Forward (D) and Backward (Ñ) Difference Operators Basic Sets of

Journal of Applied Mathematics and Physics, 2016, 4, 1630-1642 Published Online August 2016 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2016.48173

Forward (∆) and Backward (∇) Difference Operators Basic Sets of Polynomials in and Their Effectiveness in Reinhardt and Hyperelliptic Domains Saheed Abayomi Akinbode, Aderibigbe Sheudeen Anjorin Lagos State University, OJO (LASU), Apapa, Nigeria Received 7 July 2016; accepted 26 August 2016; published 29 August 2016 Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract We generate, from a given basic set of polynomials in several complex variables basic sets of polynomials operators to the set

{P ( z )} m

m ≥0

and

{Q

m

( z )}m≥0

{ Pm ( z )}m≥0 , new

generated by the application of the ∇ and ∆

{ Pm ( z )}m≥0 . All relevant properties relating to the effectiveness in Reinhardt

and hyperelliptic domains of these new sets are properly deduced. The case of classical orthogonal polynomials is investigated in details and the results are given in a table. Notations are also provided at the end of a table.

Keywords Effectiveness, Cannon Condition, Cannon Sum, Cannon Function, Reinhardt Domain, Hyperelliptic Domain

1. Introduction Recently, there has been an upsurge of interest in the investigations of the basic sets of polynomials [1]-[27]. The inspiration has been the need to understand the common properties satisfied by these polynomials, crucial to gaining insights into the theory of polynomials. For instance, in numerical analysis, the knowledge of basic sets of polynomials gives information about the region of convergence of the series of these polynomials in a given domain. Namely, for a particular differential equation admitting a polynomial solution, one can deduce the range How to cite this paper: Akinbode, S.A. and Anjorin, A.S. (2016) Forward (∆) and Backward (∇) Difference Operators Basic Sets of Polynomials in and Their Effectiveness in Reinhardt and Hyperelliptic Domains. Journal of Applied Mathematics and Physics, 4, 1630-1642. http://dx.doi.org/10.4236/jamp.2016.48173

S. A. Akinbode, A. S. Anjorin

of convergence of the polynomials set. This is an advantage in numerical analysis which can be exploited to reduce the computational time. Besides, if the basic set of polynomials satisfies the Cannon condition, then their fast convergence is guaranteed. The problem of derived and integrated sets of basic sets of polynomials in several variables has been recently treated by A. El-Sayed Ahmed and Kishka [1]. In their work, complex variables in complete Reinhardt domains and hyperelliptical regions were considered for effectiveness of the basic set. Also, recently the problem of effectiveness of the difference sets of one and several variables in disc D(R) and m polydisc ∏ i =1 ( Ri ) has been treated by A. Anjorin and M.N Hounkonnou [27]. In this paper, we investigate the effectiveness, in Reinhardt and hyperelliptic domains, of the set of polynomials generated by the forward (∆) and backward (∇) difference operators on basic sets. These operators are very important as they involve the discrete scheme used in numerical analysis. Furthermore, their composition operators form the most of second order difference equations of Mathematical Physics, the solutions of which are orthogonal polynomials [25] [26]. Let us first examine here some basic definitions and properties of basic sets, useful in the sequel. . The Definition 1.1 Let z = ( z1 , z2 , , zn ) be an element of the space of several complex variables hyperelliptic region of radii rs > 0, s ∈ I = {1, 2, , n} ), is denoted by E[r ] and its closure by E[r ] where

E[ r ] = {w ∈

: w < 1} , E[ r ] = {w ∈ : w ≤ 1}

And = w

zs ; s ∈ I. rs

, ws {w1 , w2 , , wn } =

Definition 1.2 An open complete Reinhardt domain of radii ρ s > 0, s ∈ I is denoted by Γ( ρ ) and its closure by Γ( ρ ) , where

Γ( ρ ) = Γ( ρ1 , ρ2 ,, ρn ) = {z ∈

: zs < ρ s : s ∈ I } ,

Γ( ρ ) = Γ( ρ1 , ρ2 ,, ρn ) = {z ∈

( )

: zs ≤ ρ s : s ∈ I } .

( )

The unspecified domains D Γ( ρ ) and D E( r ) are considered for both the Reinhardt and hyperelliptic domains. These domains are of radii ρ s > ρ s , rs > rs , s ∈ I . Making a contraction of this domain, we get the + where ρ s+ stands for the right-limits of ρ s , s ∈ I : domain D ( ρ ) = D ρ1+ , ρ 2+ , , ρ n+

(

) ((

( )

(

))

D Γ( ρ ) ≡ D ( ρ )

+

) =D (( ρ , ρ , , ρ )) ={z ∈ + 1

+ 2

+ n

}

: zs ≤ ρ s* : s ∈ I .

