On the Complex Zeros of Some Families of Orthogonal Polynomials

Abstract

The complex zeros of the orthogonal Laguerre polynomials 𝐿_𝑛^(𝑎)(𝑥) for 𝑎<−𝑛, ultraspherical polynomials 𝑃_𝑛^(𝜆)(𝑥) for 𝜆<−𝑛, Jacobi polynomials 𝑃_𝑛^(𝑎,𝛽)(𝑥) for 𝑎<−𝑛, 𝛽<−𝑛, 𝑎+𝛽<−2(𝑛+1), orthonormal Al-Salam-Carlitz II polynomials 𝑃_𝑛^(𝑎)(𝑥;𝑞) for 𝑎<0, 0<𝑞<1, and 𝑞-Laguerre polynomials 𝐿_𝑛^(𝑎)(𝑥;𝑞) for 𝑎<−𝑛, 0<𝑞<1 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, 𝑃_𝑛(𝑥), are the following. (P1)The orthogonal polynomials 𝑃_𝑛(𝑥) are orthogonal with respect to a weight function 𝜚(𝑥)>0 on an interval of orthogonality [𝑎,𝑏]⊆ℝ and all their zeros are real and simple and lie inside (𝑎,𝑏). (P2)Some classes of orthogonal polynomials 𝑃_𝑛(𝑥) (including some of the classes studied in the present paper) satisfy an ordinary differential equation of the form 𝜎(𝑥)𝑃_𝑛^′′(𝑥)+𝜏(𝑥)𝑃^′_𝑛(𝑥)+𝜆_𝑛𝑃_𝑛(𝑥)=0, (1.1) where 𝜎(𝑥) is a polynomial of degree at most two, 𝜏(𝑥) is a polynomial of degree exactly one, and 𝜆_𝑛 is a constant. (P3)The orthogonal polynomials 𝑃_𝑛(𝑥) satisfy a three-term recurrence relation of the form𝛼_𝑛𝑃_𝑛+1(𝑥)+𝛽_𝑛𝑃_𝑛(𝑥)+𝑐_𝑛̃𝑃_𝑛−1(𝑥)=𝑥𝑃_𝑛𝑃(𝑥),𝑛=0,1,2,…,₋₁(𝑥)=0,𝑃₀(𝑥)=1, (1.2) where 𝛼_𝑛−1𝛾_𝑛>0.

An analog to the theory of classical orthogonal polynomials has recently been developed for 𝑞-polynomials, 0<𝑞<1, which also appear in various areas of mathematics and physics. The 𝑞-polynomials satisfy also a recurrence relation of the form (1.2), but now the sequences 𝑎_𝑛, 𝛽_𝑛 and 𝑐_𝑛̃ as well as the polynomials 𝑃_𝑛(𝑥) depend on the parameter 𝑞,0<𝑞<1. On the other hand the 𝑞-polynomials do not satisfy a differential equation, but a 𝑞-difference equation which is considered as the 𝑞-analog of (1.1). For more information on classical or 𝑞-polynomials one may consult [2–6] and the references therein. Also, 𝑞-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 𝑞-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 𝑃_𝑛(𝑥). Among them are methods (M1)of real analysis utilizing the formulae of 𝑃_𝑛(𝑥) and their properties, (M2)which utilize the differential equation (1.1) when 𝑥∈ℝ, (M3)which utilize the recurrence relation (1.2) when 𝛼_𝑛, 𝛽_𝑛, and 𝛾_𝑛 are real sequences, (M4)of functional analysis which transform the problem of the zeros of 𝑃_𝑛(𝑥) 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 𝛼_𝑛, 𝛽_𝑛, and 𝛾_𝑛.

In most cases, the orthogonal polynomials depend on at least one parameter which appears in the formulae of 𝜎(𝑥) and/or 𝜏(𝑥) in (1.1), or in the formulae of 𝛼_𝑛 and/or 𝛽_𝑛 and/or 𝛾_𝑛 in (1.2), or in the formula of the weight function 𝜚(𝑥) and which influence the behavior of the zeros of 𝑃_𝑛(𝑥). In order 𝜚(𝑥) to be positive, or 𝛼_𝑛−1𝛾_𝑛>0 or even more 𝛼_𝑛, 𝛽_𝑛, and 𝛾_𝑛 to be real, the involved parameters should satisfy specific assumptions (usually simple inequalities). Otherwise, the first property (P1) of the 𝑃_𝑛(𝑥) may not hold and since 𝑃_𝑛(𝑥) are defined recursively by (1.2), if 𝛼_𝑛, 𝛽_𝑛, and 𝛾_𝑛 are complex sequences, the polynomials 𝑃_𝑛(𝑥) 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 𝑃_𝑛(𝑥) may not apply at all, when 𝑃_𝑛(𝑥) 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 𝑞-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 𝑃_𝑛(𝑥) satisfying (1.2) to the equivalent problem of the eigenvalues of a specific tridiagonal operator 𝑇. Then, by utilizing the properties of 𝑇, several properties of the zeros of 𝑃_𝑛(𝑥) can be proved.

