# On the Complex Zeros of Some Families of Orthogonal Polynomials

Authors: Eugenia N. Petropoulou
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

The complex zeros of the orthogonal Laguerre polynomialsLn(a)(x) for a<-n, ultraspherical polynomials Pn(λ)(x) for λ<-n, Jacobi polynomials Pn(a,β)(x) for a<-n, β<-n, a+β<-2(n+1), orthonormal Al-Salam-Carlitz II polynomials Pn(a)(x;q) for a<0, 0<q<1, and q-Laguerre polynomials Ln(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.


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## Body

## 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,Pn(x), are the following. (P1)
The orthogonal polynomialsPn(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 polynomialsPn(x) (including some of the classes studied in the present paper) satisfy an ordinary differential equation of the form
(1.1)σ(x)Pn′′(x)+τ(x)Pn′(x)+λnPn(x)=0,
where σ(x) is a polynomial of degree at most two, τ(x) is a polynomial of degree exactly one, and λn is a constant.(P3)
The orthogonal polynomialsPn(x) satisfy a three-term recurrence relation of the form(1.2)α̃nPn+1(x)+β̃nPn(x)+c̃nPn-1(x)=xPn(x),n=0,1,2,…,P-1(x)=0,P0(x)=1,
where α̃n-1γ̃n>0.An analog to the theory of classical orthogonal polynomials has recently been developed forq-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 ãn, β̃n and c̃n as well as the polynomials Pn(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 ofPn(x). Among them are methods (M1)
of real analysis utilizing the formulae ofPn(x) and their properties,(M2)
which utilize the differential equation (1.1) when x∈ℝ,(M3)
which utilize the recurrence relation (1.2) when α̃n, β̃n, and γ̃n are real sequences,(M4)
of functional analysis which transform the problem of the zeros ofPn(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 α̃n, β̃n, and γ̃n.In most cases, the orthogonal polynomials depend on at least one parameter which appears in the formulae ofσ(x) and/or τ(x) in (1.1), or in the formulae of α̃n and/or β̃n and/or γ̃n in (1.2), or in the formula of the weight function ϱ(x) and which influence the behavior of the zeros of Pn(x). In order ϱ(x) to be positive, or α̃n-1γ̃n>0 or even more α̃n, β̃n, and γ̃n to be real, the involved parameters should satisfy specific assumptions (usually simple inequalities). Otherwise, the first property (P1) of the Pn(x) may not hold and since Pn(x) are defined recursively by (1.2), if α̃n, β̃n, and γ̃n are complex sequences, the polynomials Pn(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 Pn(x) may not apply at all, when Pn(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 Pn(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 Pn(x) can be proved.The aim of the present paper is to provide regions (inℂ) of the location of the complex zeros of the following: (i)
Laguerre orthogonal polynomialsLn(a)(x) for a<-n,(ii)
ultraspherical orthogonal polynomialsPn(λ)(x) for λ<-n,(iii)
Jacobi orthogonal polynomialsPn(a,β)(x) for a<-n, β<-n, a+β<-2(n+1),(iv)
orthonormal Al-Salam-Carlitz II polynomialsPn(a)(x;q) for a<0, 0<q<1,(v)
q-Laguerre orthonormal polynomials Ln(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 Section2 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 orq-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 P8λ(x) are on the imaginary axis tending symmetrically to the origin as λ→-∞".The results of the present paper (specifically Theorems2.1 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 andq-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 zerosxnk(a)=Re[xnk(a)]+iIm[xnk(a)] of the Laguerre orthogonal polynomials Ln(a)(x) for a<-n satisfy the following relations:
(2.1)a+1≤Re[xnk(a)]≤2n+a-1,(2.2)|Im[xnk(a)]|≤2-n(a+1).
Moreover,
(2.3)lima→-∞xnk(a)a=1.Remark 2.2.
It is obvious from (2.1) that if a<1-2n, then Re[xnk(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
(2.4)lima→-∞|xnk(a)a|=1.
Notice that relation (2.3) is stronger than (2.4).Theorem 2.4.
The zerosxnk(λ) of the Ultraspherical orthogonal polynomials Pn(λ)(x) for λ<-n are purely imaginary. Moreover
(2.5)|xnk(λ)|≤-2λn(n+λ)(λ+n-1),(2.6)limλ→-∞xnk(λ)=0.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 P8λ(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 zerosxnk(a,β)=Re[xnk(a,β)]+iIm[xnk(a,β)] of the Jacobi orthogonal polynomials Pn(a,β)(x) for a<-n, β<-n, a+β<-2(n+1) satisfy the following relations:
(2.7)β2-a2(2n+a+β)(2n+a+β-2)≤Re[xnk(a,β)]≤β-aa+β+2,forβ>a,β-aa+β+2≤Re[xnk(a,β)]≤β2-a2(2n+a+β)(2n+a+β-2),forβ<a,(2.8)|Im[xnk(a,β)]|≤2-4(a+1)(β+1)(a+β+1)n[(2n+a+β)2-1](2n+a+β)2.
Moreover,
(2.9)lima→-∞xnk(a,β)=-1,limβ→-∞xnk(a,β)=1,(2.10)lima→-∞xnk(a,β)a=0=limβ→-∞xnk(a,β)β.Remark 2.7.
It is obvious from (2.7) that the sign of Re[xnk(a,β)] depends on the sign of β-a, that is, Re[xnk(a,β)]>0 if β<a and Re[xnk(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 xnk(λ) of the Ultraspherical orthogonal polynomials Pn(λ)(x) for λ<-n-(1/2) are purely imaginary and
(2.11)|xnk(λ)|≤2-2λn(n+λ)(λ+n-1)|λ+1/22λ+2n-1|,limλ→-∞xnk(λ)=0.”
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 zerosxnk(a;q)=Re[xnk(a;q)]+iIm[xnk(a;q)] of the orthonormal Al-Salam-Carlitz II polynomials Pn(a)(x;q) for a<0, 0<q<1 satisfy the following relations:
(2.12)a+1qn-1≤Re[xnk(a;q)]≤a+1,fora≤-1,a+1≤Re[xnk(a;q)]≤a+1qn-1,for-1<a<0,(2.13)|Im[xnk(a;q)]|≤2(-a)(1-qn)q2n-1.
Moreover,
(2.14)lima→-∞xnk(a;q)a2=0.Remark 2.11.
It is obvious from (2.12) that if -1<a<0, then Re[xnk(a;q)]≥0, whereas if a≤-1, then Re[xnk(a;q)]≤0.Theorem 2.12.
The zerosxnk(a;q)=Re[xnk(a;q)]+iIm[xnk(a;q)] of the q-Laguerre orthonormal polynomials Ln(a)(x;q) for a<-n, 0<q<1 satisfy the following relations:
(2.15)1-qa+1q2n+a-1≤Re[xnk(a;q)]≤1+qq2n+a-1-qn+a+qnqa+1,(2.16)|Im[xnk(a;q)]|≤2(1-qn)(qa+1-1)q4n+2a-1.