Thus, the function F ( z ) of the complex variables zs , s ∈ I , which is regular in E( r ) can be represented by the power series F (z) =

∞

am z m = ∑

m =0

∞

∑

(

)

a( m1 ,m2 ,,mn ) z1m1 , z2m2 , , znmn ,

( m1 ,m2 ,,mn )

(1)

where m = ( m1 , m2 , , mn ) represents the mutli indicies of non-negative integers for the function F(z). We have [1] = M ( F ( z ) , [ r ]) M = ( F ( z ) : ( r1 , r2 , , rn ) ) max F ( z )

(2)

E( r )

where r = ( r1 , r2 , , rn ) is the radius of the considered domain. Then for hyperelliptic domains E( Γ ) [1]

1 σ m = inf m t =1 t

{m} n

m 2

∏ msmS

, 2

s =1

t being the radius of convergence in the domain, m = m1 + m2 +  + mn assuming 1 ≤ σ m ≤ m /2 zs ms s = 1 , whenever m=s 0; s ∈ I . Since = ωs ; s ∈ I , we have (1) rs

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( n)

m

and

S. A. Akinbode, A. S. Anjorin

1    m   1     am am m 1  = ≤ n lim m →∞  lim m →∞  n m − ms  m m − 1  s    n  m  ∏rs  σ m ∏ ( rs )  σ mm  ∏ ( rs )   s1 s 1= =    s =1   

where also, using the above function F ( z ) of the complex variables zs , s ∈ I , which is regular in Γ( ρ ) and can be represented by the power series above (1), then we obtain

am ≤

ρ m′ ∈ ( ρ m1 , ρ m2 , , ρ mn ) and

M ( F ( z ) , ( ρ m′ ) )

, ms ≥ 0; s ∈ I ,

ρ m′

M ( F ( z ) , ( ρ ′ ) ) = max F ( z ) .

(3)

Γ( ρ ′ )

Hence, we have for the series F ( z ) ∞

= ∑ am z m m 0 =

k =



∞



m



∞



m

∑ am  ∑ π m,k Pk ( z )  ≡ ∑am  ∑ π m,k z k  , 



m 0= k 0 =

k1 , k2 , , kn ) , m ( m1 , m = (= 2 ,  , mn ) , π m ,k

arbitrary near to ρ s , s ∈ I , we conclude that





m 0= k 0 =

m , max Γ ′ F ( z ) M ( F ( z ) , ( ρ ′ ) ) . Since ρ s′ can be taken  = (ρ ) k 

 lim m →∞  am 

n

( ms − m ) 

∏ρ s

 

s =1

1 m

≤

1 n

∏ρ s

,

s =1

With lim = lim sup and ( m )  ms . Definition 1.3 A set of polynomials { Pm ( z )}m≥0 = { P0 ( z ) , P1 ( z ) ,}m≥0 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 basic set { Pm ( z )}m≥0 . Thus, according to [4], the set { Pm ( z )}m≥0 will be basic if and only if there exists a unique row-finite-matrix P such that PP = PP = 11 , where P = ( Pm ,h ) is a matrix of coefficients of the set { Pm ( z )}m≥0 ; h = ( h1 , h2 , , hn ) are multi indices of nonegative integers, P is the matrix of operators deduced from the associated set of the set { Pm ( z )}m≥0 and 11 is the infinite unit matrix of the basic set { Pm ( z )}m≥0 , the inverse of which is { Pm ( z )}m≥0 . We have

= Pm ( z )

m

m

m

m

Pm,h z h z m ∑= Pm,h Ph ( z ) ∑Pm,h Pm= ( z ) Pm ( z ) ∑ Pm,h z h . ∑=

h 0= h 0 = h 0 =

(4)

h 0 =

Thus, for the function F ( z ) given in (1), we get F ( z ) = ∑ m=0 π m,h Pm ( z ) where ∞

f(

h)

( 0 ) ≠= π

m ∞   , h=! h ( h − 1)( h − 2 ) 3 ⋅ 2 ⋅ 1 . The series ∑ m=0 π m,h Ph ( z ) hs ! k  is an associated basic series of F(z). Let N m = N m1 , N m2 , , N mn be the number of non zero coefficients Pm ,h in the representation (4). Definition 1.4 A basic set satisfying the condition

P= = π m ,h ∑ h 0 = ∑ h 0 Pm,h m ,h ah = m

m

m ,k

lim m →∞ N

1 m m

=1

Is called a Cannon basic set. If 1 m m

lim m →∞ N = a > 1,

1632

(5)

S. A. Akinbode, A. S. Anjorin

Then the set is called a general basic set. Now, let Dm = Dm1 ,m2 ,m3 ,,mn be the degree of polynomials of the highest degree in the representation (4). That is to say Dh = Dh1 ,h2 ,h3 ,,hn is the degree of the polynomial Ph ( z ) ; the Dn ≤ Dm ∀ns ≤ ms : s ∈ I and since the element of basic set are linearly independent [6], then N m ≤ 1 + 2 +  + ( Dm + 1) < λ Dm , where λ is a constant. Therefore the condition (5) for a basic set to be a Cannon set implies the following condition [6] 1 m

lim m →∞ Dm = 1.

(6)

For any function F ( z ) of several complex variables there is formally an associated basic series ∞ F ( z ) in some domain, in other words as in classical terminology of Whittaker (see [5]) the basic set of polynomials are classified according to the classes of functions represented by their associated basic series and also to the domain in which they are represented. To study the convergence property of such basic sets of polynomials in complete Reinhardt domains and in hyperelliptic regions, we consider the following notations for Cannon sum

∑ h=0 Ph ( z ) . When the associated basic series converges uniformly to

n

µ ( Pm ( z ) , ( ρ ) ) = ∏ρ s m s =1

m

∑ Pm,h h =0

M ( Ph ( z ) , ( ρ ) )

(7)

For Reinhardt domains [24], n

σ m ∏ ( rs )

m

m

∑ Pm,h h =0

s =1

M ( Ph ( z ) , ( r ) ) = Ω ( Pm ( z ) , ( r ) )

For hyperelliptic regions [1].

