Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
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 differential operator sum and product sets of basic sets of polynomials of several
complex variables in hyperelliptical regions. These results extend and improve 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 several 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 region. The purpose of this paper is to establish the effectiveness of Ruscheweyh
differential operator sum and product sets of basic sets of polynomials of several 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].
2000 Mathematics Subject Classification. 32A05, 32A15, 32A99.
Key words and phrases. Ruscheweyh differential operator, polynomials, hyperelliptical
regions.
247
248
G. F. HASSAN
Let C represent the filed of complex variables and let z = (z1 , z2 , . . . , zk ) be
an element of Ck , the space of several complex variables. To avoid lengthy
scripts, the following notations are adopted throughout this paper:
m = (m1 , m2 , . . . , mk ),
am = (am1 , am2 , . . . , amk ),
hmi = m1 + m2 + · · · + mk ,
h = (h1 , h2 , . . . , hk ),
k
X
1
|z| = {
|zs |2 } 2 ;
z
m
=
s=1
k
Y
zsms
s=1
E [r] = E [r1 ,r2 ,...,rk ] ,
D(E [r] ) = D(E [r1 ,r2 ,...,rk ] ),
r = r 1 , r2 , . . . , r k ,
[ri ] = [ri , ri , . . . , ri ],
(1)
tm =
k
Y
(2)
(k)
s
tm
s ,
s=1
α([r], [R]) = max{r1
k
Y
k
Y
Rs , rν
s=2
Rs , rk
k−1
Y
Rs },
s=1
s=1,s6=ν
where m1 , m2 , . . . , mk , h1 , h2 , . . . , hk are non-negative integers and
ν = {2, 3, 4, . . . , k − 1}.
In Ck , E[r] denotes an open hyperellipse region
k
P
s=1
|zs |2
rs2
< 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]:
(1.1)
D(E [r] ) = {w0 : |w| ≤ 1},
(1.2)
E[r] = {w : |w| < 1},
(1.3)
E [r] = {w : |w| ≤ 1},
where w = (w1 , w2 , . . . , wk ); ws = zrss and w0 = (w10 , w20 , . . . , wk0 ); ws0 = rz+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
am z
m
=
∞
X
am1 ,m2 ,...,mk z1m1 z2m2 . . . zkmk .
(m1 ,m2 ,...,mk )=0
For the function f (z), we have from [13,14] that
(1.5)
M (f, E [r] ) = sup |f (z)|,
E [r]
RUSCHEWEYH DIFFERENTIAL OPERATOR SETS
then it follows that