The aim of the present paper is to provide regions (in ℂ) of the location of the complex zeros of the following: (i)Laguerre orthogonal polynomials 𝐿_𝑛^(𝑎)(𝑥) for 𝑎<−𝑛, (ii)ultraspherical orthogonal polynomials 𝑃_𝑛^(𝜆)(𝑥) for 𝜆<−𝑛,(iii)Jacobi orthogonal polynomials 𝑃_𝑛^(𝑎,𝛽)(𝑥) for 𝑎<−𝑛, 𝛽<−𝑛, 𝑎+𝛽<−2(𝑛+1),(iv)orthonormal Al-Salam-Carlitz II polynomials 𝑃_𝑛^(𝑎)(𝑥;𝑞) for 𝑎<0, 0<𝑞<1, (v)𝑞-Laguerre orthonormal polynomials 𝐿_𝑛^(𝑎)(𝑥;𝑞) for 𝑎<−𝑛, 0<𝑞<1.

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 𝑞-polynomials or their limit relations. More precisely, in the thesis [17] and the paper [18], the behavior of the complex zeros of the Laguerre, 𝑞-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, 𝑞-Laguerre and Jacobi polynomials are proved using their explicit formulae and their recurrence relations. Also in [19], the zeros of the hypergeometric polynomial 𝐹(−𝑛,𝑏;2𝑏;𝑧), for 𝑏>−1/2 are studied. These results are then applied in order to obtain information for the zeros of the Ultraspherical (for 𝜆<−𝑛) and Jacobi (for 𝛽=−(1/2)(𝑎+1)−𝑛, 𝑎>−2 and for 𝑎=−2𝛽−2𝑛−1, 𝛽>−1) 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 𝜆 decreases from −1/2 to 1−𝑛. Several useful figures created using Mathematica illustrate these trajectories when 𝑛=8. In the end, the authors conclude that “as 𝜆 descends below −7, all 8 zeros of 𝑃^𝜆₈(𝑥) are on the imaginary axis tending symmetrically to the origin as 𝜆→−∞".

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 𝑞-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 𝑥_𝑛𝑘(𝑎)=Re[𝑥_𝑛𝑘(𝑎)]+𝑖Im[𝑥_𝑛𝑘(𝑎)] of the Laguerre orthogonal polynomials 𝐿_𝑛^(𝑎)(𝑥) for 𝑎<−𝑛 satisfy the following relations: 𝑎+1≤Re[𝑥_𝑛𝑘(𝑎)]≤2𝑛+𝑎−1,|Im[𝑥_𝑛𝑘(𝑎)]|≤2−𝑛(𝑎+1). (2.1)(2.2) Moreover, lim_{𝑎→−∞}𝑥_𝑛𝑘(𝑎)𝑎=1. (2.3)

Remark 2.2. It is obvious from (2.1) that if 𝑎<1−2𝑛, then Re[𝑥_𝑛𝑘(𝑎)]≤0.

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 lim_{𝑎→−∞}|𝑥_𝑛𝑘(𝑎)𝑎|=1. (2.4) Notice that relation (2.3) is stronger than (2.4).

Theorem 2.4. The zeros 𝑥_𝑛𝑘(𝜆) of the Ultraspherical orthogonal polynomials 𝑃_𝑛^(𝜆)(𝑥) for 𝜆<−𝑛 are purely imaginary. Moreover |𝑥_𝑛𝑘(𝜆)|≤−2𝜆𝑛,(𝑛+𝜆)(𝜆+𝑛−1)lim_{𝜆→−∞}𝑥_𝑛𝑘(𝜆)=0. (2.5)(2.6)

Remark 2.5. In [19], as a consequence of a more general result regarding the zeros of the hypergeometric function 𝐹(−𝑛,𝑏;2𝑏;𝑧), it was proved that all zeros of the Ultraspherical polynomials are purely imaginary for 𝜆<1−𝑛, which is slightly stronger than the inequality 𝜆<−𝑛. 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 𝜆 decreases from −1/2 to 1−𝑛. Several useful figures created using Mathematica illustrate these trajectories when 𝑛=8. In the end the authors conclude that “as 𝜆 descends below −7, all 8 zeros of 𝑃^𝜆₈(𝑥) are on the imaginary axis tending symmetrically to the origin as 𝜆→−∞", which is in accordance with the more general result (2.6).