2. Basic Sets of Polynomials in

Generated by ∇ and ∆ Operators

Now, we define the forward difference operator ∆ acting on the monomial z m such that J ( ∆ ) =∆ n z m ∆ n ≡ ( E= + ( −11) ) n

n

n

k =0

 

∑  k  E n−k ( −11)

k

by binomial expansion.

where E is the shift operator and 11 -the identity operator. Then ∆ n Pm= (z)

n

n

∑  k  ( −11)

k

 

n

E n−k Pm= (z)

n

∑  k  ( −11)

k 0= k 0 =

k

 

Pm ( z + n − k ) .

So, considering the monomial z m ∆n z m =

n

n

n

n

k m k m ∑  k  ( −11) ( z + n − k ) = ∑  k  ( −11) ( z + α ) ,

 

= k 0= k 0

(z +α )

m

 

α = n − k.

m m = ∑  j  z m − jα j . j =0  

Hence n

 nm k j m− j  ( −11) α z . 0  j 

m

∑∑  k  

= ∆n z m

= k 0 =j

Since α= n − k , = αj

j

 j

j

 j

t − k) ( n= ( −k ) ∑   n j −t ( −1) ( k ) ∑  t  n j −= t j

 

t

=t 0= k 0

Hence

1633

 

t

t

.

(8)

S. A. Akinbode, A. S. Anjorin n

 nm j  t +k t s j −t    × ζ ( −11) n ( k ) z , 0   j  t 

j m− j

m

∑∑∑∑  k  

= ∆n z m

k= 0 =j 0 =t 0 =s

m − j m− j m− j = ∑ s =0 ζ z s by definition. Similarly, we define the backward difference operator where ζ =   and z  s  ∇ acting on the monomial z m such that

J ( ∇ ) = ∇ n z m , ∇ = 11 − E −1.

(9)

Equivalently, in terms of lag operator L defined as LF = ( z ) f ( z − 1) , we get ∇ = 11 − L . Remark that the advantage which comes from defining polynomials in the lag operator stems from the fact that they are isomorphic to the set of ordinary algebraic polynomials. Thus, we can rely upon what we know about ordinary polynomials to treat problems concerning lag-operator polynomials. So, = ∇n

(11 − E= ) ∑  k  ( − E= ) ∑  k  ( −11)     n

−1

n

n

n

n

k

−1

k

E −k .

(10)

k 0= k 0 =

The Cannon functions for the basic sets of polynomils in complete Reinhardt domain and in hyperelliptical regions [1], are defined as follows, respectively: 1

µ ( P ( z ) , ( ρ ) ) = lim m →∞ {µ ( Pm ( z ) , ( ρ ) )} m

{

}

Ω ( P ( z ) , ( r )) = lim m →∞ Ω ( Pm ( z ) , ( r ) )

1 m

.

Concerning the effectiveness of the basic set { Pm ( z )}m≥0 in complete Reinhardt domain we have the following results: Theorem 2.1 A necessary and sufficient condition [7] for a Cannon set { Pm ( z )}m≥0 to be: 1. effective in Γ( ρ ) is that µ ( P ( z ) , ( ρ ) ) = ∏ s =1 ρ s ; n

( )

2. effective in D Γ( ρ )

(

( )) = ∏

is that µ P ( z ) , ρ +

n s =1

ρs .

Theorem 2.2 The necessary and sufficient condition for the Cannon basic set several complex variables to be effective [1] in the closed hyperelliptic E( r ) where r = ( r1 , r2 , , rn ) .

{Pm ( z )}m≥0 of polynomials of n is that Ω ( P ( z ) , ( r ) ) = ∏ s=1 rs

{Pm ( z )}m≥0 of polynomials of several complex variables will be effective in D ( E( r ) ) n Ω ( P ( z ) , ( r + )) = ∏ s=1 rs . See also [1]. We also get for a given polynomial set {Qm ( z )}m≥0 :

The Cannon basic set if and only if

∇ n Qm ( z ) =

n

n

∑  k  ( −11)  

k

E − k Qm ( z ) ∇ n Qm ( z ) =

n

n

∑  k  ( −11)

k 0= k 0 =

 

k

Qm ( z − k ) .

So, considering the monomial z m , zm ∇ n=

n

n

m− j n

= k 0

 

=s 0 = k 0 =j

 nm j +k j s  × ζ ( −11) k z . 0  j 

m

k m ) ∑∑∑    ∑  k  ( −11) ( z − k= k

Let’s prove the following statement: Theorem 2.3 The set of polynomials

{P ( z )} and {Q ( z )} {P ( z )} {P ( J ( ∆ )( z ) )} = {Q ( J ( ∇ )( z ) )} = {Q ( z )} m

m

m≥0

m

m≥0

m

m≥0

m≥0

m

m

m≥0 m≥0

Are basic. Proof: To prove the first part of this theorem, it is sufficient to to show that the initial sets of polynomials

1634

{

{Pm ( z )}m≥0

}

and {Qm ( z )}m≥0 , from which Pm ( z ) and m≥0 dent. Suppose there exists a linear relation of the form

{Q

m

( z )}m≥0

S. A. Akinbode, A. S. Anjorin

are generated, are linearly indepen-

m

∑ ci Pm,i ( z ) ≡ 0, ci ≠ 0

(11)

i =0

For at least one i, i ∈ I . Then

( P ( J ( ∆ ) ) )  ∑ c P m

m

i =0

i m ,i



( z )  ≡ 0. 