lim sup
hmi→∞
 1
hmi



|am |
≤ 1/
k


Q

 σm {rs }hmi−ms 

k
Y
249
rs ,
s=1
s=1
where
σm
{hmi}(1/2)hmi
1
, (see [14] and [27])
= inf m = k
Q (1/2)ms
|t|=1 t
ms
s=1
1 ≤ σm
³√ ´hmi
(1/2)ms
≤
k
on the assumption ms
= 1, whenever ms = 0, s ∈ I.
Definition 1.1 ([25,26]). A set of polynomials
{Pm [z]} = {P0 [z], P1 [z], . . . , Pn [z], . . .}
is said to be basic when every polynomial in the complex variables zs , s ∈ I,
can be uniquely expressed as a finite linear combination of the elements of the
set {Pm [z]}.
Thus according to [25, Th.5] the set {Pm [z]} will be basic if and only if
there exists a unique row-finite matrix P such that
P P = P P = I,
(1.6)
where P = {Pm,h } is the matrix of coefficients, P = {P m,h } is the matrix of
operators of the set {Pm [z]} and I is the infinite unit matrix.
For the basic set {Pm [z]} and its inverse {P m [z]}, we have
X
(1.7)
Pm [z] =
Pm,h zh ,
m
(1.8)
z
=
X
h
P m,h Ph [z] =
h
(1.9)
P m [z] =
X
X
Pm,h P h [z],
h
P m,h zh .
h
Thus, (1.4) becomes
f (z) =
X
Πm Pm [z],
m
where
X
X
f (h) (0)
,
h
s!
h
h
P
where h! = h(h − 1)(h − 2) · · · 3 · 2 · 1. The series
Πm Pm [z] is the associated
Πm =
basic series of f (z).
P h,m ah =
P h,m
m
250
G. F. HASSAN
P
Definition 1.2 ([13,14]). The associated basic series
Πm Pm [z] is said to
m
P
represent f (z) in E [r] (resp. E[r] , D(E [r] )) when
Πm Pm [z] converges unim
formly to f (z) in E [r] (resp. E[r] , some hyperellipse surrounding the hyperellipse E [r] , not necessarily the former hyperellipse).
Definition 1.3 ([13,14]). The basic set {Pm [z]} is said to be effective in E [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,
(1.10)
M (Pm , E [r] ) = sup |Pm [z]| ,
(1.11)
X¯
¯
¯P m,h ¯ M (Ph , E [r] ),
G(Pm , E [r] ) =
E [r]
h
(1.12)
Ω(Pm , E [r] ) = σm
k
Y
{rs }hmi−ms G(Ph , E [r] ),
s=1
where Ω(Pm , E [r] ) is called the Cannon sum of the basic set {Pm [z]} for
the hyperellipse E [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)
1
Ω(P, E [r] ) = lim sup {Ω(Pm , E [r] )} hmi .
hmi→∞
Let Nm = Nm1 ,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)
1
lim {Nm } hmi = a,
hmi→∞
a > 1,
is called general basic set and if a = 1, then the basic set is called Cannon set
[25,26]
Now, let Dm = Dm1 ,m2 ,...,mk be the degree of the polynomial of the highest
degree in the representation (1.8). That is to say, if Dh = Dh1 ,h2 ,...,hk is the
degree of the polynomial Ph [z], then Dh ≤ Dm ∀ hs ≤ ms ; s ∈ I, and since
the elements of the basic set are linearly independent, then
Nm ≤ 1 + 2 + · · · + (Dm + 1) ≤ λ1 D2m ,
where λ1 be a constant.
Therefore, the condition (1.14) for a basic set to be Cannon set implies the
following condition:
RUSCHEWEYH DIFFERENTIAL OPERATOR SETS
251
1
(1.15)
lim {Dm } hmi = 1 (see[30])
hmi→∞
Results on the effectiveness of the basic set {Pm [z]} in hyperelliptical regions
are:
Theorem 1.1 ([13,14]). The necessary and sufficient condition for a Cannon
basic set {Pm [z]} of polynomials of several complex variables to be effective in
k
Q
the closed hyperellipse E [r] is that Ω(P, E [r] ) =
rs .
s=1
Theorem 1.2 ([13,14]). The necessary and sufficient condition for a Cannon
basic set {Pm [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 {Pm [z]} of polynomials of several complex variables will be effective in D(E [r] ), if and only if,
Ω(P, D(E [r] )) =
k
Y
rs .
s=1
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 zm is
defined by:
 k
[ P Dns ]zm ; m 6= 0,
n m
D z = s=1 zs

1;
m=0
where
zs
(2.1)
Dznss zsms =
(z ns +ms −1 )(ns ) , (see [29])
ns ! s
the derivatives are repeated ns -times; s ∈ I. Thus,
!
 Ã
k

P
n
+
m
−
1
s
s

zm ; m 6= 0
n m
(2.2)
D z = s=1
ns


1;
m = 0.
Applying Dn into (1.8) we get
!
 Ã
k

P
ns + ms − 1 m P
(n)


z =
P m,h Ph [z]; m 6= 0
ns
s=1
h
P

(n)

1 =
P 0,h Ph [z];
m=0
h
252
G. F. HASSAN
where
(n)
Pm
[z] = Dn Pm [z]
= Pm,0 +
=
X
X
h≥1
¶
k µ
X
n s + hs − 1 h
Pm,h
z
ns
s=1
αn,h Pm,h zh
h
and
αn,h
!
 Ã
k

