pm^∗(n)[z]} of polynomials associated with the set {Pm[z]} will be effective in E[r].

Theorem 4.5. If the Cannon basic set {P_m[z]} of polynomials of the sev- eral complex variables z_s, s ∈ I is effective in the region D(E_[r]), then the Ruscheweyh differential operator product set {Pm^∗(n)[z]} of polynomials associ- ated with the set {Pm[z]} will be effective in D(E[r]).

Before obtaining effectiveness conditions of basic sets of polynomials in hy- perspherical regions from our results, we introduce the following notations:

The open hypersphere is defined by:

Sr={z∈C^k: Ã _k

X

s=1

|zs|²

!¹₂

< r}, the closed hypersphere is defined by:

S_r ={z∈C^k : Ã _k

X

s=1

|z_s|²

!¹₂

≤r}.

The region D[S_r] means unspecified domain containing the closed hyper- sphere S_r.

To get the results concerning the effectiveness in hyperspherical regions (cf.

[15]) as special cases from the results concerning effectiveness in hyperelliptical regions, putrs =r, s∈I in Theorem 3.1, Theorem 4.1, Theorem 4.2, Theorem 4.3, Theorem 4.4 and Theorem 4.5 we can arrive to the following result Corollary 4.1. The effectiveness of the set {P_m[z]} in the equiellipse

1. E_[r]^∗ yields the effectiveness of the sets {Pm⁽ⁿ⁾[z]} and {Pm^∗(n)[z]} in the hypersphere S_r

2. E_[r]^∗ yields the effectiveness of the sets {Pm⁽ⁿ⁾[z]} and {Pm^∗(n)[z]} in the hypersphere S_r

3. D(E_[r]^∗) yields the effectiveness of the sets {Pm⁽ⁿ⁾[z]} and {Pm^∗(n)[z]} in the region D(Sr) where [r]^∗ = (r, r, r, . . . , r), r is repeated k-times.

Now, suppose that J_N(Dⁿ) is a polynomial of the operator Dⁿ as given in (2.2) andJ_N(D^∗n) is a polynomial of the operatorD^∗n as given in (4.12) such that J_N(Dⁿ) = P^N

j=1

λ_j(Dⁿ)^j and J_N(D^∗n) = P_N

j=1λ_j(D^∗n)^j, where (Dⁿ)^j = (Dⁿ)^j−1Dⁿ, (D^∗n)^j = (D^∗n)^j−1D^∗n, j be a finite positive integer and λj are constants 6= 0.

It is worthy to ensure that Theorem 3.1, Theorem 4.1, Theorem 4.2, Theo- rem 4.3, Theorem 4.4, Theorem 4.5 and Corollary 4.1 will be still true when we replace the set{DⁿP_m(z)} and {D^∗nP_m(z)} by the sets{J_N(Dⁿ)P_m(z)}

and {J_N(D^∗n)P_m(z)}, respectively.

Remark 4.1. Similar results for the sets

{DⁿP_m(z)},{D^∗nP_m(z)},{J_N(Dⁿ)P_m(z)}

and{JN(D^∗n)Pm(z)}in hyperelliptical regions can be obtained when the orig- inal set{Pm(z)} is general basic set.

Remark 4.2. The effectiveness of Ruscheweyh differential operator sets of poly- nomials of several complex variables in complete Reinhardt domains was stud- ied in [15].

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Received April 9, 2005.

Department of Mathematics

Faculty of Science, University of Assiut, Assiut 71516, Egypt

E-mail address: [email protected]