Hence, it follows that ∑ i =0 ci Pm,i ( z ) ≡ 0 . This means that { Pm ( z )}m≥0 would not be linearly independent. Then the set would not be basic. Consequently (11) is impossible. Since 1, z1 , z 2 , , z n are polynomials, each m of them can be represented in the form z = ∑ π m , k Pk ( z ) . Hence, we write m

∑ π 0,k Pk ( z )

= m1 0,= 1

k

∑ π1,k Pk ( z )

= m2 1,= z

k

∑ π 2,k Pk ( z )

= m3 2,= z

k

 = mn n= , zn

∑π n,k Pk ( z ) k

= with z

z1 , z2 , , zn ) , k ( k1 , k2 , , kn ) . (=

In general, given any polynomial Pm ( z ) = ∑ i =0 ci z i and using m

m  i  Pm ( z ) = ∑ci  ∑π i , j Pj ( z )  =i 0 = j 0 

=

∑ci (π i ,0 P0 ( z ) + ) m

i =0

=

∑ciπ i ,0 P0 ( z ) + ciπ i ,1P1 ( z ) +  i =0

m

cm Pm ( z ) ; m (= m1 , m2 , , mn ) ; z ( z1 , z2 , , zn ) . ∑=

=

mn = m1

n

n

{

}

Hence the representation is unique. So, the set Pm ( z ) is a basic set. Changing ∆ to ∇ leads to the same m≥0 conclusion. We obtain the following result. of polynomials in several complex variables zs ; s ∈ I is EffecTheorem 2.4 The Cannon set Pm ( z )

{

}

m≥0

tive in the closed complete Reinhardt domain Γ( ρ ) and in the closed Reinhardt region D Γ( ρ ) . Proof: In a complete Reinhardt domain for the forward difference operator ∆, the Cannon sum of the monomial z m is given by

( )

n

µ ( Pm ( z ) , ( ρ ) ) =

∏ ( ρs ) s =1 m− j n

m − ms

∑ Pm,h

(

M Pm,h ( z ) , ( ρ s )

 nm j  t +k ∑∑∑∑  k   j   t  ζ ( −1) n j −t k t =s 0 k= 0 =j 0 =t 0      m

j

Then

(

)

h

)

M Pm ,h= ( z ) , ( ρ ) max P ( z ) ≤ N m M ( Pm,h ( z ) , ( ρ ) ) Γ( ρ )

where N m is a constant. Therefore,

1635

.

S. A. Akinbode, A. S. Anjorin

µ ( Pm,h ( z ) , ( ρ ) )

 n  m − ms Nm  ∏ ( ρs ) ∑ Pm,h M ( Pm,h ( z ) , ( ρ ) )  h  ≤ m− js =j1 n m  nm j  t +k j −t t ∑∑∑∑  k   j   t  ζ ( −1) n k =s 0 =t 0 = k 0 =j 0     

which implies that  n  m − ms Nm  ∏ ( ρs ) µ ( Pm ( z ) , ( ρ ) )   . ≤ m− j j n s =m1  nm j  t +k j −t t ∑∑∑∑  k   j   t  ζ ( −1) n k s =t 0 = k 0=j 0     

µ ( Pm ( z ) , ( ρ ) )

Then the Cannon function

µ ( P ( z ) , ( ρ ) ) ≤ lim m →∞ µ ( Pm ( z ) , ( ρ ) ) m 1

µ ( P ( z ) , ( ρ ) ) ≤ ∏ρ s ; s ∈ I . n

s =1

)

(

But µ P ( z ) , ( ρ ) ≥ 0 . Hence

µ ( P ( z ) , ( ρ ) ) = lim m →∞ µ ( Pm ( z ) , ( ρ ) ) m 1

µ ( P ( z )= , ( ρ ))

n

∏ρ s ; s ∈ I . s =1

Similarly, for the backward difference operator ∇, the Cannon sum n

(

∏ ( ρs )

)

µ Q m ( z ) , ( ρ ) =

m − ms

s =1

M ( Qm, j ( z ) , ( ρ ) )

j m− j n

 nm k+ j j  ζ ( −1) k j 0  

m

∑∑∑  k   s = k 0 =j

n

(

∑ Qm, j

∏ ( ρs )

)

µ Q m ( z ) , ( ρ ) =

m − ms

s =1

∑ Qm, j

(

M Q m, j ( z ) , ( ρ )

j

.

 nm k ∑∑∑ ( −1)  k   j  ζ ( −1) k j s = k 0=j 0   

m− j n

m

)

j

Then n

(

)

µ Q m ( z ) , ( ρ ) ≤

∏ ( ρs ) s =1 m− j n

m − ms

( K µ (Q m

m

∑∑∑ ( −1)

j+k

=s 0 = k 0=j 0

m

( z ) , ( ρ )))

 n m j    ζ k  k  j 

.

