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Abstract and Applied Analysis
VolumeΒ 2010, Article IDΒ 263860, 14 pages
http://dx.doi.org/10.1155/2010/263860
Research Article

On the Complex Zeros of Some Families of Orthogonal Polynomials

Division of Applied Mathematics and Mechanics, Department of Engineering Sciences, University of Patras, 26500 Patras, Greece

Received 8 March 2010; Accepted 6 May 2010

Academic Editor: RomanΒ Ε imon Hilscher

Copyright Β© 2010 Eugenia N. Petropoulou. 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

The complex zeros of the orthogonal Laguerre polynomials for , ultraspherical polynomials for , Jacobi polynomials for , , , orthonormal Al-Salam-Carlitz II polynomials for , , and -Laguerre polynomials for , are studied. Several inequalities regarding the real and imaginary properties of these zeros are given, which help locating their position. Moreover, a few limit relations regarding the asymptotic behavior of these zeros are proved. The method used is a functional analytic one. The obtained results complement and improve previously known results.

1. Introduction

Orthogonal polynomials appear naturally in various problems of physics and mathematics and are considered as one of the basic tools in confronting problems of mathematical physics. Also, orthogonal polynomials have many important applications in problems of numerical analysis, such as interpolation or optimization. For a survey on applications and computational aspects of orthogonal polynomials, see [1] and the references therein.

Some of the most important properties of orthogonal polynomials, , are the following. (P1)The orthogonal polynomials are orthogonal with respect to a weight function on an interval of orthogonality and all their zeros are real and simple and lie inside . (P2)Some classes of orthogonal polynomials (including some of the classes studied in the present paper) satisfy an ordinary differential equation of the form where is a polynomial of degree at most two, is a polynomial of degree exactly one, and is a constant. (P3)The orthogonal polynomials satisfy a three-term recurrence relation of the form where .

An analog to the theory of classical orthogonal polynomials has recently been developed for -polynomials, , which also appear in various areas of mathematics and physics. The -polynomials satisfy also a recurrence relation of the form (1.2), but now the sequences , and as well as the polynomials depend on the parameter . On the other hand the -polynomials do not satisfy a differential equation, but a -difference equation which is considered as the -analog of (1.1). For more information on classical or -polynomials one may consult [2–6] and the references therein. Also, -polynomials arise in the context of indeterminate moment problems. In this case, there are some classes of orthogonal polynomials for which the corresponding measure of orthogonality is not unique. This may give rise to various types of -polynomials, other than the ones studied in the present paper. For more information see [4, 7–9] and the references therein.

Due to their importance, orthogonal polynomials have drawn the attention of many researchers and there is a renewed interest for them and their properties during the last 20–30 years. Many of the results regarding orthogonal polynomials, concern the properties of their zeros, such as their monotonicity, concavity, or convexity. This interest in the zeros of the orthogonal polynomials stimulates due to the physical interpretation of their zeros, such as their electrostatic interpretation and their appearance in various physical problems; see [4] and the references therein.

There are several ways to deal with problems involving the properties of the zeros of . Among them are methods (M1)of real analysis utilizing the formulae of and their properties, (M2)which utilize the differential equation (1.1) when , (M3)which utilize the recurrence relation (1.2) when , , and are real sequences, (M4)of functional analysis which transform the problem of the zeros of to the equivalent problem of the eigenvalues of a specific linear operator by using (1.2), regardless of the type (complex or real) of the sequences , , and .

In most cases, the orthogonal polynomials depend on at least one parameter which appears in the formulae of and/or in (1.1), or in the formulae of and/or and/or in (1.2), or in the formula of the weight function and which influence the behavior of the zeros of . In order to be positive, or or even more , , and to be real, the involved parameters should satisfy specific assumptions (usually simple inequalities). Otherwise, the first property (P1) of the may not hold and since are defined recursively by (1.2), if , , and are complex sequences, the polynomials will no longer be real polynomials. In this case, their zeros will no longer be exclusively real and there is a need to locate their position. Moreover, the usual methods (M1)–(M3) mentioned before for the study of the zeros of may not apply at all, when are complex, or they may need serious modifications. Instead, the (M4) method can be used directly.

Such a functional analytic method was introduced in [10] and was successfully used in a series of papers by the authors of [10] and their collaborators, including paper [11], where results were given regarding the real part of the complex zeros of a class of polynomials including the generalized Bessel polynomials. The most recent application of this method was in [12, 13], where convexity results and differential inequalities were deduced for the largest and lowest zeros and functions involving these zeros of several -polynomials. This method is also used in the present paper and it is briefly presented in Section 3. The main idea is to transform the problem of the zeros of satisfying (1.2) to the equivalent problem of the eigenvalues of a specific tridiagonal operator . Then, by utilizing the properties of , several properties of the zeros of can be proved.