Theorem 2.6. The zeros 𝑥_𝑛𝑘(𝑎,𝛽)=Re[𝑥_𝑛𝑘(𝑎,𝛽)]+𝑖Im[𝑥_𝑛𝑘(𝑎,𝛽)] of the Jacobi orthogonal polynomials 𝑃_𝑛^(𝑎,𝛽)(𝑥) for 𝑎<−𝑛, 𝛽<−𝑛, 𝑎+𝛽<−2(𝑛+1) satisfy the following relations: 𝛽²−𝑎²(2𝑛+𝑎+𝛽)(2𝑛+𝑎+𝛽−2)≤Re[𝑥_𝑛𝑘(𝑎,𝛽)]≤𝛽−𝑎𝑎+𝛽+2,for𝛽>𝑎,𝛽−𝑎𝑎+𝛽+2≤Re[𝑥_𝑛𝑘𝛽(𝑎,𝛽)]≤²−𝑎²(2𝑛+𝑎+𝛽)(2𝑛+𝑎+𝛽−2),for𝛽<𝑎,|Im[𝑥_𝑛𝑘(𝑎,𝛽)]|≤2−4(𝑎+1)(𝛽+1)(𝑎+𝛽+1)𝑛[(2𝑛+𝑎+𝛽)²−1](2𝑛+𝑎+𝛽)². (2.7)(2.8) Moreover, lim_{𝑎→−∞}𝑥_𝑛𝑘(𝑎,𝛽)=−1,lim_{𝛽→−∞}𝑥_𝑛𝑘(𝑎,𝛽)=1,lim_{𝑎→−∞}𝑥_𝑛𝑘(𝑎,𝛽)𝑎=0=lim_{𝛽→−∞}𝑥_𝑛𝑘(𝑎,𝛽)𝛽. (2.9)(2.10)

Remark 2.7. It is obvious from (2.7) that the sign of Re[𝑥_𝑛𝑘(𝑎,𝛽)] depends on the sign of 𝛽−𝑎, that is, Re[𝑥_𝑛𝑘(𝑎,𝛽)]>0 if 𝛽<𝑎 and Re[𝑥_𝑛𝑘(𝑎,𝛽)]<0 if 𝛽>𝑎.

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 [0,𝑎] and then letting 𝑎→∞. Thus, it is obvious that Theorem 2.1 cannot be obtained from Theorem 2.6, since that would require taking the limit for 𝑎→∞, which cannot hold since 𝑎<−𝑛<0.

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 𝑎=𝛽=𝜆−(1/2). By applying Theorem 2.6 for 𝑎=𝛽=𝜆−(1/2) one obtains that “the zeros 𝑥_𝑛𝑘(𝜆) of the Ultraspherical orthogonal polynomials 𝑃_𝑛^(𝜆)(𝑥) for 𝜆<−𝑛−(1/2) are purely imaginary and |𝑥_𝑛𝑘(𝜆)|≤2−2𝜆𝑛|(𝑛+𝜆)(𝜆+𝑛−1)𝜆+1/22𝜆+2𝑛−1|,lim_{𝜆→−∞}𝑥_𝑛𝑘(𝜆)=0.” (2.11) 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 𝑥_𝑛𝑘(𝑎;𝑞)=Re[𝑥_𝑛𝑘(𝑎;𝑞)]+𝑖Im[𝑥_𝑛𝑘(𝑎;𝑞)] of the orthonormal Al-Salam-Carlitz II polynomials 𝑃_𝑛^(𝑎)(𝑥;𝑞) for 𝑎<0, 0<𝑞<1 satisfy the following relations: 𝑎+1𝑞^𝑛−1≤Re[𝑥_𝑛𝑘(𝑎;𝑞)]≤𝑎+1,for𝑎≤−1,𝑎+1≤Re[𝑥_𝑛𝑘(𝑎;𝑞)]≤𝑎+1𝑞^𝑛−1,for−1<𝑎<0,|Im[𝑥_𝑛𝑘(𝑎;𝑞)]|≤2(−𝑎)(1−𝑞^𝑛)𝑞^2𝑛−1. (2.12)(2.13) Moreover, lim_{𝑎→−∞}𝑥_𝑛𝑘(𝑎;𝑞)𝑎²=0. (2.14)

Remark 2.11. It is obvious from (2.12) that if −1<𝑎<0, then Re[𝑥_𝑛𝑘(𝑎;𝑞)]≥0, whereas if 𝑎≤−1, then Re[𝑥_𝑛𝑘(𝑎;𝑞)]≤0.