## 3. The Method

A sequence{Pn(x)}n=0∞ of orthogonal polynomials satisfies a three-term recurrence relation of the following form:(3.1)α̃nPn+1(x)+β̃nPn(x)+c̃nPn-1(x)=xPn(x),n=0,1,2,…,P-1(x)=0,P0(x)=1.
However, after specific transformations, relation (3.1) can take the following form:(3.2)αnQn+1(x)+αn-1Qn-1(x)+βnQn(x)=xQn(x),n=1,2,…,Q0(x)=0,Q1(x)=1.
The same holds for q-polynomials. The only difference is that now the sequences ãn, β̃n, c̃n, αn and βn as well as the polynomials Pn(x) and Qn(x) depend also on the parameter q, 0<q<1.The method used in this paper for the study of the zeros of the polynomialsPn(x) is a functional-analytic one, based on the equivalent transformation of the problem of the zeros of Pn(x) to the problem of the eigenvalues of a specific linear operator. More precisely, let {ek}k=1n be an orthonormal base in a finite dimensional Hilbert space Hn with inner product denoted as usual by (·,·) and let V be the truncated shift operator: (3.3)Vek=ek+1,k=1,…,n-1,Ven=0.
The adjoint of V is the shift operator V* defined by (3.4)V*ek=ek-1,k=2,…,n,V*e1=0.
Let also A and B be the diagonal operators: (3.5)Aek=αkek,Bek=βkek,k=1,…,n.It is known (see, e.g., [10]) that the zeros of the polynomials Qn+1(x) defined by (3.2) are the eigenvalues of the operator T=AV*+VA+B, that is,(3.6)Tfk=xnkfk,∥fk∥=1
and vice versa. In the case where the polynomials Qn+1(x) depend on a parameter, the eigenvalues xnk and the corresponding eigenvectors fk depend also on the same parameter.For technical reasons one may choose, instead, the orthonormal base{Ek}k=0n-1 of Hn. In this case the operators V, V*, A, and B are defined as follows: (3.7)VEk=Ek+1,k=0,…,n-2,VEn-1=0,V*Ek=Ek-1,k=1,…,n-1,V*E0=0,AEk=αkEk,BEk=βkEk,k=0,…,n-1,
and the zeros of the polynomials Qn(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 sequencesαn and βn 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
(3.8)xnk=(Tfk,fk).
From (3.8), several useful information and inequalities regarding xnk can be deduced.(ii)
Ifαn is purely imaginary and βn 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
(3.9)xnk=(Bfk,fk)+i(Sfk,fk).
Since S and B are both self-adjoint operators, the inner products (Bfk,fk) and (Sfk,fk) are both real. From (3.9), several useful information and inequalities regarding xnk 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 αn and βn.


