A pair (𝒰,𝒱) of Hermitian regular linear functionals on the unit circle is said to be a (1,1)-coherent pair if their corresponding sequences of monic orthogonal polynomials {ϕn(x)}n≥0 and {ψn(x)}n≥0 satisfy ϕn[1](z)+anϕn-1[1](z)=ψn(z)+bnψn-1(z), an≠0, n≥1, where ϕn[1](z)=ϕn+1′(z)/(n+1). In this contribution, we consider the cases when 𝒰
is the linear functional associated with the Lebesgue and Bernstein-Szegő measures, respectively, and we obtain a classification of the situations where 𝒱 is associated with either a positive nontrivial measure or its rational spectral transformation.

1. Introduction

A pair (𝒰,𝒱) of regular linear functionals on the linear space of polynomials with real coefficients ℙ is a (1,1)-coherent pair if and only if their corresponding sequences of monic orthogonal polynomials (SMOP) {Pn(x)}n≥0 and {Qn(x)}n≥0 satisfy (1)Pn+1′(x)n+1+anPn′(x)n=Qn(x)+bnQn-1(x),an≠0,n≥1.
This concept is a generalization of the notion of coherent pair, for us (1,0)-coherent pair, introduced by Iserles et al. in [1], where bn=0, for every n≥1.

In the work by Delgado and Marcellán [2], the notion of a generalized coherent pair of measures, in short, (1,1)-coherent pair of measures, arose as a necessary and sufficient condition for the existence of an algebraic relation between the SMOP {Sn(x;λ)}n≥0 associated with the Sobolev inner product
(2)〈p(x),r(x)〉λ=∫ℝp(x)r(x)dμ0+λ∫ℝp′(x)r′(x)dμ1,λ>0,p,r∈ℙ,
and the SMOP {Pn(x)}n≥0 associated with the positive Borel measure μ0 in the real line as follows:
(3)Sn+1(x;λ)+cn(λ)Sn(x;λ)=Pn+1(x)+n+1nanPn(x),n≥1,
where {cn(λ)}n≥1 are rational functions in λ>0. Besides, they obtained the classification of all (1,1)-coherent pairs of regular functionals (𝒰,𝒱) and proved that at least one of them must be semiclassical of class at most 1, and 𝒰 and 𝒱 are related by a rational type expression. This is a generalization of the results of Meijer [3] for the (1,0)-coherence case (when bn=0, n≥1), where either 𝒰 or 𝒱 must be a classical linear functional.

The most general case of the notion of coherent pair was studied by de Jesus et al. in [4] (see also [5]), the so-called (M,N)-coherent pairs of order (m,k), where the derivatives of order m and k of two SMOP {Pn(x)}n≥0 and {Qn(x)}n≥0 with respect to the regular linear functionals 𝒰 and 𝒱 are related by
(4)∑i=0Man-i,n,mPn+m-i(m)(x)=∑i=0Nbn-i,n,kQn+k-i(k)(x),n≥0,
where M, N, m, k∈ℤ+∪{0} and the real numbers an-i,n,m,bn-i,n,k satisfy some natural conditions. They showed that the regular linear functionals 𝒰 and 𝒱 are related by a rational factor, and, when m≠k, those linear functionals are semiclassical. Besides, they proved that if (μ0,μ1) is a (M,N)-coherent pair of order (m,0) of positive Borel measures on the real line, then
(5)∑j=0max{M,N}cn-j,n,m(λ)Sn-j+m,m(x;λ)=∑j=0Man-j,n,mPn-j+m(x),n≥0,
holds, where cn-j,n,m(λ), 0<j≤max{M,N}, n≥0, are rational functions in λ such that cn-j,n,m(λ)=0 for n<j≤max{M,N}, and {Sn,m(x;λ)}n≥0 is the Sobolev SMOP with respect to the inner product
(6)〈p(x),r(x)〉λ,m=∫ℝp(x)r(x)dμ0+λ∫ℝp(m)(x)r(m)(x)dμ1,λ>0,m∈ℤ+,p, r∈ℙ. Also, they showed that (M,max{M,N})-coherence of order (m,0) is a necessary condition for the algebraic relation (5). For a historical summary about coherent pairs on the real line, see, for example, the introductory sections in the recent papers of de Jesus et al. [6] and of Marcellán and Pinzón-Cortés [7].