Theorem 2.12. The zeros 𝑥_𝑛𝑘(𝑎;𝑞)=Re[𝑥_𝑛𝑘(𝑎;𝑞)]+𝑖Im[𝑥_𝑛𝑘(𝑎;𝑞)] of the 𝑞-Laguerre orthonormal polynomials 𝐿_𝑛^(𝑎)(𝑥;𝑞) for 𝑎<−𝑛, 0<𝑞<1 satisfy the following relations: 1−𝑞^𝑎+1𝑞^{2𝑛+𝑎−1}≤Re[𝑥_𝑛𝑘(𝑎;𝑞)]≤1+𝑞𝑞^{2𝑛+𝑎−1}−𝑞^𝑛+𝑎+𝑞^𝑛𝑞^𝑎+1,|Im[𝑥_𝑛𝑘(𝑎;𝑞)]|≤2(1−𝑞^𝑛)(𝑞^𝑎+1−1)𝑞^{4𝑛+2𝑎−1}. (2.15)(2.16)

3. The Method

A sequence {𝑃_𝑛(𝑥)}^∞_𝑛=0 of orthogonal polynomials satisfies a three-term recurrence relation of the following form:𝛼_𝑛𝑃_𝑛+1(𝑥)+𝛽_𝑛𝑃_𝑛(𝑥)+𝑐_𝑛̃𝑃_𝑛−1(𝑥)=𝑥𝑃_𝑛𝑃(𝑥),𝑛=0,1,2,…,₋₁(𝑥)=0,𝑃₀(𝑥)=1. (3.1) However, after specific transformations, relation (3.1) can take the following form:𝛼_𝑛𝑄_𝑛+1(𝑥)+𝛼_𝑛−1𝑄_𝑛−1(𝑥)+𝛽_𝑛𝑄_𝑛(𝑥)=𝑥𝑄_𝑛𝑄(𝑥),𝑛=1,2,…,₀(𝑥)=0,𝑄₁(𝑥)=1. (3.2) The same holds for 𝑞-polynomials. The only difference is that now the sequences 𝑎_𝑛, 𝛽_𝑛, 𝑐_𝑛̃, 𝛼_𝑛 and 𝛽_𝑛 as well as the polynomials 𝑃_𝑛(𝑥) and 𝑄_𝑛(𝑥) depend also on the parameter 𝑞, 0<𝑞<1.

The method used in this paper for the study of the zeros of the polynomials 𝑃_𝑛(𝑥) is a functional-analytic one, based on the equivalent transformation of the problem of the zeros of 𝑃_𝑛(𝑥) to the problem of the eigenvalues of a specific linear operator. More precisely, let {𝑒_𝑘}^𝑛_𝑘=1 be an orthonormal base in a finite dimensional Hilbert space 𝐻_𝑛 with inner product denoted as usual by (⋅,⋅) and let 𝑉 be the truncated shift operator: 𝑉𝑒_𝑘=𝑒_𝑘+1,𝑘=1,…,𝑛−1,𝑉𝑒_𝑛=0. (3.3) The adjoint of 𝑉 is the shift operator 𝑉^∗ defined by 𝑉^∗𝑒_𝑘=𝑒_𝑘−1,𝑘=2,…,𝑛,𝑉^∗𝑒₁=0. (3.4) Let also 𝐴 and 𝐵 be the diagonal operators: 𝐴𝑒_𝑘=𝛼_𝑘𝑒_𝑘,𝐵𝑒_𝑘=𝛽_𝑘𝑒_𝑘,𝑘=1,…,𝑛. (3.5)

It is known (see, e.g., [10]) that the zeros of the polynomials 𝑄_𝑛+1(𝑥) defined by (3.2) are the eigenvalues of the operator 𝑇=𝐴𝑉^∗+𝑉𝐴+𝐵, that is,𝑇𝑓_𝑘=𝑥_𝑛𝑘𝑓_𝑘,‖𝑓_𝑘‖=1 (3.6) and vice versa. In the case where the polynomials 𝑄_𝑛+1(𝑥) depend on a parameter, the eigenvalues 𝑥_𝑛𝑘 and the corresponding eigenvectors 𝑓_𝑘 depend also on the same parameter.