## 4. Proofs

Proof of Theorem2.1.
The Laguerre polynomialsLn(a)(x) satisfy the recurrence relation (3.1) for
(4.1)α̃n=-(n+1),β̃n=a+2n+1,c̃n=-(a+n).
However, by setting first Lna(x)=Rn+1(a)(x) and then Rn(a)(x)=(-1)n+1UnQn(a)(x), where U0=0, U1=1, and Un=((n+a-1)/(n-1))Un-1, for n≥2, one obtains as in [21] a relation for Qn(a)(x) of the form (3.2) with
(4.2)αn=n(n+a),βn=2n+a-1.
For a<-n, the sequence αn is purely imaginary and can be rewritten as αn=in(-n-a). It is obvious from what already mentioned that the zeros xnk(a) of Ln(a)(x) are the same with the zeros of Qn+1(a)(x) and, as a consequence, are the same as the eigenvalues of the operator T=AV*+VA+B, where
(4.3)Aek=ik(-k-a)ek,Bek=(2k+a-1)ek,k=1,…,n.
Due to (3.9), it is obvious that
(4.4)Re[xnk(a)]=(Bfk,fk),Im[xnk(a)]=(Sfk,fk).
Since
(4.5)(Bfk,fk)=∑l=1nβl|(fk,el)|2,a+1≤βl≤2n+a-1,
it follows that
(4.6)(a+1)∥fk∥2≤(Bfk,fk)≤(2n+a-1)∥fk∥2⇒a+1≤Re[xnk(a)]≤2n+a-1.
Regarding the imaginary part ofxnk(a), it follows using the Schwarz inequality that(4.7)|Im[xnk(a)]|≤∥S∥≤2∥A∥=2supk-k(a+k)≤2-n(a+1).
From (2.1) and (2.2) it follows that (since a<0)(4.8)1+2n-1a≤Re[xnk(a)]a≤1+1a2-n(a+1)a≤Im[xnk(a)]a≤-2-n(a+1)a}⇒lima→-∞xnk(a)a=1.Proof of Theorem2.4.
The Ultraspherical polynomialsPn(λ)(x) satisfy the recurrence relation (3.1) for
(4.9)α̃n=n+12(n+λ),β̃n=0,c̃n=n+2λ-12(n+λ).
However, by setting first Pnλ(x)=Rn+1(λ)(x) and then Rn(λ)(x)=nUn/(n+λ-1)Qn(λ)(x), where U0=0, U1=λ, and Un=((n+2λ-2)/n)Un-1, for n≥2, one obtains as in [22] a relation for Qn(λ)(x) of the form (3.2) with
(4.10)αn=12n(n+2λ-1)(n+λ)(n+λ-1),βn=0.
For λ<-n, the sequence αn is purely imaginary and can be rewritten as αn=i(1/2)(-n)(2λ+n-1)/(n+λ)(n+λ-1). As before the zeros xnk(λ) of Pn(λ)(x) are the same with the eigenvalues of the operator T=AV*+VA+B, where
(4.11)Aek=i(-k)(2λ+k-1)(k+λ)(k+λ-1)ek,Bek=0,k=1,…,n.
Due to (3.9), it is obvious that the zeros xnk(λ) of Pn(λ)(x) are purely imaginary and that
(4.12)xnk(λ)=i(Sfk,fk).
As before, it follows using the Schwarz inequality that
(4.13)|xnk(λ)|≤∥S∥≤2∥A∥=2supk12(-k)(2λ+k-1)(k+λ)(k+λ-1)≤-2λn(n+λ)(λ+n-1).
Relation (2.6) follows immediately from (2.5).Proof of Theorem2.6.
The Jacobi polynomialsPn(a,β)(x) satisfy the recurrence relation (3.1) for
(4.14)α̃n=2(n+1)(n+a+β+1)(2n+a+β+1)(2n+a+β+2),β̃n=β2-a2(2n+a+β+2)(2n+a+β),c̃n=2(n+a)(n+β)(2n+a+β+1)(2n+a+β).
However, by setting Pn(a,β)(x)=UnQn(a,β)(x), where U-1=0, U0=1, and
(4.15)Un+1=(n+a+1)(n+β+1)(2n+a+β+1)(n+1)(n+a+β+1)(2n+a+β+3)Un,forn≥1,
one obtains as in [23] a relation for Qn(a,β)(x) of the form (3.1) but now with
(4.16)α̃n=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)=an,β̃n=β2-a2(2n+a+β+2)(2n+a+β)=βn,c̃n=an-1.
For a<-n, β<-n, a+β<-2(n+1) the sequence αn is purely imaginary and can be rewritten as
(4.17)αn=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).