On the other hand, the notion of coherent pair was extended to the theory of orthogonal polynomials in a discrete variable by Area et al. in [8–10]. They used the difference operator Dω as well as the q-derivative operator Dq defined by
(7)(Dωp)(x)=p(x+ω)-p(x)ω,ω∈ℂ∖{0},(Dqp)(x)=p(qx)-p(x)(q-1)xforx≠0,(Dqp)(0)=p′(0),q∈ℂ∖{0,1},
instead of the usual derivative operator D. In this way, they obtained similar results to those by Meijer and similar classification as a limit case when either ω→0 or q→1, respectively. Likewise, Marcellán and Pinzón-Cortés in [11, 12] studied the analogue of the generalized coherent pairs introduced by Delgado and Marcellán, that is, (1,1)-Dω-coherent pairs and (1,1)-Dq-coherent pairs. Finally, Álvarez-Nodarse et al. [13] analyzed the more general case, (M,N)-Dω-coherent pairs of order (m,k) and (M,N)-Dq-coherent pairs of order (m,k), proving the analogue results to those in [4].

Furthermore, Branquinho et al. in [14] extended the concept of coherent pair to Hermitian linear functionals associated with nontrivial probability measures supported on the unit circle. They studied (3) in the framework of orthogonal polynomials on the unit circle (OPUC). Also, they concluded that if (𝒰,𝒱) is a (1,0)-coherent pair of Hermitian regular linear functionals, then {Pn(z)}n≥0 is semiclassical and {Qn(z)}n≥0 is quasiorthogonal of order at most 6 with respect to the functional [zA(z)+(1/z)A¯(1/z)]𝒰, A∈ℙ. Besides, they analyzed the cases when either 𝒰 or 𝒱 is the Lebesgue measure or 𝒰 is the Bernstein-Szegő measure.

Later on, Branquinho and Rebocho in [15] obtained that if the sequences {Pn(z)}n≥0 and {Qn(z)}n≥0 satisfy, for n≥0,
(8)∑j=0M1αn,jPn+1+M1-j′(z)n+1+M1-j+∑j=0M2ηn,j(Pn+M2-j*(z))′=∑j=0N1βn,jQn+N1-j(z)+∑j=0N2γn,jQn+N2-j*(z),
with N1=M1, max{M2,N2}<N1, and some extra conditions, then {Pn(z)}n≥0 and {Qn(z)}n≥0 are semiclassical sequences of OPUC. Moreover, when Pn(z)=Qn(z) for all n and under some extra conditions, (8) is a necessary condition for the semiclassical character of {Pn(z)}n≥0. Finally, they analyzed the (0,1)-coherence case (Pn+1′(z))/(n+1)=Qn(z)+bnQn-1(z), bn≠0, n≥1, when 𝒰 is the linear functional associated with either the Lebesgue measure or the Bernstein-Szegő measure.

The aim of our contribution is to describe the (1,1)-coherence pair (𝒰, 𝒱) when 𝒰 and 𝒱 are regular linear functionals, focusing our attention on the cases when 𝒰 is either the Lebesgue or the Bernstein-Szegő linear functional. The structure of this work is as follows. In Section 2, we state some definitions and basic results which will be useful in the forthcoming sections. In Section 3, we introduce the concept of (1,1)-coherent pair of Hermitian regular linear functionals, and we obtain some results that will be applied in the sequel. In Section 4, we analyze (1,1)-coherent pairs when 𝒰 is the linear functional associated with the Lebesgue measure on the unit circle. We determine the cases when the linear functional 𝒱 is associated with a positive measure on the unit circle, or a rational spectral transformation of it. Finally, in Section 5, we deal with a similar analysis for the case when 𝒰 is the linear functional associated with the Bernstein-Szegő measure.

2. Preliminaries

Let us consider the unit circle 𝕋={z∈ℂ:|z|=1}, the linear space of Laurent polynomials with complex coefficients Λ=span{zn:n∈ℤ}, and a linear functional 𝒰:Λ→ℂ. We can associate with 𝒰 a sequence of moments {cn}n∈ℤ defined by cn=〈𝒰,zn〉, n∈ℤ, and a bilinear form as follows:
(9)〈p(z),q(z)〉=〈𝒰,p(z)q¯(1z)〉,
where p, q∈ℙ, the linear space of polynomials with complex coefficients. Its Gram matrix with respect to {zn}n≥0 is an infinite Toeplitz matrix (cj-k)j,k≥0 with leading principal minors given by Δn=det((cj-k)j,k=0n), n∈ℤ+∪{0}.

The linear functional 𝒰 is said to be Hermitian if c-n=c¯n, quasidefinite or regular if Δn≠0 for all n∈ℤ+∪{0}, and positive definite if Δn>0 for all n∈ℤ+∪{0}. We will denote by ℋ the set of Hermitian linear functionals defined on Λ.