P
n
+
h
−
1
s
s

; h 6= 0
= s=1
ns


1;
h=0
(n)
The set {Pm [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 {Pm [z]} then can we say that
(n)
{Pm [z]} is still basic? In fact the aim of the following section is to give an
answer to this question.
(n)
To construct the basic property of the set {Pm [z]} we write
X (n)
X
Pm,h zh .
αn,h Pm,h zh =
Dn Pm [z] =
h
h
The matrix of coefficients P (n) of this set are
P (n) = (αn,h Pm,h ).
(n)
Also the matrix of operators P
follows from the representation
X (n) (n)
1 X
(n)
zm =
P m,h Ph [z] =
P m,h Ph [z],
αn,m h
h
that is to say
µ
P
(n)
Therefore
P (n) P
(n)
=
=
1
αn,m
Ã
X
¶
P m,h .
!
(n) (n)
Pm,h P h,k
h
=
Ã
X
αn,h Pm,h
h
= P P = I.
where I is the infinite unit matrix.
1
αn,h
!
P h,k
RUSCHEWEYH DIFFERENTIAL OPERATOR SETS
253
Also,
P
(n)
P (n) =
Ã
X
Ã
!
(n)
P m,h
(n)
Ph,k
h
αn,k X
=
P m,h Ph,k
αn,m h
¶
µ
αn,k m
= I.
δ
=
αn,m k
!
Hence the basic property of Ruscheweyh differential operator sum set
(n)
{Pm [z]} follows directly from Theorem 5 of [25].
3. Effectiveness of Ruscheweyh differential operator sum set
of polynomials in closed hyperellipse
Let {Pm [z]} is a basic set of polynomials of several complex variables and
(n)
{Pm [z]} is the Ruscheweyh differential operator sum set. Consider the next
question: If the set {Pm [z]} is effective in closed hyperellipse E [r] , do the set
(n)
{Pm [z]} still effective in E [r] ? In this section we will give the answer of this
question.
(n)
(n)
Let Ω(Pm , E [r] ) be the Cannon sum of the set {Pm [z]} for the hyperellipse
E [r] , then
(n)
,
Ω(Pm
E [r] ) = σm
k
Y
{rs }hmi−ms
X ¯¯ (n) ¯¯
(n)
¯P m,h ¯ M (Ph , E [r] )
s=1
(3.1)
=
σm
αn,m
k
Y
h
{rs }hmi−ms
s=1
X¯
¯
¯P m,h ¯ M (P (n) , E [r] )
h
h
where,
(n)
M (Ph ,
¯
¯
¯ (n) ¯
E [r] ) = max ¯Ph [z]¯ .
E [r]
254
G. F. HASSAN
Now, we let, Dm be the degree of the polynomial of the highest degree in
the representation (1.8). Hence by Cauchy’s inequality we see that
¯ (n) ¯ X ¯¯ (n) ¯¯
(n)
M (Pm , E [r] ) = max ¯Pm [z]¯ ≤
¯Pm,h ¯
E [r]
=
σh
{rs }hs
αn,h |Pm,h | s=1
h
(3.2)
{rs }hs
s=1
h
k
Q
X
k
Q
"
σh
= M (Pm , E [r] ) 1 +
X
X
αn,h
h
#
αn,h
h≥1
"
= M (Pm , E [r] )
≤ M (Pm , E [r] )
¶#
k µ
XX
n s + hs − 1
1+
ns
h≥1 s=1
≤ KNm Dnm M (Pm , E [r] )
≤ K Dn+2
m M (Pm , E [r] )
where K be a constant and the power n here because we differentiated ns -times.
Thus the relation between the Cannon sums of the two sets {Pm [z]} and
(n)
{Pm [z]} can be obtained from the relations (3.1) and (3.2) as follows
(3.3)
(n)
Ω(Pm
, E [r] ) ≤ K
Dn+2
m
Ω(Pm , E [r] ).
αn,m
Consider condition (1.15), we obtain that
1
(n)
Ω(P (n) , E [r] ) = lim sup{Ω(Pm
, E [r] )} hmi
hmi→∞
(3.4)
1
Dn+2
≤ lim sup{K m Ω(Pm , E [r] )} hmi
hmi→∞
αn,m
≤ Ω(P, E [r] ) =
k
Y
rs .
s=1
But
Ω(P
(n)
, E [r] ) ≥
k
Y
rs (see [14]).
s=1
Then,
(3.5)
Ω(P (n) , E [r] ) =
k
Y
s=1
rs .
RUSCHEWEYH DIFFERENTIAL OPERATOR SETS
255
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
(n)
Ruscheweyh differential operator sum set {Pm [z]} in E [r] .
Hence, we obtain the following theorem
Theorem 3.1. If the Cannon basic set {Pm [z]} of polynomials of the several complex variables zs , s ∈ I for which the condition (1.15) is satisfied, is
effective in the closed hyperellipse E [r] , then the Ruscheweyh differential oper(n)
ator sum set {Pm [z]} of polynomials associated with the set {Pm [z]} will be
effective in E [r] .
(n)
If, condition (1.15) is not satisfied then the set {Pm [z]} can not be effective
in E [r] , where the set {Pm [z]} is effective in E [r] . To ensure this, we give the
following example.
Example 1. Consider the set {Pm [z]} of polynomials of the several complex
variables zs ∈ I given by