The aim of the present paper is to provide regions (in ) of the location of the complex zeros of the following: (i)Laguerre orthogonal polynomials for , (ii)ultraspherical orthogonal polynomials for ,(iii)Jacobi orthogonal polynomials for , , ,(iv)orthonormal Al-Salam-Carlitz II polynomials for , , (v)-Laguerre orthonormal polynomials for , .

These regions are given in the form of inequalities regarding the real and imaginary properties of the zeros of the polynomials under consideration. Moreover, a few limit relations regarding the asymptotic behavior of these zeros are given. All these results are stated in Section 2 and proved in Section 4. The reason for choosing the above mentioned five classes of orthogonal polynomials, apart from pure mathematical curiosity, is the fact that their zeros and especially the zeros of the Jacobi and Laguerre polynomials admit a very interesting electrostatic interpretation (see, e.g., [14–16], [6, page 140] and the references therein).

To the best of the author's knowledge there are very few results concerning the location of the complex zeros of the classical or -polynomials or their limit relations. More precisely, in the thesis [17] and the paper [18], the behavior of the complex zeros of the Laguerre, -Laguerre, and Jacobi polynomials is primarily studied. Among others, an inequality regarding the real part of the zeros of the Laguerre polynomials and limit relations regarding the zeros of the Laguerre, -Laguerre and Jacobi polynomials are proved using their explicit formulae and their recurrence relations. Also in [19], the zeros of the hypergeometric polynomial , for are studied. These results are then applied in order to obtain information for the zeros of the Ultraspherical (for ) and Jacobi (for , and for , ) polynomials. Finally in [20], the zeros of the Ultraspherical polynomials are further investigated. More precisely, the authors give a description of the trajectories of the zeros as decreases from to . Several useful figures created using Mathematica illustrate these trajectories when . In the end, the authors conclude that β€œas descends below , all zeros of are on the imaginary axis tending symmetrically to the origin as ".

The results of the present paper (specifically Theorems 2.1 and 2.4) complement and improve the results of [17, 19, 20].

2. Main Results

In this section, several theorems are stated regarding the complex zeros of the orthogonal Laguerre, Ultraspherical and Jacobi, as well as the orthonormal Al-Salam-Carlitz II and -Laguerre polynomials. In each case, a region of the complex plane is given where these zeros lie, as well as a few limit relations regarding their asymptotic behavior. The proofs of these theorems are given in Section 4.

Theorem 2.1. The zeros of the Laguerre orthogonal polynomials for satisfy the following relations: Moreover,

Remark 2.2. It is obvious from (2.1) that if , then .

Remark 2.3. In [17, pages 112–131], using the explicit formula for the Laguerre polynomials and their recurrence relation, the inequality (2.1) was obtained, among other interesting relations. Moreover it was proved that Notice that relation (2.3) is stronger than (2.4).

Theorem 2.4. The zeros of the Ultraspherical orthogonal polynomials for are purely imaginary. Moreover

Remark 2.5. In [19], as a consequence of a more general result regarding the zeros of the hypergeometric function , it was proved that all zeros of the Ultraspherical polynomials are purely imaginary for , which is slightly stronger than the inequality . In [20], the zeros of the Ultraspherical polynomials are further investigated. More precisely, the authors give a description of the trajectories of the zeros as decreases from to . Several useful figures created using Mathematica illustrate these trajectories when . In the end the authors conclude that β€œas descends below , all zeros of are on the imaginary axis tending symmetrically to the origin as ", which is in accordance with the more general result (2.6).

Theorem 2.6. The zeros of the Jacobi orthogonal polynomials for , , satisfy the following relations: Moreover,

Remark 2.7. It is obvious from (2.7) that the sign of depends on the sign of , that is, if and if .

Remark 2.8. It is well known, see, for example, [4, page 99], that the Laguerre polynomials are a limiting case of the Jacobi polynomials, by first putting the Jacobi weight function on and then letting . Thus, it is obvious that Theorem 2.1 cannot be obtained from Theorem 2.6, since that would require taking the limit for , which cannot hold since .

Remark 2.9. It is well known, see, for example, [4, page 94] or [5, page 40], that the Ultraspherical polynomials are Jacobi polynomials for . By applying Theorem 2.6 for one obtains that β€œthe zeros of the Ultraspherical orthogonal polynomials for are purely imaginary and This result, however, is slightly worse than Theorem 2.4 and this is the reason that the Ultraspherical polynomials are treated in their own and not as a specific case of the Jacobi polynomials.