𝒰∈ℋ is regular if and only if there exists a (unique) sequence of monic orthogonal polynomials on the unit circle (OPUC) {ϕn(z)}n≥0; this is, it satisfies that deg(ϕn(z))=n and 〈ϕm(z),ϕn(z)〉=κnδm,n, with κn≠0, for n, m∈ℤ+∪{0}. Every monic OPUC ϕn(z) has an explicit representation, the so-called Heine’s formula, as follows:(10)ϕn(z)=1Δn-1|c0c1⋯cn⋮⋮⋮⋮c-(n-1)c-(n-2)⋯c11z⋯zn|,n≥1,ϕ0(z)=1.
Besides, they satisfy the forward and backward Szegő recurrence relations(11)ϕn(z)=zϕn-1(z)+αnϕn-1*(z),ϕn(z)=(1-αn|2)zϕn-1(z)+αnϕn*(z),n≥1,ϕ0(z)=1,
where αn=ϕn(0), n≥1, are said to be the Verblunsky (reflection, Schur, Szegő, or Geronimus) coefficients and ϕn*(z)=znϕ¯n(1/z), n∈ℤ+∪{0}, is called the reversed polynomial of ϕn(z). Conversely, if {ϕn(z)}n≥0 is a sequence of monic polynomials which satisfies (11) and |αn|≠1 for n≥1, then {ϕn(z)}n≥0 is the sequence of monic OPUC with respect to some Hermitian regular linear functional.

If 𝒰 is a Hermitian regular (resp., positive definite) linear functional, then (see [16–18]) |αn|≠1 (resp., |αn|<1), for n≥1.

A positive definite Hermitian linear functional 𝒰 has an integral representation (see [19])
(12)〈p(z),q(z)〉=〈𝒰,p(z)q¯(1z)〉=12π∫02πp(z)q¯(1z)dμ(θ),z=eiθ,p,q∈ℙ,
where μ is a nontrivial probability measure supported on an infinite subset of 𝕋. A measure μ belongs to the Nevai class (see [20, 21]) if limn→∞|ϕn(0)|=0.

On the other hand (see [19]), an analytic function F(z), defined on 𝔻={z∈ℂ:|z|<1}, is said to be a Carathéodory function if and only if F(0)=1 and ReF(z)>0 on 𝔻. If μ is a probability measure on 𝕋, then
(13)F(z)=12π∫02πeiθ+zeiθ-zdμ(θ)
is a Carathéodory function. Conversely, the Herglotz representation theorem claims that every Carathéodory function F(z) has a representation given by (13) for a unique probability measure μ on 𝕋.

Besides (see [22]), a Carathéodory function (13) admits the expansions
(14)F(z)=c0+2∑n=1∞c-nzn,|z|<1,F(z)=-c0-2∑n=1∞cnz-n,|z|>1,
where {cn}n≥0 are the moments of the measure associated with F(z).

To complete this section, we state the following definitions. Let {ϕn(z)}n≥0 be a sequence of monic OPUC with corresponding Verblunsky coefficients {αn}n≥1, and let N∈ℤ+∪{0}. The polynomials defined by
(15)ϕn(N)(z)=zϕn-1(z)+αn+Nϕn-1*(z),n≥1,ϕ0(N)(z)=1,
are called the associated polynomials of {ϕn(z)}n≥0 of order N. Similarly, given a finite set of complex numbers {γn}n=1N, with |γn|≠1, n=1,2,…,N, let us define the new Verblunsky coefficients {α~n}n≥1={γ1,…,γN,α1,α2,…}. Then the monic OPUC defined by the forward Szegő relation associated with {α~n}n≥1 are said to be the antiassociated polynomials of {ϕn(z)}n≥0 of order N.

3. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M199"><mml:mo mathvariant="bold">(</mml:mo><mml:mn>1,1</mml:mn><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>-Coherent Pairs on the Unit Circle

A pair of Hermitian regular linear functionals (𝒰,𝒱) defined on the linear space of Laurent polynomials is said to be a (1,1)-coherent pair if their corresponding sequences of monic OPUC, {ϕn(z)}n≥0 and {ψn(z)}n≥0, are related by
(16)ϕn[1](z)+anϕn-1[1](z)=ψn(z)+bnψn-1(z),an≠0,n≥1,
where ϕn[1](z)=(ϕn+1′(z))/(n+1), for n∈ℕ. In such a case, the pair {ϕn(z)}n≥0 and {ψn(z)}n≥0 is also said to be a (1,1)-coherent pair. If bn=0 for every n≥1, then (𝒰,𝒱) is called a (1,0)-coherent pair.

Lemma 1.

If (𝒰,𝒱) satisfies (16), then, one has the following.

a1≠b1 if and only if ϕn[1](z)≠ψn(z), for every n≥1.