where K m =∇ =constant as ∇ is bounded for the Reinhardt domain is complete. Thus, m

1

µ ( Q ( z ) , ( ρ ) ) = lim m →∞ µ ( Q m ( z ) , ( ρ ) ) m 1

n

≤ lim m →∞ µ ( Qm ( z ) , ( ρ ) ) m ≤ ∏ρ s . s =1

But

µ ( Q ( z ) , [ ρ ]) ≥ ∏ρ s . n

s =1

1636

S. A. Akinbode, A. S. Anjorin

Hence, we deduce that µ ( Qm ( z ) , ( ρ ) ) = ∏ s =1 ρ s . n

Theorem 2.5 If the Cannon basic set { Pm ( z )}≥0 (resp. {Qm ( z )}≥0 ) of polynomials of the several complex variables zs , s ∈ I for which the condition (5) is satisfied, is effective in E( r ) , then the (∆) and (∇)-set Pm ( z ) (resp. Q m ( z ) ) of polynomials associated with the set { Pm ( z )}m≥0 (resp. {Qm ( z )}m≥0 ) will

{

}

{

m≥0

}

m≥0

be effective in E( r ) .  P ( z ) , ( r ) The Cannon sum Ω have the form

(

)

of the forward difference operator ∆ of the set n

(

σ m ∏ ( rs )

)

(

m

M Pm,h ( z ) , ( r )

∑ Pm,h

m − ms

 P ( z ) , ( r ) = Ω j  nm j  k +t ∑∑∑  k   j   t  ζ * ( −1) n j −k k t k= 0 =j 0 =t 0     

ζ* where=

∑ s =0 × ζ m− j

h =0

s =1 n m

{P ( z )} m

≥0

in E( r ) will

)

and n

∏r h

s

(

)

M Pm= max Ph ( z ) ≤ ∑ Pm,h ,h ( z ) , ( r )

s =1

σh

E( r )

σ h M ( Pm,h ( z ) , ( r ) ) ∏r hs n

≤

n

m

j

∑∑∑ ( −1)

k +t

k= 0 =j 0 =t 0

 n   m   j  * j −t t      ζ n k × k  j  t 

s =1

n

∏r

hs

σ

h

s =1

≤ K1 N m Dm2 M ( Pm,h ( z ) , ( r ) ) ≤ K1 Dmm+2 M ( Pm,h ( z ) , ( r ) ) where K1 is a constant. Then n

(

)

 P [ z ] , [ r ] ≤ Ω m

K1σ m Dmn+2 ∏ [ rs ]

m − ms

m

∑ Pm,h h =0

s =1

 nm j  k +t ∑∑∑  k   j   t  ζ * ( −1) n j −t k t k= 0 =j 0 =t 0      n

j

m

n

= K 2 ∏ [ rs ]

m − ms

m

∑ Pm,h h =0

s =1

where

K2 =

K1σ m Dmn+2 n m j  nm j  t +k ∑∑∑  k   j   t  × ζ * ( −1) n j −t k= 0 =j 0 =t 0     

So, by similar argument as in the case of Reinhardt domain we obtain

(

( (

)

 P ( z ) , ( r ) =  P ( z ) , ( r ) Ω lim m →∞ Ω m n

))

1 m 1 m

s α) = lim = m →∞ ( K 2 ∏ ( rs )

m −m

n

∏rs ,

=s 1 =s 1

(

)

 P ( z ) , ( r ) ≥ ∏ n r . Similarly, for the where α = ∑ h Pm,h , since the Cannon function is such that [1] Ω s =1 s backward difference operator

σ m ∏rs m −ms ∑ Qm,h M ( Q m,h ( z ) , ( r ) ) n

(

)

m

h =0 s =1  Q ( z ) , ( r ) = Ω m m− j n m nm j +k ∑∑∑  k   j  ζ ( −1) k j =s 0 = k 0 =j 0    

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S. A. Akinbode, A. S. Anjorin

Such that the Cannon function writes as

(

( (

)

 Q ( z ) , ( r ) ≤ lim m →∞ Ω  Q ( z ) , ( r ) Ω m

))

1 m

n

= ∏rs . s =1

But

(

)

n

 Q ( z ) , ( r ) ≥ ∏r Ω s s =1

Since the Cannon function is non-negative. Hence

(

)

n

 Q ( z ) , ( r ) = Ω ∏rs . s =1

3. Examples Let us illustrate the effectiveness in Reinhardt and hyperelliptic domains, taking some examples. First, suppose that the set of polynomials { Pm ( z )}m≥0 is given by P0 ( z ) = 1 Pm ( z ) = z m + 2m

2

for m ≥ 1.