For technical reasons one may choose, instead, the orthonormal base {𝐸_𝑘}^𝑛−1_𝑘=0 of 𝐻_𝑛. In this case the operators 𝑉, 𝑉^∗, 𝐴, and 𝐵 are defined as follows: 𝑉𝐸_𝑘=𝐸_𝑘+1,𝑘=0,…,𝑛−2,𝑉𝐸_𝑛−1𝑉=0,_∗𝐸_𝑘=𝐸_𝑘−1,𝑘=1,…,𝑛−1,𝑉^∗𝐸₀=0,𝐴𝐸_𝑘=𝛼_𝑘𝐸_𝑘,𝐵𝐸_𝑘=𝛽_𝑘𝐸_𝑘,𝑘=0,…,𝑛−1, (3.7) and the zeros of the polynomials 𝑄_𝑛(𝑥) are the eigenvalues of the operator 𝑇. 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 𝛼_𝑛 and 𝛽_𝑛 are real, then the operator 𝑇=𝐴𝑉^∗+𝑉𝐴+𝐵 is a self-adjoint operator and thus its eigenvalues are all real. Moreover, it follows easily from (3.6) that 𝑥_𝑛𝑘=(𝑇𝑓_𝑘,𝑓_𝑘). (3.8) From (3.8), several useful information and inequalities regarding 𝑥_𝑛𝑘 can be deduced. (ii)If 𝛼_𝑛 is purely imaginary and 𝛽_𝑛 real, then 𝑇^∗=𝑉𝐴^∗+𝐴^∗𝑉^∗+𝐵 and thus 𝑇 is obviously not a self-adjoint operator. However, by setting 𝑆=(1/𝑖)(𝐴𝑉^∗+𝑉𝐴), 𝑇 takes the form 𝑇=𝑖𝑆+𝐵, where both 𝑆 and 𝐵 are self-adjoint operators. Then it follows easily from (3.6) that 𝑥_𝑛𝑘=(𝐵𝑓_𝑘,𝑓_𝑘)+𝑖(𝑆𝑓_𝑘,𝑓_𝑘). (3.9) Since 𝑆 and 𝐵 are both self-adjoint operators, the inner products (𝐵𝑓_𝑘,𝑓_𝑘) and (𝑆𝑓_𝑘,𝑓_𝑘) are both real. From (3.9), several useful information and inequalities regarding 𝑥_𝑛𝑘 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 𝛼_𝑛 and 𝛽_𝑛.

4. Proofs

Proof of Theorem 2.1. The Laguerre polynomials 𝐿_𝑛^(𝑎)(𝑥) satisfy the recurrence relation (3.1) for 𝛼_𝑛=−(𝑛+1),𝛽_𝑛=𝑎+2𝑛+1,𝑐_𝑛̃=−(𝑎+𝑛). (4.1) However, by setting first 𝐿^𝑎_𝑛(𝑥)=𝑅^(𝑎)_𝑛+1(𝑥) and then 𝑅_𝑛^(𝑎)(𝑥)=(−1)^𝑛+1𝑈_𝑛𝑄_𝑛^(𝑎)(𝑥), where 𝑈₀=0, 𝑈₁=1, and 𝑈_𝑛=((𝑛+𝑎−1)/(𝑛−1))𝑈_𝑛−1, for 𝑛≥2, one obtains as in [21] a relation for 𝑄_𝑛^(𝑎)(𝑥) of the form (3.2) with 𝛼_𝑛=𝑛(𝑛+𝑎),𝛽_𝑛=2𝑛+𝑎−1. (4.2) For 𝑎<−𝑛, the sequence 𝛼_𝑛 is purely imaginary and can be rewritten as 𝛼_𝑛=𝑖𝑛(−𝑛−𝑎). It is obvious from what already mentioned that the zeros 𝑥_𝑛𝑘(𝑎) of 𝐿_𝑛^(𝑎)(𝑥) are the same with the zeros of 𝑄^(𝑎)_𝑛+1(𝑥) and, as a consequence, are the same as the eigenvalues of the operator 𝑇=𝐴𝑉^∗+𝑉𝐴+𝐵, where 𝐴𝑒_𝑘=𝑖𝑘(−𝑘−𝑎)𝑒_𝑘,𝐵𝑒_𝑘=(2𝑘+𝑎−1)𝑒_𝑘,𝑘=1,…,𝑛. (4.3) Due to (3.9), it is obvious that Re[𝑥_𝑛𝑘(𝑎)]=(𝐵𝑓_𝑘,𝑓_𝑘),Im[𝑥_𝑛𝑘(𝑎)]=(𝑆𝑓_𝑘,𝑓_𝑘). (4.4) Since (𝐵𝑓_𝑘,𝑓_𝑘)=^𝑛_𝑙=1𝛽_𝑙|(𝑓_𝑘,𝑒_𝑙)|²,𝑎+1≤𝛽_𝑙≤2𝑛+𝑎−1, (4.5) it follows that (𝑎+1)‖𝑓_𝑘‖²≤(𝐵𝑓_𝑘,𝑓_𝑘)≤(2𝑛+𝑎−1)‖𝑓_𝑘‖²⇒𝑎+1≤Re[𝑥_𝑛𝑘(𝑎)]≤2𝑛+𝑎−1. (4.6)
Regarding the imaginary part of 𝑥_𝑛𝑘(𝑎), it follows using the Schwarz inequality that|Im[𝑥_𝑛𝑘(𝑎)]|≤‖𝑆‖≤2‖𝐴‖=2sup_𝑘−𝑘(𝑎+𝑘)≤2−𝑛(𝑎+1). (4.7)
From (2.1) and (2.2) it follows that (since 𝑎<0)1+2𝑛−1𝑎≤Re[𝑥_𝑛𝑘(𝑎)]𝑎1≤1+𝑎2−𝑛(𝑎+1)𝑎≤Im[𝑥_𝑛𝑘(𝑎)]𝑎≤−2−𝑛(𝑎+1)𝑎}⇒lim_{𝑎→−∞}𝑥_𝑛𝑘(𝑎)𝑎=1. (4.8)