k
k
Q
Q


zsms + σam
zsams , m 6= 0,
Pm [z] = σm
s=1
s=1
(3.6)
k
Q


Pm [z] = σm
zsms ,
otherwise,
s=1
hmi
where a = b
, b > 1, then
k
Y
zsms = zm =
s=1
1
[Pm [z] − Pam [z]],
σm
and the Cannon sum Ω(Pm , E [r] ) will given by
Ω(Pm , E [r] ) =
k
Y
[rshmi + 2 rshmi+(a−1)ms ],
s=1
which leads to
1
Ω(P, E [1] ) ≤ lim sup{Ω(Pm , E [1] )} hmi = 1;
hmi→∞
i.e. the set {Pm [z]} is effective in E [r] for rs = 1, s ∈ I.
(n)
Now, construct Ruscheweyh differential operator sum set {Pm [z]} as follows;

k
Q
P (n) [z] = σ α
zm + σ α
z ams ; (m) 6= 0
m

(n)
Pm [z]
m n,m
am
n,am
s=1
s
= σm αn,m zm
otherwise.
Thus it follows that,
1
(n)
zm =
[P (n) [z] − Pam
[z]]
σm αn,m m
256
G. F. HASSAN
(n)
and the Cannon sum Ω(Pm , E [r] ) will given by
(n)
Ω(Pm
,
E [r] ) = σm
k
Y
{rs }
hmi−ms
X ¯¯ (n) ¯¯
(n)
¯P m,h ¯ M (Ph , E [r] )
s=1
=
=
1
αn,m
k
Y
h
[αn,m
k
Y
rshmi
+ 2αn,am
s=1
rshmi+(a−1)ms ]
s=1
s=1
rshmi + ζ(a)
k
Y
k
Y
rshmi+(a−1)ms ,
s=1
where ζ(a) > 1 is a constant depending only on a and
Ω(P (n) , E [r] ) = 1 + ζ(a) > 1,
(n)
that is to say that the Ruscheweyh differential operator sum set {Pm [z]} is
not effective in E [r] for rs = 1, s ∈ I, although the original set {Pm [z]} is
effective in E [r] . The reason for this, obviously, that condition (1.15) is not
satisfied by the set {Pm [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(n)
erator sum set {Pm [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 < Rs < rs , s ∈ I.
(s)
Construct the sets of numbers {ri , s ∈ I} , (cf. [14]), in such a way that
(s)
0 < r0 < rs , s ∈ I and
(s)
r0
(4.2)
(4.3)
(j)
r0
=
rs
; s, j ∈ I,
rj
1
(s)
(s)
ri+1 = (rs + ri ); s ∈ I; i ≥ 0.
2
It follows, easily, from (4.2) and (4.3) that
(s)
(4.4)
ri
(j)
ri
=
rs
; s, j ∈ I; i ≥ 0.
rj
Therefore it follows that
(s)
Rs < ri < rs ; s ∈ I; i ≥ 0.
RUSCHEWEYH DIFFERENTIAL OPERATOR SETS
257
Now, since the set {Pm [z]} accord to (4.1), in view of (1.12) and (1.13), then
(s)
corresponding to the numbers ri , s ∈ I, there exists a constant K ≥ 1 such
that
σm
k
Y
(s)
{ri }hmi−ms
G(Pm , E [ri ] ) <
(1)
K{ri+1
s=1
k
Y
(s)
ri }hmi ,
s=2
from which we get, in view of (4.4), the following inequality
k
(1)
K ri+1 hmi Y (s) ms
G(Pm , E [ri ] ) <
{
}
{ri }
σm ri(1)
s=1
k
(1)
K Y ri+1 (s) ms
{ (1) ri }
=
σm s=1 ri
(4.5)
=
k
(s)
K Y ri+1 (s) ms
{ (s) ri }
σm s=1 ri
k
K Y (s) ms
=
{r } ; (ms ≥ 0; s ∈ I).
σm s=1 i+1
Now, for the numbers rs , Rs ; s ∈ I, we have at least one of the following
cases:
(i)
(ii)
(iii)
R1
Rs
Rν
Rs
Rk
Rs
≤
≤
≤
r1
; s ∈ I or
rs
rν
; s ∈ I, ν
rs
rk
; s ∈ I.
rs
= 2 or 3 or . . . or k − 1 or
Suppose now, that relation (i) is satisfied, then from the construction of the
(s)