Then = z m Pm ( z ) − 2m

2

⋅1

Mm (ρ ) = sup Pk ( z ) = ρ sm + 2m , s ∈ I 2

z < ρs

= M 0 ( P0= P0 ( z ) 2m . ( z ) , ( ρ ) ) sup 2

z ≤ρ

Hence m

µ ( Pm ( z ) , ( ρ ) )= ρ sm + 2 ⋅ 2m =

∑ Pm,h

2

h =0

M ( Pm ( z ) , ( ρ ) )

which implies 1

µ ( P ( z ) , ( ρ ) ) = lim m →∞ {µ ( Pm ( z ) , ( ρ ) )} m

{

= lim m →∞ ρ + 2 ⋅ 2 m s

m2

}

1 m

for ρ s < 2 ; µ ( P ( z ) , ( ρ ) ) = ∞ . Now consider the new polynomial from the polynomial defined above: n m z ∆=

=

n

n

k =0

 

k m ∑  k  ( −11) ( z + n − k ) n

m

 nm j  * k +t t s j −t    ζ ( −11) n ( k ) z . 0   j  t 

j

∑∑∑  k   k= 0 =j 0 =t

Hence by Theorem 2.4, n

µ ( Pm ( z ) , ( ρ ) ) =

∏ ( ρs ) s =1 n m

m − ms

n

∑ Pm,h h =0

(

M Pm,h ( z ) , ( ρ )

 nm j  k +t ∑∑∑  k   j   t  ζ * ( −1) n j −t k t k= 0 =j 0 =t 0      j

1638

) .

S. A. Akinbode, A. S. Anjorin

where

(

)

M Pm ,h= ( z ) , ( ρ ) max P ( z ) ≤ N m M ( Pm,h ( z ) , ( ρ ) ) Γ( ρ )

where N m is a constant. Hence,

µ ( Pm,h ( z ) , ( ρ ) )

 n  m − ms Nm  ∏ ( ρs ) ∑ Pm,h M ( Pm,h ( z ) , ( ρ ) )  h  s =1  ≤ j n m  nm j  * k +t ∑∑∑  k   j   t  ζ ( −1) n j −t k t =t 0 = k 0=j 0     

The Cannon function 1

µ ( P ( z ) , ( ρ ) ) ≤ lim

n   n  m m − ms )∑ Pm,h M ( Pm,h ( z ) , ( ρ ) )    Nm  ∏ ( ρs ) h =0    s =1

m →∞

  nm j  * k +t  j −t t  ∑∑∑       ζ ( −1) n k  =t 0 =k 0=j 0  k   j   t   j

n

m

1 m

.

which implies

(

µ ( P ( z ) , ( ρ ) ) = K1µ ( Pm ( z ) , ( ρ ) )

)

1 m

=∞

as m → ∞

where 1 m

K1 =

Nm j

n

n

∏ ( ρs )

m − ms

s =1

 nm j  * k +t j −t t    ζ ( −1) n k 0  j  t 

m

∑∑∑  k  

k 0=j =t 0 =

and

µ ( Pm ( z ) , ( ρ ) ) = ∑ Pm,h M ( Pm,h ( z ) , ( ρ ) ) . m

h =0

Hence

µ ( P ( z ) , ( ρ )) = ∞. Then µ ( P ( z ) , ( ρ ) ) = ∞. Similarly, for the operator ∇, we have

µ ( Q ( z ) , ( ρ ) ) ≤

N

1 m m

1

1

 n m − ms  m  m  ∏ ( ρs )   ∑ Qm,h M ( Qm ( z ) , ( ρ ) )    s =1   h =0 1

m  m− j n m  n   m  j +k j   ∑∑∑     ζ ( −1) k   =s 0 =k 0=j 0  k   j  

Since

∑ Q m,h

1 m

( M (Q

m ,h

1

( z ) , ( ρ ))) m

(

= µ ( Qm ( z ) , ( ρ ) )

Then

µ ( Q ( z ) , ( ρ ) ) = ∞; m → ∞

1639

)

1 m

= ∞.

S. A. Akinbode, A. S. Anjorin

Table 1. Region of effectiveness: (1) Disc λ ( R ) = ∏ j =1 R j ; (2) Hyperelliptic λ ( R ) = ∏ s =1 rs ; Reinhardt domain n

n

λ ( R ) = ∏ j =1 ρ sˆ . n

Polynomials

∆ n Pm [ z ]

(P ) m

∇ n Pm [ z ]

Monomials

∑ ∑ ∑ ∑

Zm

ζ ( −1) k t

n

m

j

m− j

k =0

j =0

t =0

s =0

× Ck C j Ct ( n ) Z n

m

j −t

j

k +t

∑ ∑ ∑ n

m

m− j n

k 0 =j 0=s 0 =

s

Ck mC j ζ k j ( −1)

k+ j

Zs

Chebyshev (first kind) [ m 2]

Pm ( Z )

∑

=

[ m 2] m µ =0

C2 µ Z

m−2 µ

(Z

2

− 1)

∑ ∑ ∑ ∑ µ

n

µ

× ( −1)

s+k

(n − k )

s +t

[ m 2]

∑ ∑ ∑

m−2 µ

= µ 0 = k 0=s 0 =j 0

η1 × mC2 µ m−2 µ C j nCk µ Cs

n

m−2 µ

= µ 0 = k 0=j 0

η2

× nCk mC2 µ m−2 µ C j × ( −1)

Zq

q +v+ j

Zq

Chebyshev (second kind) [ m 2]

∑ µ ( −1) =0

µ m− µ

Cµ ( 2 Z )

[ m 2]

n

m−2 µ

µ =0

k =0

q =0

∑ ∑ ∑

m−2 µ

n

m−2 µ n

µ =0

k =0

s =0

× m−2 µ Csη 4 ( −1)

×η3 2m−2 µ ( −1) ( n − k ) Z s k

[ m 2]