Proof of Theorem 2.4. The Ultraspherical polynomials 𝑃_𝑛^(𝜆)(𝑥) satisfy the recurrence relation (3.1) for 𝛼_𝑛=𝑛+1,𝛽2(𝑛+𝜆)^𝑛=0,𝑐_𝑛̃=𝑛+2𝜆−1.2(𝑛+𝜆) (4.9) However, by setting first 𝑃^𝜆_𝑛(𝑥)=𝑅^(𝜆)_𝑛+1(𝑥) and then 𝑅_𝑛^(𝜆)(𝑥)=𝑛𝑈_𝑛/(𝑛+𝜆−1)𝑄_𝑛^(𝜆)(𝑥), where 𝑈₀=0, 𝑈₁=𝜆, and 𝑈_𝑛=((𝑛+2𝜆−2)/𝑛)𝑈_𝑛−1, for 𝑛≥2, one obtains as in [22] a relation for 𝑄_𝑛^(𝜆)(𝑥) of the form (3.2) with 𝛼_𝑛=12𝑛(𝑛+2𝜆−1)(𝑛+𝜆)(𝑛+𝜆−1),𝛽_𝑛=0. (4.10) For 𝜆<−𝑛, the sequence 𝛼_𝑛 is purely imaginary and can be rewritten as 𝛼_𝑛=𝑖(1/2)(−𝑛)(2𝜆+𝑛−1)/(𝑛+𝜆)(𝑛+𝜆−1). As before the zeros 𝑥_𝑛𝑘(𝜆) of 𝑃_𝑛^(𝜆)(𝑥) are the same with the eigenvalues of the operator 𝑇=𝐴𝑉^∗+𝑉𝐴+𝐵, where 𝐴𝑒_𝑘=𝑖(−𝑘)(2𝜆+𝑘−1)𝑒(𝑘+𝜆)(𝑘+𝜆−1)^𝑘,𝐵𝑒_𝑘=0,𝑘=1,…,𝑛. (4.11) Due to (3.9), it is obvious that the zeros 𝑥_𝑛𝑘(𝜆) of 𝑃_𝑛^(𝜆)(𝑥) are purely imaginary and that 𝑥_𝑛𝑘(𝜆)=𝑖(𝑆𝑓_𝑘,𝑓_𝑘). (4.12) As before, it follows using the Schwarz inequality that |𝑥_𝑛𝑘(𝜆)|≤‖𝑆‖≤2‖𝐴‖=2sup_𝑘12(−𝑘)(2𝜆+𝑘−1)≤(𝑘+𝜆)(𝑘+𝜆−1)−2𝜆𝑛.(𝑛+𝜆)(𝜆+𝑛−1) (4.13)
Relation (2.6) follows immediately from (2.5).

