International Journal of Mathematics and Mathematical Sciences

VolumeΒ 2011, Article IDΒ 794593, 23 pages

http://dx.doi.org/10.1155/2011/794593

## Division Problem of a Regular Form: The Case

Department of Mathematics, Institut SupΓ©rieur des Sciences AppliquΓ©es et de Technologie, Rue Omar Ibn El Khattab, GabΓ¨s 6072, Tunisia

Received 13 December 2010; Revised 25 February 2011; Accepted 17 March 2011

Academic Editor: HeinrichΒ Begehr

Copyright Β© 2011 M. Mejri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present a systematic study of a regular linear functional to find all regular forms which satisfy the equation , . We also give the second-order recurrence relation of the orthogonal polynomial sequence with respect to and study the semiclassical character of the found families. We conclude by treating some examples.

#### 1. Introduction

In the present paper, we intend to study the following problem: let be a regular form (linear functional), and and nonzero polynomials. Find all regular forms satisfying This problem has been studied in some particular cases. In fact the product of a linear form by a polynomial () is studied in [1β3] and the inverse problem , is considered in [4β7]. More generally, when and have nontrivial common factor the authors of [8] found necessary and sufficient conditions for to be a regular form. The case where is treated in [4, 9β11]. The aim of this contribution is to analyze the case in which and , . We remark that and have a common factor and (see also [7]). In fact, the inverse problem is studied in [12]. On the other hand, this situation generalize the case treated in [13] (see (2.9)). In Section 1, we will give the regularity conditions and the coefficients of the second-order recurrence relation satisfied by the monic orthogonal polynomial sequence (MOPS) with respect to . We will study the case where is a symmetric form; thus regularity conditions become simpler. The particular case when is a symmetric positive definite form is analyzed. The second section is devoted to the case where is semi-classical form. We will prove that is also semi-classical and some results concerning the class of are given. In the last section, some examples will be treated. The regular forms found in theses examples are semi-classical of class [14]. The integral representations of these regular forms and the coefficients of the second-order recurrence satisfied by the MOPS with respect to are given.

#### 2. The Problem

Let be the vector space of polynomials with coefficients in and its algebraic dual. We denote by the action of on . In particular, we designate by , , the moments of . For any form , any polynomial , any , , let , , , and be the forms defined by duality: where ; .

We define a left multiplication of a form by a polynomial as Let us recall that a form is called regular if there exists a monic polynomial sequence , , such that We have the following result.

Lemma 2.1 (see [15]). *Let , , and . The following formulas hold:
**
where , .*

We consider the following problem: given a regular form , find all regular forms satisfying with constraints , . From (2.6) we can deduce that Then the form depends on two arbitrary parameters and .

We notice that when , we encounter the problem studied in [13] again.

We suppose that the form has the following integral representation: where is a locally integrable function with rapid decay, continuous at the origin; then the form is represented by where [16, 17] Let denote the sequence of monic orthogonal polynomials with respect to ; we have with When is regular, let be the corresponding MOPS: From (2.7), we know that the existence of the sequence is among all the strictly quasi-orthogonal sequences of order two with respect to ( is not necessarily a regular form) [15, 18β20]. That is, with , .

From (2.16), we have

Lemma 2.2. *Let be a sequence of polynomials satisfying (2.16) where , , and are complex numbers such that for all . The sequence is orthogonal with respect to if and only if
*

*Proof. *The conditions (2.19) are necessary from the definition of the orthogonality of with respect to .

For , we have (by (2.7))
and from (2.16), we get
Taking into account the orthogonality of , we obtain
By (2.19), it follows that
Consequently, the previous relations and (2.22) prove that is orthogonal with respect to , which proves the Lemma.

*Remark 2.3. *When is regular, from Theorem 5.1 in [21], there exist complex numbers , and such that

From (2.16), (2.24), and (2.15) we obtain the following relations: Taking into account (2.16), (2.18) and (2.19), we get with the initial conditions: If we denote from the Cramer rule we have

Lemma 2.4. *The following formulas hold:
**
where , , and .*

*Proof. *Equations (2.32) and (2.33) are deduced, respectively, from (2.9) and (2.8).

We have
Using (2.4), we get
From (2.9), we obtain
According to (2.5) and (2.37), we can deduce (2.34).

