Forward ( ∆ ) and Backward ( ∇ ) Difference Operators Basic Sets of Polynomials in and Their Effectiveness in Reinhardt and Hyperelliptic Domains

We generate, from a given basic set of polynomials in several complex variables ( ) { } m m P z ≥0 , new basic sets of polynomials ( ) { } m m P z  ≥0 and ( ) { } m m Q z  ≥0 generated by the application of the ∇ and ∆ operators to the set ( ) { } m m P z ≥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.


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 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 polydisc ( ) 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.) ( ) where , , , n m m m m =  represents the mutli indicies of non-negative integers for the function F(z).We have [1] ( , : , , , max where , , , n r r r r =  is the radius of the considered domain.Then for hyperelliptic domains ( ) { } ( ) ( ) where also, using the above function ( ) F z of the complex variables , s z s I ∈ , which is regular in ( ) ρ Γ and can be represented by the power series above (1), then we obtain , , , Hence, we have for the series ( ) . Since s ρ′ can be taken arbitrary near to , s s I ρ ∈ , we conclude that ( ) Thus, for the function

, , , , n h h h h h D D =
 is the degree of the polynomial ( ) h P z ; the : ∈ and since the element of basic set are linearly independent [6], then ( ) , where λ is a constant.Therefore the condition (5) for a basic set to be a Cannon set implies the following condition [6] 1 lim 1.
For any function

( )
F z of several complex variables there is formally an associated basic series ( ) . When the associated basic series converges uniformly to

( )
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 For Reinhardt domains [24], For hyperelliptic regions [1].

Basic Sets of Polynomials in Generated by ∇ and ∆ Operators
Now, we define the forward difference operator ∆ acting on the monomial m z such that where E is the shift operator and 11-the identity operator.Then by definition.Similarly, we define the backward difference operator ∇ acting on the monomial m z such that ( ) Equivalently, in terms of lag operator L defined as ( ) ( ) − .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, ( ) ( ) ( ) The Cannon functions for the basic sets of polynomils in complete Reinhardt domain and in hyperelliptical regions [1], are defined as follows, respectively: = ∏ .See also [1].We also get for a given polynomial set Let's prove the following statement: Theorem 2. 3 The set of polynomials

Are basic.
Proof: To prove the first part of this theorem, it is sufficient to to show that the initial sets of polynomials would not be linearly independent.Then the set would not be basic.Consequently (11) is impossible.Since .Hence, we write ,0 0 0 ,0 0 ,1 1 0 where m N is a constant.Therefore, .
as ∇ is bounded for the Reinhardt domain is complete.Thus,

∑∑∑
So, by similar argument as in the case of Reinhardt domain we obtain Such that the Cannon function writes as ( ) , lim , = .
Since the Cannon function is non-negative.Hence where m N is a constant.Hence, ( , , as 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

Definition 1 . 2
An open complete Reinhardt domain of radii 0, considered for both the Reinhardt and hyperelliptic domains.These domains are of radii , ,

t
being the radius of convergence in the domain, of polynomials of the highest degree in the representation (4).That is to say 1 2 3

≥Theorem 2 . 2
in complete Reinhardt domain we have the following results: Theorem 2.1 A necessary and sufficient condition[7] for a Cannon set The necessary and sufficient condition for the Cannon basic set several complex variables to be effective[1] in the closed hyperelliptic ( )


are generated, are linearly independent.Suppose there exists a linear relation of the form set.Changing ∆ to ∇ leads to the same conclusion.We obtain the following result.In a complete Reinhardt domain for the forward difference operator ∆, the Cannon sum of the monomial m z is given by the backward difference operator ∇, the Cannon sum ∈ for which the condition (5) is satisfied, is effective in ( ) r E , then the (∆) and (∇)-set Let us illustrate the effectiveness in Reinhardt and hyperelliptic domains, taking some examples.First, suppose that the set of polynomials new polynomial from the polynomial defined above: new sets are nowhere effective since the parents sets are nowhere effective.By changing 1 effectiveness as in Reinhart domain for both operators ∆ and ∇ in the hyperelliptic domain.The following notations are relevant to the table below.
in the hiperelliptic domain.