∑ ∑ ∑

× n Ck m − 2 µ Cq m − µ C µ t

s+k

Ck m− µ Cµ ( −1)

k

2m−2 µ Z s

Hermite [ m 2]

∑ ( −1) k!( 2 z ) k

k =0

m−2 k

( m − 2k ) !

n

[ m 2]

m−2 k

j =0

k =0

t =0

∑ ∑ ∑ × Cj n

m−2 k

∑

η5

Ct ( −1) 2 j

m−2 k

k!( m − 2k )!× ( n − j ) Z s

s

m − 2 k −t s =0

×j Z t

n

m−2 k

[ m 2] n

j =0

t =0

k =0

∑ ∑ ∑

C j ( −1)

j +t m − 2 k

Ct 2m−2 k k!( m − 2k ) !

s

Implication: The new sets are nowhere effective since the parents sets are nowhere effective. By changing m −m ∏ s=1 ρ s s in Reinhart domain to σ m ∏ s=1 [ rs ]

n

n

m − ms

, where σ m = inf t =1

1 tm

{m}

m 2

∏ s=1msm n

S 2

, we obtain the

same condition of effectiveness as in Reinhart domain for both operators ∆ and ∇ in the hyperelliptic domain. The following notations are relevant to the table below. 2( µ − s ) m − 2 s −t − j

∑ ∑

η1 =

=t 0 = q 0

η= 2

 2µ − 2s   m − 2s − t − j     t q   

µ 2( µ − q ) m − 2 q − j

q + j +v ∑ ∑ ∑ ( −1) ,η=3

q 0 =s 0 =

v 0 =

η4 =

η5 =

q =0

m − 2 k −t − s

∑ s =0

∑

=s 0

 m − 2µ − q   , s  

(13)

 m − 2µ − s   . q  

(14)

 m − 2k − t − s    s  

(15)

m−2 µ − s

∑

m−2 µ −q

(12)

Finally, for the classical orthogonal polynomials, the explicit results of computation are given in a Table 1 below. Thus, in this paper, we have provided new sets of polynomials in C, generated by ∇ and ∆ operators, which satisfy all properties of basic sets related to their effectiveness in specified regions such as in hyperelliptic and Reinhardt domains. Namely, the new basic sets are effective in complete Reinhardt domain as well as in closed Reinhardt domain. Furthermore, we have proved that if the Cannon basic set { Pm ( z )}m≥0 is effective in hyperis also effective in the hiperelliptic domain. elliptic domain, then the new set Pm ( z )

{

}

m≥0

References [1]

El-Sayed Ahmed, A. and Kishka, Z.M.G. (2003) On the Effectiveness of Basic Sets of Polynomials of Several Complex Variables in Elliptical Regions. Proceedings of the Third International ISAAC Congress, 1, 265-278.

[2]

El-Sayed Ahmed, A. (2013) On the Convergence of Certain Basic Sets of Polynomials. Journal of Mathematics and Computer Science, 3, 1211-1223.

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S. A. Akinbode, A. S. Anjorin

[3]

El-Sayed Ahmed, A. (2006) Extended Results of the Hadamard Product of Simple Sets of Polynomials in Hypersphere. Annales Societatis Mathematicae Polonae, 2, 201-213.

[4]

Adepoju, J.A. and Nassif, M. (1983) Effectiveness of Transposed Inverse Sets in Faber Regions. International Journal of Mathematics and Mathematical Sciences, 6, 285-295.

[5]

Whittaker, J.M. (1949) Sur les séries de bases de polynômes quelconques. Quaterly Journal of Mathematics (Oxford), 224-239.

[6]

Sayed, K.A.M. (1975) Basic Sets of Polynomials of Two Complex Variables and Their Convergences Properties. Ph D Thesis, Assiut University, Assiut.

[7]

Sayed, K.A.M. and Metwally, M.S. (1998) Effectiveness of Similar Sets of Polynomials of Two Complex Variables in Polycylinders and in Faber Regions. International Journal of Mathematics and Mathematical Sciences, 21, 587-593.

[8]

Abdul-Ez, M.A. and Constales, D. (1990) Basic Sets of Polynomials in Clifford Analysis. Journal of Complex Variables and Applications, 14, 177-185.

[9]

Abdul-Ez, M.A. and Sayed, K.A.M. (1990) On Integral Operators Sets of Polynomials of Two Complex Variables. Bug. Belg. Math. Soc. Simon Stevin, 64, 157-167.