We have
Then (by (2.39))
It follows that
hence (2.35).

Proposition 2.5. *One has
**
where
**
with
*

*Proof. *Using (2.13), we, respectively, obtain
Taking into account previous relations, we obtain for (2.28) the following:
that is,
Let ; based on the relations (2.32)β(2.34), it follows that
From (2.48) and (2.47), we obtain the desired results.

Proposition 2.6. *The form is regular if and only if , . Then, the coefficients of the three-term recurrence relation (2.15) are given by
*

*Proof**Necessity*

From (2.27) and Lemma 2.4, we get
and again with (2.27) and (2.42), we can deduce that
Moreover, is orthogonal with respect to , therefore it is strictly quasiorthogonal of order two with respect to , and then it satisfies (2.16) with , . This implies , . Otherwise, if there exists an such that , from (2.29), , which is a contradiction.*Sufficiency*

Let
We get
We have .

From (2.56) and (2.57) we get
On account of (2.54), we can deduce that .

Then we had just proved that the initial conditions (2.27) are satisfied.

Furthermore, the system (2.26) is a Cramer system whose solution is given by (2.29), (2.30), and (2.31); with all these numbers , , and (), define a sequence polynomials by (2.16). Then it follows from (2.26) and Lemma 2.2 that is regular and is the corresponding MOPS.

Moreover, by (2.22) we get
Making in (2.60), it follows that
Based on relations (2.58), (2.60), (2.61), and (2.29), we, respectively, obtain
We have proved (2.49) and (2.50).

When is orthogonal, we have
By (2.16) and the orthogonality of , we get
By virtue of (2.13) and the regularity of we obtain
and consequently, we get the second result in (2.51) from (2.58), and (2.64).

From (2.16), and the orthogonality of , we have
Using (2.13), (2.16), and the the orthogonality of , we have
Taking into account the previous relation, (2.66) becomes
From (2.60) and (2.29), we have
Last equation and (2.68) give (2.52).

Moreover, if the form is regular, for (2.29), (2.30), and (2.31), we get where In the sequel, we will assume that is a symmetric linear form.

We need the following lemmas.

Lemma 2.7. *If and are sequences of complex numbers fulfilling
**
then
*

Lemma 2.8. *When given by (2.13) is symmetric, one has
*

*Proof. *As is symmetric, then , , and therefore from (2.13) we have
Now, it is sufficient to use Lemma 2.7 in order to obtain the desired results.

Let

Corollary 2.9. *If is a symmetric form, one has
**
where
*

*Proof. *Following Lemma 2.8, for (2.43) we have
As a consequence, relations (2.81) and (2.42) yield (2.79).

Theorem 2.10. *The form is regular if and only if , , where is defined in (2.80). **In this case one has
**
where , .*

*Proof. *From Proposition 2.6 and Corollary 2.9, we can deduce that is regular if and only if , .

Moreover, from (2.70) we can deduce (2.82).

By (2.49), (2.51), (2.78), and (2.79), for (2.55), (2.56), and (2.57) we get
When by Lemma 2.8, for (2.73) we get
Taking into account (2.79), (2.80), and (2.86)-(2.87), relations (2.70), (2.71) and (2.72) give (2.82)β(2.84).

As a result of relations (2.82)β(2.84) and Proposition 2.6 we get (2.85).

Corollary 2.11. * If is a symmetric positive definite form, then the form is regular when . ** When is regular, it is positive definite form if and only if
*

*Proof. *(1) If is positive definite, then , , therefore , and so , under the hypothesis of the corollary.

(2) If is regular, it is positive definite if and only if , . By Theorem 2.10, we conclude the desired results.

#### 3. Some Results on the Semiclassical Case

Let us recall that a form is called semiclassical when it is regular and its formal Stieltjes function satisfies [15] where monic, , and are polynomials with The class of the semi-classical form is if and only if the following condition is satisfied [22]: where , that is, , , and are coprime.

In the sequel, we will suppose that the form is semi-classical of class satisfying (3.1).

Proposition 3.1. *When is regular, it is also semi-classical and satisfies
**
where
**
Moreover, the class of depends on the zero of .*

*Proof. *We need the following formula:
From (2.7), we have . Using (3.6), we get
Differentiating the previous equation, we obtain
By (3.1) we can deduce (3.4) and (3.5).