It is obvious from what already mentioned that the zeros xnk(a,β) of Pn(a,β)(x) are the same with the eigenvalues of the operator T=AV*+VA+B, where
(4.18)AEk=i-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)Ek,k=0,…,n-1,BEk=(β2-a2)(2k+a+β+2)(2k+a+β)Ek,k=0,…,n-1.
Due to (3.9), it is obvious that
(4.19)Re[xnk(a)]=(Bfk,fk),Im[xnk(a)]=(Sfk,fk).
Since
(4.20)(Bfk,fk)=∑l=0n-1βl|(fk,El)|2,β2-a2(2n+a+β)(2n+a+β-2)≤βl≤β-aa+β+2,forβ>a,β-aa+β+2≤βl≤β2-a2(2n+a+β)(2n+a+β-2),forβ<a,
relation (2.7) follows as in the proof of Theorem 2.1.
Regarding the imaginary part ofxnk(a,β), it follows using the Schwarz inequality that(4.21)|Im[xnk(a,β)]|≤∥S∥≤2∥A∥⇒|Im[xnk(a,β)]|≤2supk-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[xnk(a,β)]|≤2-4(a+1)(β+1)(a+β+1)n[(2n+a+β)2-1](2n+a+β)2.
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 Theorem2.10.
The Al-Salam Carlitz II orthonormal polynomialsPn(a)(x;q) satisfy the recurrence relation (3.1) for
(4.22)α̃n=a(1-qn+1)q2n+1,β̃n=a+1qn,c̃n=ãn-1,
where 0<q<1. For a<0, the sequence α̃n is purely imaginary and can be rewritten as α̃n=i((-a)(1-qn+1))/q2n+1. It is obvious from what already mentioned that the zeros xnk(a;q) of Pn(x;q) are the same with the eigenvalues of the operator T=AV*+VA+B, where
(4.23)AEk=i(-a)(1-qk+1)q2k+1Ek,BEk=a+1qkEk,k=0,…,n-1.
Due to (3.9), it is obvious that
(4.24)Re[xnk(a;q)]=(Bfk,fk),Im[xnk(a;q)]=(Sfk,fk).
Since
(4.25)(Bfk,fk)=∑l=0n-1βl|(fk,el)|2,a+1qn-1≤βl≤a+1,fora≤-1,a+1≤βl≤a+1qn-1,for-1<a<0,
relation (2.12) follows as in the proof of Theorem 2.1.
Regarding the imaginary part ofxnk(a;q), it follows using the Schwarz inequality that(4.26)|Im[xnk(a;q)]|≤∥S∥≤2∥A∥=2supk(-a)(1-qk+1)q2k+1≤2(-a)(1-qn)q2n-1.
Relation (2.14) follows as in the proof of Theorem 2.1.Proof of Theorem2.12.
Theq-Laguerre orthonormal polynomials Ln(a)(x;q) satisfy the recurrence relation (3.1) for
(4.27)α̃n=(1-qn+1)(1-qn+a+1)q4n+2a+3,β̃n=1-qn+1+q-qn+a+1q2n+a+1,c̃n=ãn-1,
where 0<q<1. For a<-n, the sequence α̃n is purely imaginary and can be rewritten as α̃n=i((1-qn+1)(qn+a+1-1))/q4n+2a+3. It is obvious from what already mentioned that the zeros xnk(a;q) of Ln(a)(x;q) are the same with the eigenvalues of the operator T=AV*+VA+B, where
(4.28)AEk=i(1-qk+1)(qk+a+1-1)q4k+2a+3Ek,BEk=1-qk+1+q-qk+a+1q2k+a+1Ek,k=0,…,n-1.
Due to (3.9), it is obvious that
(4.29)Re[xnk(a;q)]=(Bfk,fk),Im[xnk(a;q)]=(Sfk,fk).
Since
(4.30)(Bfk,fk)=∑l=0n-1βl|(fk,el)|2,1-qa+1q2n+a-1≤βl≤1+qq2n+a-1-qn+a+qnqa+1,
relation (2.15) follows as in the proof of Theorem 2.1.
Regarding the imaginary part ofxnk(a;q), it follows using the Schwarz inequality that(4.31)|Im[xnk(a;q)]|≤∥S∥≤2∥A∥=2supk(1-qk+1)(qk+a+1-1)q4k+2a+3⇒|Im[xnk(a;q)]|≤2(1-qn)(qa+1-1)q4n+2a-1.



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*Source: 263860-2010-07-11.xml*