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We generate, from a given basic set of polynomials in several complex variables , new basic sets of polynomials and generated by the application of the Δ and ∇ operators to the set . 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.

Recently, there has been an upsurge of interest in the investigations of the basic sets of polynomials [

In this paper, we investigate the effectiveness, in Reinhardt and hyperelliptic domains, of the set of polynomials generated by the forward (D) and backward (Ñ) 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 [

Let us first examine here some basic definitions and properties of basic sets, useful in the sequel.

Definition 1.1 Let

And

Definition 1.2 An open complete Reinhardt domain of radii

The unspecified domains

Thus, the function

where

where

t being the radius of convergence in the domain,

where also, using the above function

Hence, we have for the series

arbitrary near to

With

Definition 1.3 A set of polynomials

Thus, according to [

Thus, for the function

is an associated basic series of F(z). Let

Definition 1.4 A basic set satisfying the condition

Is called a Cannon basic set. If

Then the set is called a general basic set.

Now, let

For any function

For Reinhardt domains [

For hyperelliptic regions [

Now, we define the forward difference operator D acting on the monomial

where E is the shift operator and

So, considering the monomial

Hence

Since

Hence

where

Ñ acting on the monomial

Equivalently, in terms of lag operator L defined as

The Cannon functions for the basic sets of polynomils in complete Reinhardt domain and in hyperelliptical regions [

Concerning the effectiveness of the basic set

Theorem 2.1 A necessary and sufficient condition [

1. effective in

2. effective in

Theorem 2.2 The necessary and sufficient condition for the Cannon basic set

several complex variables to be effective [

The Cannon basic set

if and only if

So, considering the monomial

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

For at least one i,

Hence, it follows that

In general, given any polynomial

Hence the representation is unique. So, the set

Theorem 2.4 The Cannon set

in the closed complete Reinhardt domain

Proof: In a complete Reinhardt domain for the forward difference operator D, the Cannon sum of the monomial

Then

where

which implies that

Then the Cannon function

But

Similarly, for the backward difference operator Ñ, the Cannon sum

Then

where

But

Hence, we deduce that

Theorem 2.5 If the Cannon basic set

be effective in

The Cannon sum

where

where

where

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

where

Such that the Cannon function writes as

But

Since the Cannon function is non-negative. Hence

Let us illustrate the effectiveness in Reinhardt and hyperelliptic domains, taking some examples. First, suppose that the set of polynomials

Then

Hence

which implies

for

Now consider the new polynomial from the polynomial defined above:

Hence by Theorem 2.4,

where

where

The Cannon function

which implies

where

and

Hence

Similarly, for the operator Ñ, we have

Since

Then

Polynomials | ||
---|---|---|

Monomials | ||

Chebyshev (first kind) | ||

Chebyshev (second kind) | ||

Hermite | ||

Implication: The new sets are nowhere effective since the parents sets are nowhere effective. By changing

same condition of effectiveness as in Reinhart domain for both operators D and Ñ in the hyperelliptic domain.

The following notations are relevant to the table below.

Finally, for the classical orthogonal polynomials, the explicit results of computation are given in a

Thus, in this paper, we have provided new sets of polynomials in C, generated by Ñ and D 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

Saheed Abayomi Akinbode,Aderibigbe Sheudeen Anjorin, (2016) Forward (Δ) and Backward (∇) Difference Operators Basic Sets of Polynomials in and Their Effectiveness in Reinhardt and Hyperelliptic Domains. Journal of Applied Mathematics and Physics,04,1630-1642. doi: 10.4236/jamp.2016.48173

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