Since is a semi-classical, satisfies (3.1) where , and are coprime.

Let be a zero of different from 0, which implies that . We know that .

If , then . if , then . Hence .

Corollary 3.2. *Introducing
*(1)*if , then ;*(2)*if and , then ;*(3)*if and or , then .*

*Proof. *(1) From (3.9) and (3.5), we obtain , . Therefore, it is not possible to simplify, which means that the class of is .

(2) If , then from (3.5) we have . Consequently, (3.4)β(3.6) is divisible by . Thus, fulfils (3.4) with
If , it is not possible to simplify, which means that the class of is .

(3) When , then it is possible to simplify (3.4)β(3.10) by . Thus, fulfils (3.4) with
Since we have , , then we can deduce that if or , it is not possible to simplify, which means that the class of is .

#### 4. Some Examples

In the sequel the examples treated generalize some of the cases studied in [13].

##### 4.1. the Generalized Hermite Form

Let us describe the case , where is the generalized Hermite form. Here is [1] From (4.1), we get We want , .

But from (4.1) and (4.2), we have , with fulfilling and so Then we get Table 1.

Proposition 4.1. *If is the generalized Hermite form, then the form given by (2.9) has the following integral representation:
**
It is a quasi-antisymmetric (, ) and semi-classical form of class satisfying the following functional equation:
*

*Proof. *It is well known that the generalized Hermite form possesses the following integral representation [1]:
Following (2.11), we obtain (4.6). Also the form is quasi-antisymmetric because it satisfies
since is symmetric by hypothesis.

When , is classical and satisfies (3.4) with [22]
Then, , .

Now, it is sufficient to use Corollary 3.2 and Proposition 3.1 in order to obtain (4.7).

If , the form is semi-classical of class one and satisfies (3.4) with [23]
Therefore , , .

By Proposition 3.1 and Corollary 3.2 we can deduce (4.8).

##### 4.2. the Corecursive of the Second Kind Chebychev Form

Let us describe the case ; it is the corecursive of the second kind Chebychev functional. Here is [1] In this case we have the following result.

Lemma 4.2. *For , one has
*

*Proof. *The proof is analogous for the demonstration of Lemma 2.8.

Following Lemma 4.2, for (2.44) we have Therefore, we get for (2.42) Then we obtain where On account of Proposition 2.6, we can deduce that the form given by (2.9) is regular if and only if , , .

In the sequel, we suppose that the last condition is satisfied.

By virtue of (4.17) and Lemma 4.2, relations (2.49)β(2.52), and (2.55)β(2.57), (2.70)β(2.72) give Table 2.

Proposition 4.3. *If is the corecursive of the second kind Chebychev form, then the form given by (2.9) has the following integral representation:
**
It is a semi-classical form of class satisfying the following functional equation:
*

*Proof. *It is well known that possesses the following integral representation [1]:
From (2.11) we easily obtain (4.19).

The form satisfies (3.4) with [15]
Therefore, , , .

Now, we can simply use Proposition 3.1 and Corollary 3.2 in order to obtain (4.20).

Corollary 4.4. *When and , one has
*

*Proof. *From Table 2, we reach the desired results.

*Remarks 4.2. *(1) One has the form , where is studied in [24].

(2) Let [15, 19] be the first associated sequence of orthogonal with respect to and , the coefficients of the three-term recurrence relations; we have

The sequence is a second-order self-associated sequence; that is, is identical to its associated orthogonal sequence of second kind (see [25]).

#### Acknowledgments

Sincere thanks are due to the referee for his valuable comments and useful suggestions and his careful reading of the manuscript. The author is indebted to the proofreader the English teacher Hajer Rebai who checked the language of this work.

#### References

- T. S. Chihara,
*An Introduction to Orthogonal Polynomials*, Mathematics and Its Applications, Vol. 1, Gordon and Breach, New York, NY, USA, 1978. - E. B. Christoffel, βΓber die Gaussiche quadratur und eine Verallgemeinerung derselben,β
*Journal fΓΌr die reine und angewandte Mathematik*, vol. 55, pp. 61β82, 1858.