Proof of Theorem 2.6. The Jacobi polynomials 𝑃_𝑛^(𝑎,𝛽)(𝑥) satisfy the recurrence relation (3.1) for 𝛼_𝑛=2(𝑛+1)(𝑛+𝑎+𝛽+1),𝛽(2𝑛+𝑎+𝛽+1)(2𝑛+𝑎+𝛽+2)_𝑛=𝛽²−𝑎²,𝑐̃(2𝑛+𝑎+𝛽+2)(2𝑛+𝑎+𝛽)_𝑛=2(𝑛+𝑎)(𝑛+𝛽).(2𝑛+𝑎+𝛽+1)(2𝑛+𝑎+𝛽) (4.14) However, by setting 𝑃_𝑛^(𝑎,𝛽)(𝑥)=𝑈_𝑛𝑄_𝑛^(𝑎,𝛽)(𝑥), where 𝑈₋₁=0, 𝑈₀=1, and 𝑈_𝑛+1=(𝑛+𝑎+1)(𝑛+𝛽+1)(2𝑛+𝑎+𝛽+1)𝑈(𝑛+1)(𝑛+𝑎+𝛽+1)(2𝑛+𝑎+𝛽+3)^𝑛,for𝑛≥1, (4.15) one obtains as in [23] a relation for 𝑄_𝑛^(𝑎,𝛽)(𝑥) of the form (3.1) but now with 𝛼_𝑛=4(𝑛+1)(𝑛+𝑎+1)(𝑛+𝛽+1)(𝑛+𝑎+𝛽+1)(2𝑛+𝑎+𝛽+2)²(2𝑛+𝑎+𝛽+1)(2𝑛+𝑎+𝛽+3)=𝑎_𝑛,𝛽_𝑛=𝛽²−𝑎²(2𝑛+𝑎+𝛽+2)(2𝑛+𝑎+𝛽)=𝛽_𝑛,𝑐_𝑛̃=𝑎_𝑛−1. (4.16) For 𝑎<−𝑛, 𝛽<−𝑛, 𝑎+𝛽<−2(𝑛+1) the sequence 𝛼_𝑛 is purely imaginary and can be rewritten as 𝛼_𝑛=𝑖−4(𝑛+1)(𝑛+𝑎+1)(𝑛+𝛽+1)(𝑛+𝑎+𝛽+1)(2𝑛+𝑎+𝛽+2)²(2𝑛+𝑎+𝛽+1)(2𝑛+𝑎+𝛽+3). (4.17) It is obvious from what already mentioned that the zeros 𝑥_𝑛𝑘(𝑎,𝛽) of 𝑃_𝑛^(𝑎,𝛽)(𝑥) are the same with the eigenvalues of the operator 𝑇=𝐴𝑉^∗+𝑉𝐴+𝐵, where 𝐴𝐸_𝑘=𝑖−4(𝑘+1)(𝑘+𝑎+1)(𝑘+𝛽+1)(𝑘+𝑎+𝛽+1)(2𝑘+𝑎+𝛽+2)²𝐸(2𝑘+𝑎+𝛽+1)(2𝑘+𝑎+𝛽+3)^𝑘,𝑘=0,…,𝑛−1,𝐵𝐸_𝑘=(𝛽²−𝑎²)𝐸(2𝑘+𝑎+𝛽+2)(2𝑘+𝑎+𝛽)^𝑘,𝑘=0,…,𝑛−1. (4.18) Due to (3.9), it is obvious that Re[𝑥_𝑛𝑘(𝑎)]=(𝐵𝑓_𝑘,𝑓_𝑘),Im[𝑥_𝑛𝑘(𝑎)]=(𝑆𝑓_𝑘,𝑓_𝑘). (4.19) Since (𝐵𝑓_𝑘,𝑓_𝑘)=^𝑛−1_𝑙=0𝛽_𝑙|(𝑓_𝑘,𝐸_𝑙)|²,𝛽²−𝑎²(2𝑛+𝑎+𝛽)(2𝑛+𝑎+𝛽−2)≤𝛽_𝑙≤𝛽−𝑎𝑎+𝛽+2,for𝛽>𝑎,𝛽−𝑎𝑎+𝛽+2≤𝛽_𝑙≤𝛽²−𝑎²(2𝑛+𝑎+𝛽)(2𝑛+𝑎+𝛽−2),for𝛽<𝑎, (4.20) relation (2.7) follows as in the proof of Theorem 2.1.
Regarding the imaginary part of 𝑥_𝑛𝑘(𝑎,𝛽), it follows using the Schwarz inequality that|Im[𝑥_𝑛𝑘(𝑎,𝛽)]|≤‖𝑆‖≤2‖𝐴‖⇒|Im[𝑥_𝑛𝑘(𝑎,𝛽)]|≤2sup_𝑘−4(𝑘+1)(𝑘+𝑎+1)(𝑘+𝛽+1)(𝑘+𝑎+𝛽+1)(2𝑘+𝑎+𝛽+2)²(2𝑘+𝑎+𝛽+1)(2𝑘+𝑎+𝛽+3)⇒|Im[𝑥_𝑛𝑘(𝑎,𝛽)]|≤2−4(𝑎+1)(𝛽+1)(𝑎+𝛽+1)𝑛[(2𝑛+𝑎+𝛽)²−1](2𝑛+𝑎+𝛽)². (4.21) 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 𝑃_𝑛^(𝑎)(𝑥;𝑞) satisfy the recurrence relation (3.1) for 𝛼_𝑛=𝑎(1−𝑞^𝑛+1)𝑞^2𝑛+1,𝛽_𝑛=𝑎+1𝑞^𝑛,𝑐_𝑛̃=𝑎_𝑛−1, (4.22) where 0<𝑞<1. For 𝑎<0, the sequence 𝛼_𝑛 is purely imaginary and can be rewritten as 𝛼_𝑛=𝑖((−𝑎)(1−𝑞^𝑛+1))/𝑞^2𝑛+1. It is obvious from what already mentioned that the zeros 𝑥_𝑛𝑘(𝑎;𝑞) of 𝑃_𝑛(𝑥;𝑞) are the same with the eigenvalues of the operator 𝑇=𝐴𝑉^∗+𝑉𝐴+𝐵, where 𝐴𝐸_𝑘=𝑖(−𝑎)(1−𝑞^𝑘+1)𝑞^2𝑘+1𝐸_𝑘,𝐵𝐸_𝑘=𝑎+1𝑞^𝑘𝐸_𝑘,𝑘=0,…,𝑛−1. (4.23) Due to (3.9), it is obvious that Re[𝑥_𝑛𝑘(𝑎;𝑞)]=(𝐵𝑓_𝑘,𝑓_𝑘),Im[𝑥_𝑛𝑘(𝑎;𝑞)]=(𝑆𝑓_𝑘,𝑓_𝑘). (4.24) Since (𝐵𝑓_𝑘,𝑓_𝑘)=^𝑛−1_𝑙=0𝛽_𝑙|(𝑓_𝑘,𝑒_𝑙)|²,𝑎+1𝑞^𝑛−1≤𝛽_𝑙≤𝑎+1,for𝑎≤−1,𝑎+1≤𝛽_𝑙≤𝑎+1𝑞^𝑛−1,for−1<𝑎<0, (4.25) relation (2.12) follows as in the proof of Theorem 2.1.
Regarding the imaginary part of 𝑥_𝑛𝑘(𝑎;𝑞), it follows using the Schwarz inequality that|Im[𝑥_𝑛𝑘(𝑎;𝑞)]|≤‖𝑆‖≤2‖𝐴‖=2sup_𝑘(−𝑎)(1−𝑞^𝑘+1)𝑞^2𝑘+1≤2(−𝑎)(1−𝑞^𝑛)𝑞^2𝑛−1. (4.26) Relation (2.14) follows as in the proof of Theorem 2.1.