sets {ri , s ∈ I}, we see that
(1)
(4.6)
r
R1
r1
; s ∈ I.
≤
= i+1
(s)
Rs
rs
ri+1
258
G. F. HASSAN
(n)
Thus the Cannon sum of the set {Pm [z]} for the hyperellipse E [R] , in view of
(4.5) and (4.6) lead to
(n)
,
Ω(Pm
E [R] ) = σm
k
Y
hmi−ms
{Rs }
X ¯¯ (n) ¯¯
(n)
¯P m,h ¯ M (Ph , E [R] )
s=1
σm
=
{Rs }hmi−ms X ¯
¯
s=1
¯P m,h ¯ M (P (n) , E [R] )
h
αn,m
h
σm
<L
k
Q
{Rs }hmi−ms X ¯
¯
s=1
¯P m,h ¯ M (Ph , E [r ] )
i
αn,m
h
σm
=L
<
h
k
Q
k
Q
{Rs }hmi−ms
s=1
G(Pm , E [ri ] )
αn,m
k
KL Y
(s)
{Rs }hmi−ms {ri+1 }ms
αn,m s=1
k
k
KL Y (s) ms R1 ms Y
=
{ri+1 } { }
{Rs }hmi
αn,m s=1
Rs
s=2
k
k
KL Y (s) ms r1 ms Y
≤
{r } { }
{Rs }hmi
αn,m s=1 i+1
rs
s=2
k
k
(1)
KL Y (s) ms ri+1 ms Y
{r } { (s) }
=
{Rs }hmi
αn,m s=1 i+1
ri+1
s=2
=
k
Y
KL
(1)
{ri+1
Rs }hmi ;
αn,m
s=2
which implies that
1
Ω(P (n) , E [R] ) = lim sup {Ω(Pm , E [R] )} hmi
hmi→∞
(4.7)
≤
(1)
ri+1
k
Y
Rs < r 1
s=2
k
Y
Rs ,
s=2
where
¶ k
k µ
(s)
X X
ns + hs − 1 Y Ri hs
L=1+
{ (s) }
ns
r
(h)≥1 s=1
s=1
i
∀ 0 < Rs < rs ; s ∈ I.
RUSCHEWEYH DIFFERENTIAL OPERATOR SETS
259
Also, if relation (ii) is satisfied for ν = 2 or 3 or . . . or k − 1, then we shall
have
(ν)
r
Rν
rν
≤
= i+1
; s ∈ I.
(s)
Rs
rs
ri+1
(4.8)
Thus (4.5) and (4.8) leads
(n)
Ω(Pm
,
k
KL Y
(s)
{Rs }hmi−ms {ri+1 }ms
E [R] ) <
αn,m s=1
k
k
Y
KL Y (s) ms Rν ms
=
{r } { } {
Rs }hmi
αn,m s=1 i+1
Rs
s=1,s6=ν
k
k
Y
KL Y (s) ms rν ms
≤
{ri+1 } { } {
Rs }hmi
αn,m s=1
rs
s=1,s6=ν
=
k
k
(ν)
Y
KL Y (s) ms ri+1 ms
Rs }hmi
{ri+1 } { (s) } {
αn,m s=1
ri+1
s=1,s6=ν
k
Y
KL
(ν)
=
Rs }hmi
{r
αn,m i+1 s=1,s6=ν
Therefore
(4.9)
n
Ω(P , E [R] ) ≤
(ν)
ri+1
k
Y
k
Y
Rs < rν
Rs
s=1,s6=ν
s=1,s6=ν
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] ) < rk
k−1
Y
Rs .
s=1
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(n)
ential operator sum set {Pm [z]} is effective in the open hyperellipse E[r] when
the original set {Pm [z]} is effective in E[r] .
Hence, we obtain the following theorem:
Theorem 4.1. If the Cannon basic set {Pm [z]} of polynomials of the several
complex variables zs , s ∈ I is effective in the open hyperellipse E[r] , then the
(n)
Ruscheweyh differential operator sum set {Pm [z]} of polynomials associated
with the set {Pm [z]} will be effective in E[r] .
260
G. F. HASSAN
Now, using a similar proof as done to Theorem 4.1, the following relation
follows
k
k
Y
Y
(n)
Ω[P , D(E [r] ] =
rs when Ω[P, D(E [r] ] =
rs
s=1
s=1
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 several complex variables zs , s ∈ I is effective in the region D(E [r] ), then the
(n)
Ruscheweyh differential operator sum set {Pm [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 zm , such that
·
¸
k
 Q
Dznss zm ; m 6= 0,
D∗n zm =
s=1