Theorem 2.10. The zeros of the orthonormal Al-Salam-Carlitz II polynomials for , satisfy the following relations: Moreover,

Remark 2.11. It is obvious from (2.12) that if , then , whereas if , then .

Theorem 2.12. The zeros of the -Laguerre orthonormal polynomials for , satisfy the following relations:

3. The Method

A sequence of orthogonal polynomials satisfies a three-term recurrence relation of the following form: However, after specific transformations, relation (3.1) can take the following form: The same holds for -polynomials. The only difference is that now the sequences , , , and as well as the polynomials and depend also on the parameter , .

The method used in this paper for the study of the zeros of the polynomials is a functional-analytic one, based on the equivalent transformation of the problem of the zeros of to the problem of the eigenvalues of a specific linear operator. More precisely, let be an orthonormal base in a finite dimensional Hilbert space with inner product denoted as usual by and let be the truncated shift operator: The adjoint of is the shift operator defined by Let also and be the diagonal operators:

It is known (see, e.g., [10]) that the zeros of the polynomials defined by (3.2) are the eigenvalues of the operator , that is, and vice versa. In the case where the polynomials depend on a parameter, the eigenvalues and the corresponding eigenvectors depend also on the same parameter.

For technical reasons one may choose, instead, the orthonormal base of . In this case the operators , , , and are defined as follows: and the zeros of the polynomials are the eigenvalues of the operator . In the rest of the paper it will be obvious from the text which base (and as a consequence which definition of the above mentioned operators) is used.

It worths mentioning at this point that. (i)If the sequences and are real, then the operator is a self-adjoint operator and thus its eigenvalues are all real. Moreover, it follows easily from (3.6) that From (3.8), several useful information and inequalities regarding can be deduced. (ii)If is purely imaginary and real, then and thus is obviously not a self-adjoint operator. However, by setting , takes the form , where both and are self-adjoint operators. Then it follows easily from (3.6) that Since and are both self-adjoint operators, the inner products and are both real. From (3.9), several useful information and inequalities regarding can be deduced, like those mentioned in Theorems 2.1–2.12. The proofs of these theorems rely on relation (3.9) and inequalities depending on the formulae of and .

4. Proofs

Proof of Theorem 2.1. The Laguerre polynomials satisfy the recurrence relation (3.1) for However, by setting first and then , where , , and , for , one obtains as in [21] a relation for of the form (3.2) with For , the sequence is purely imaginary and can be rewritten as . It is obvious from what already mentioned that the zeros of are the same with the zeros of and, as a consequence, are the same as the eigenvalues of the operator , where Due to (3.9), it is obvious that Since it follows that
Regarding the imaginary part of , it follows using the Schwarz inequality that
From (2.1) and (2.2) it follows that (since )

Proof of Theorem 2.4. The Ultraspherical polynomials satisfy the recurrence relation (3.1) for However, by setting first and then , where , , and , for , one obtains as in [22] a relation for of the form (3.2) with For , the sequence is purely imaginary and can be rewritten as . As before the zeros of are the same with the eigenvalues of the operator , where Due to (3.9), it is obvious that the zeros of are purely imaginary and that As before, it follows using the Schwarz inequality that
Relation (2.6) follows immediately from (2.5).

Proof of Theorem 2.6. The Jacobi polynomials satisfy the recurrence relation (3.1) for However, by setting , where , , and one obtains as in [23] a relation for of the form (3.1) but now with For , , the sequence is purely imaginary and can be rewritten as It is obvious from what already mentioned that the zeros of are the same with the eigenvalues of the operator , where Due to (3.9), it is obvious that Since relation (2.7) follows as in the proof of Theorem 2.1.
Regarding the imaginary part of , it follows using the Schwarz inequality that Relation (2.9) follows immediately from relations (2.7) and (2.8). Relation (2.10) follows as in the proof of Theorem 2.1.

Proof of Theorem 2.10. The Al-Salam Carlitz II orthonormal polynomials satisfy the recurrence relation (3.1) for where . For , the sequence is purely imaginary and can be rewritten as . It is obvious from what already mentioned that the zeros of are the same with the eigenvalues of the operator , where Due to (3.9), it is obvious that Since relation (2.12) follows as in the proof of Theorem 2.1.
Regarding the imaginary part of , it follows using the Schwarz inequality that Relation (2.14) follows as in the proof of Theorem 2.1.

Proof of Theorem 2.12. The -Laguerre orthonormal polynomials satisfy the recurrence relation (3.1) for where . For , the sequence is purely imaginary and can be rewritten as . It is obvious from what already mentioned that the zeros of are the same with the eigenvalues of the operator , where Due to (3.9), it is obvious that Since relation (2.15) follows as in the proof of Theorem 2.1.
Regarding the imaginary part of , it follows using the Schwarz inequality that

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