Proof of Theorem 2.12. The 𝑞-Laguerre orthonormal polynomials 𝐿_𝑛^(𝑎)(𝑥;𝑞) satisfy the recurrence relation (3.1) for 𝛼_𝑛=(1−𝑞^𝑛+1)(1−𝑞^𝑛+𝑎+1)𝑞^{4𝑛+2𝑎+3},𝛽_𝑛=1−𝑞^𝑛+1+𝑞−𝑞^𝑛+𝑎+1𝑞^2𝑛+𝑎+1,𝑐_𝑛̃=𝑎_𝑛−1, (4.27) where 0<𝑞<1. For 𝑎<−𝑛, the sequence 𝛼_𝑛 is purely imaginary and can be rewritten as 𝛼_𝑛=𝑖((1−𝑞^𝑛+1)(𝑞^𝑛+𝑎+1−1))/𝑞^{4𝑛+2𝑎+3}. It is obvious from what already mentioned that the zeros 𝑥_𝑛𝑘(𝑎;𝑞) of 𝐿_𝑛^(𝑎)(𝑥;𝑞) are the same with the eigenvalues of the operator 𝑇=𝐴𝑉^∗+𝑉𝐴+𝐵, where 𝐴𝐸_𝑘=𝑖(1−𝑞^𝑘+1)(𝑞^𝑘+𝑎+1−1)𝑞^{4𝑘+2𝑎+3}𝐸_𝑘,𝐵𝐸_𝑘=1−𝑞^𝑘+1+𝑞−𝑞^𝑘+𝑎+1𝑞^2𝑘+𝑎+1𝐸_𝑘,𝑘=0,…,𝑛−1. (4.28) Due to (3.9), it is obvious that Re[𝑥_𝑛𝑘(𝑎;𝑞)]=(𝐵𝑓_𝑘,𝑓_𝑘),Im[𝑥_𝑛𝑘(𝑎;𝑞)]=(𝑆𝑓_𝑘,𝑓_𝑘). (4.29) Since (𝐵𝑓_𝑘,𝑓_𝑘)=^𝑛−1_𝑙=0𝛽_𝑙|(𝑓_𝑘,𝑒_𝑙)|²,1−𝑞^𝑎+1𝑞^{2𝑛+𝑎−1}≤𝛽_𝑙≤1+𝑞𝑞^{2𝑛+𝑎−1}−𝑞^𝑛+𝑎+𝑞^𝑛𝑞^𝑎+1, (4.30) relation (2.15) follows as in the proof of Theorem 2.1.
Regarding the imaginary part of 𝑥_𝑛𝑘(𝑎;𝑞), it follows using the Schwarz inequality that|Im[𝑥_𝑛𝑘(𝑎;𝑞)]|≤‖𝑆‖≤2‖𝐴‖=2sup_𝑘(1−𝑞^𝑘+1)(𝑞^𝑘+𝑎+1−1)𝑞^{4𝑘+2𝑎+3}⇒|Im[𝑥_𝑛𝑘(𝑎;𝑞)]|≤2(1−𝑞^𝑛)(𝑞^𝑎+1−1)𝑞^{4𝑛+2𝑎−1}. (4.31)