1;
m = 0,
where Dznss is defined as in (2.1). Thus,
!
 Ã
k
Q

n
+
m
−
1
s
s

zm ;
(4.12)
D∗n zm = s=1
ns


1;
m 6= 0
m = 0.
Thus inserting the operator D∗n into (1.8) we get
!
 Ã
k
Q
P

n
+
m
−
1
s
s
(n)

zm = h P m,h Ph [z]; m 6= 0
ns
s=1

P

(n)
1 = h P 0,h Ph [z];
m = 0,
where
∗(n)
Pm
[z] = D∗n Pm [z]
= Pm,0 +
=
X
X
Pm,h
h≥1
¶
k µ
Y
n s + hs − 1
zh
ns
s=1
γn,h Pm,h zh
h
where,
γn,h
∗(n)
!
 Ã
k

Q
n s + hs − 1

; h 6= 0
= s=1
ns


1;
h=0
The set {Pm [z]} is said be Ruscheweyh differential operator product set of
polynomials of several complex variables.
RUSCHEWEYH DIFFERENTIAL OPERATOR SETS
261
Similarly we may proceed as in Section 2 to prove the basic property of this
∗(n)
set {Pm [z]}, such that
P ∗(n) P
∗(n)
=P
∗(n)
P ∗(n) = I.
∗(n)
For {Pm [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 zs , s ∈ I for which the condition (1.15) is satisfied, is effective in the closed hyperellipse E [r] , then the Ruscheweyh differential operator
∗(n)
product set {Pm [z]} of polynomials associated with the set {Pm [z]} will be
effective in E [r] .
Theorem 4.4. If the Cannon basic set {Pm [z]} of polynomials of the several
complex variables zs , s ∈ I is effective in the open hyperellipse E[r] , then the
∗(n)
Ruscheweyh differential operator product sum set {Pm [z]} of polynomials
associated with the set {Pm [z]} will be effective in E[r] .
Theorem 4.5. If the Cannon basic set {Pm [z]} of polynomials of the several complex variables zs , s ∈ I is effective in the region D(E [r] ), then the
∗(n)
Ruscheweyh differential operator product set {Pm [z]} of polynomials associated with the set {Pm [z]} will be effective in D(E [r] ).
Before obtaining effectiveness conditions of basic sets of polynomials in hyperspherical regions from our results, we introduce the following notations:
The open hypersphere is defined by:
à k
! 12
X
< r},
Sr = {z ∈ Ck :
|zs |2
s=1
the closed hypersphere is defined by:
S r = {z ∈ Ck :
à k
X
! 12
|zs |2
≤ r}.
s=1
The region D[S r ] means unspecified domain containing the closed hypersphere 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, put rs = 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 {Pm [z]} in the equiellipse
(n)
∗(n)
1. E [r]∗ yields the effectiveness of the sets {Pm [z]} and {Pm [z]} in the
hypersphere S r
262
G. F. HASSAN
(n)
∗(n)
2. E[r]∗ yields the effectiveness of the sets {Pm [z]} and {Pm [z]} in the
hypersphere Sr
(n)
∗(n)
3. D(E [r]∗ ) yields the effectiveness of the sets {Pm [z]} and {Pm [z]} in
the region D(S r ) where [r]∗ = (r, r, r, . . . , r), r is repeated k-times.
Now, suppose that JN (Dn ) is a polynomial of the operator Dn as given in
(2.2) and JN (D∗n ) is a polynomial of the operator D∗n as given in (4.12) such
N
P
P
∗n j
n j
that JN (Dn ) =
λj (Dn )j and JN (D∗n ) = N
j=1 λj (D ) , where (D ) =
j=1
(Dn )j−1 Dn , (D∗n )j = (D∗n )j−1 D∗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, Theorem 4.3, Theorem 4.4, Theorem 4.5 and Corollary 4.1 will be still true when
we replace the set {Dn Pm (z)} and {D∗n Pm (z)} by the sets {JN (Dn )Pm (z)}
and {JN (D∗n )Pm (z)}, respectively.
Remark 4.1. Similar results for the sets
{Dn Pm (z)}, {D∗n Pm (z)}, {JN (Dn )Pm (z)}
and {JN (D∗n )Pm (z)} in hyperelliptical regions can be obtained when the original set {Pm (z)} is general basic set.
Remark 4.2. The effectiveness of Ruscheweyh differential operator sets of polynomials of several complex variables in complete Reinhardt domains was studied 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: gamal6@yahoo.com