[10] Abdul-Ez, M.A. (1996) On the Representation of Clifford Valued Functions by the Product System of Polynomials. Journal of Complex Variables and Applications, 29, 97-105. [11] Abul-Dahab, M.A., Saleem, A.M. and Kishka, Z.M. (2015) Effectiveness of Hadamard Product of Basic Sets of Polynomials of Several Complex Variables in Hyperelliptical Regions. Electronic Journal of Mathematical Analysis and Applications, 3, 52-65. [12] Mursi, M. and Makar, B.H. (1955) Basic Sets of Polynomials of Several Complex Variables i. The Second Arab Sci. Congress, Cairo, 51-60. [13] Mursi, M. and Makar, B.H. (1955) Basic Sets of Polynomials of Several Complex Variables ii. The Second Arab Sci. Congress, Cairo, 61-68. [14] Nasif, M. (1971) Composite Sets of Polynomials of Several Complex Variables. Publicationes Mathematics Debrecen, 18, 43-53. [15] Nasif, M. and Adepoju, J.A. (1984) Effectiveness of the Product of Simple Sets of Polynomials of Two Complex Variables in Polycylinders and in Faber Regions. Journal of Natural Sciences and Mathematics, 24, 153-171. [16] Metwally, M.S. (1999) Derivative Operator on a Hypergeometric Function of the Complex Variables. Southwest Journal of Pure and Applied Mathematics, 2, 42-46. [17] Mikhall, N.N. and Nassif, M. (1954) On the Differences and Sum of Simple Sets of Polynomials. Assiut Universit Bulletin of Science and Technology. (In Press) [18] Makar, R.H. (1954) On Derived and Integral Basic Sets of Polynomials. Proceedings of the American Mathematical Society, 5, 218-225. [19] Makar, R.H. and Wakid, O.M. (1977) On the Order and Type of Basic Sets of Polynomials Associated with Functions of Trices. Periodica Mathematica Hungarica, 8, 3-7. http://dx.doi.org/10.1007/BF02018040 [20] Boss, R.P. and Buck, R.C. (1986) Polynomial Expansions of Analytic Functions II. In: Ergebinsse der Mathematika Hung ihrer grenz gbicle. Neue Foge., Vol. 19 of Moderne Funktionetheorie, Springer Verlag, Berlin. [21] Kumuyi, W.F. and Nassif, M. (1986) Derived and Integrated Sets of Simple Sets of Polynomials in Two Complex Variables. Journal of Approximation Theory, 47, 270-283. http://dx.doi.org/10.1016/0021-9045(86)90018-3 [22] New, W.F. (1953) On the Representation of Analytic Functions by Infinite Series. Philosophical Transactions of the Royal Society A, 245, 439-468. [23] Kishka, Z.M.G. (1993) Power Set of Simple Set of Polynomials of Two Complex Variables. Bulletin de la Société Royale des Sciences de Liège, 62, 361-372. [24] Kishka, Z.M.G. and El-Sayed Ahmed, A. (2003) On the Order and Type of Basic and Composite Sets of Polynomials in Complete Reinhardt Domains. Periodica Mathematica Hungarica, 46, 67-79. http://dx.doi.org/10.1023/A:1025705824816 [25] Hounkonnou, M.N., Hounga, C. and Ronveaux, A. (2000) Discrete Semi-Classical Orthogonal Polynomials: Generalized Charlier. Journal of Computational and Applied Mathematics, 114, 361-366. http://dx.doi.org/10.1016/S0377-0427(99)00275-7 [26] Lorente, M. (2001) Raising and Lowering Operators, Factorization and Differential/Difference Operators of Hypergeometric Type. Journal of Physics A: Mathematical and General, 34, 569-588. http://dx.doi.org/10.1088/0305-4470/34/3/316 [27] Anjorin, A. and Hounkonnou, M.N. (2006) Effectiveness of the Difference Sets of One and Several Variables in Disc D(R) and Polydisc. (ICMPA) Preprint 07.

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Appendix

Key Notations

( ) = Cannon sum of the new ∆-set in Reinhardt domain. µ ( Q [ z ] , [ ρ ]) = Cannon sum of the new ∇-set in Reinhardt domain. µ ( P [ z ] , [ r ]) = Cannon sum of the new ∆-set in Hyperelliptic domain. µ ( Q [ z ] , [ r ]) = Cannon sum of the new ∇-set in Hyperelliptic domain. µ ( P [ z ] , [ ρ ]) = Cannon function of the new ∆-set in Reinhardt domain. µ ( Q [ z ] , [ ρ ]) = Cannon function of the new ∇-set in Reinhardt domain. µ ( P [ z ] , [ r ]) = Cannon sum of the new ∆-set in Hyperelliptic domain. µ ( Q [ z ] , [ r ]) = Cannon sum of the new ∇-set in Hyperelliptic domain.

1) µ Pm,h [ z ] , [ ρ ] 2) 3) 4) 5) 6) 7) 8)

m ,h

m ,h

m ,h

m ,h

m ,h

9) M ( F ( z ) , [ r ′]) = max Γ

F (z) .

[ r ′]

10) M ( F ( z ) , [ ρ ′]) = max Γ

[ ρ ′]

F (z) .

11) M ( Pm,= max P [ z ] ≤ N m M ( Pm ,h [ z ] , [ r ]) h [ z ] , [ r ]) Γ[ r ]

(

)

(

)

(

)

M Pm ,h= [ z ] , [ ρ ] max P [ z ] ≤ N m M ( Pm,h [ z ] , [ ρ ]) Γ[ ρ ]

M Q m= max Q [ z ] ≤ N m M ( Qm,h [ z ] , [ r ]) ,h [ z ] , [ r ] Γ[ r ]

M Q m,h= [ z ] , [ ρ ] max Q [ z ] ≤ N m M ( Qm,h [ z ] , [ ρ ]) Γ[ ρ ]

m where N m is a constant. Qm,h = Q     is a coefficient corresponding to polynomials set h 

m Pm,h = P     is a coefficient corresponding to polynomial set h  or Qm,h = Pm,h .

{Pm ( z )}m≥0 . We should note that

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{Qm ( z )}m≥0 , Qm ,h ≠ Pm ,h