For n≥1, one has
(17)ϕn[1](z)=ψn(z)+(bn-an)ψn-1(z)+∑k=0n-2(-1)n-(k+1)anan-1⋯ak+2(bk+1-ak+1)ψk(z),(18)ψn(z)=ϕn[1](z)+(an-bn)ϕn-1[1](z)+∑k=0n-2(-1)n-(k+1)bnbn-1⋯bk+2(ak+1-bk+1)ϕk[1](z).

Proof.

From (16) it is easy to check that a1=b1 if and only if there exists N∈ℕ, N≥1, such that ϕN[1](z)=ψN(z). Also, from (16) and using induction on n, it is immediate to prove (17) and (18).

Corollary 2.

If (𝒰,𝒱) is a (1,1)-coherent pair given by (16), then
(19)〈𝒱,ϕn[1](z)〉=(-1)n(a1-b1)∏j=2naj〈𝒱,1〉,n≥1,
where ∏j=k1k2aj=1 whenever k2<k1.

We will study the (1,1)-coherence relations when 𝒰 is the linear functional associated with basic positive measures on the unit circle, namely, the Lebesgue and Bernstein-Szegő measures.

The Lebesgue linear functional is the linear functional associated with the Lebesgue measure dμ(θ)=dθ/2π, and its corresponding sequence of monic OPUC is ϕn(z)=zn, for n∈ℤ+∪{0}. Besides, the reversed polynomials are ϕn*(z)=1, n∈ℤ+∪{0}, and its Verblunsky coefficients are αn=ϕn(0)=0, for n≥1. Furthermore, its moments are cn=δn,0, for n∈ℤ+∪{0}, and its Carathéodory function is F(z)=1.

The Bernstein-Szegő linear functional is associated with the measure dμ(θ)=((1-|C|2)/|1+Ceiθ|2)(dθ/2π), with C∈ℂ and |C|<1. Its corresponding monic OPUC are ϕn(z)=zn-1(z+C) for n≥1 and ϕ0(z)=1. Its reversed polynomials are ϕn*(z)=1+C¯z, for n≥1, and its Verblunsky coefficients are αn=ϕn(0)=0, for n≥2 and α1=C. Besides, its moments are cn=(-C)n for n∈ℤ+∪{0}, and its Carathéodory function is F(z)=(1-zC)/(1+zC).

We begin by analyzing the first one.

4. The Lebesgue Linear FunctionalTheorem 3.

Let (𝒰,𝒱) be a (1,1)-coherent pair on the unit circle such that their corresponding monic OPUC satisfy (16), and let 𝒰 be the Lebesgue linear functional.

If a1=b1, then 𝒱 is also the linear functional associated with the Lebesgue measure, and an=bn for n≥1.

If a1≠b1 and |ψn(0)|=|βn|≠1, n≥1, then
(20)vn=(-a2)n-1(b1-a1)v0,n≥1,(21)a2=b2(1-|β1|2)+β1,an=a2,bn=bn-11-|βn-1|2=b2∏k=2n-1(1-|βk|2),n≥3,(22)β1=a1-b1,βn=(-1)n-1bn⋯b2β1=-bnβn-1,n≥2,(23)ψ1(z)=z+a1-b1,andforn≥2,ψn(z)=zn+(a2-bn)zn-1+∑k=1n-2(-1)n-k-1bn⋯bk+2(a2-bk+1)zk+βn,
where {vn}n≥0 is the sequence of moments associated with 𝒱.

Proof.

Since ϕn[1](z)=zn for n∈ℤ+∪{0}, then (16) becomes
(24)zn+anzn-1=ψn(z)+bnψn-1(z),an≠0,n≥1.
Thus, applying the linear functional 𝒱 on the previous expression, we get
(25)vn=-anvn-1=(-1)n-1an⋯a2(b1-a1)v0,n≥2,v1=(b1-a1)v0.

(i) If a1=b1, then from (25) we have vn=0 for n≥1. Thus, ψn(z)=zn for n≥1, and, as a consequence, from (24) we obtain an=bn for every n≥1.

(ii) From (18), we have
(26)ψn(z)=zn+(an-bn)zn-1+∑k=0n-2(-1)n-(k+1)bnbn-1⋯bk+2(ak+1-bk+1)zk.
Multiplying (26) by z-1 and applying 𝒱, we obtain
(27)0=vn-1+(an-bn)vn-2+∑k=0n-2(-1)n-(k+1)bnbn-1⋯bk+2(ak+1-bk+1)vk-1.
Thus, multiplying this equation by bn+1 and adding it to the previous equation for n+1, we get
(28)0=vn+an+1vn-1=(25)-anvn-1+an+1vn-1=(an+1-an)vn-1,n≥2.
Since an≠0, n≥1, and a1≠b1, (25) yields vn≠0 for n≥1. Thus, from (28), we conclude that an+1=an for n≥2 or, equivalently, an+1=a2 for n≥2. Therefore, (25) becomes (20).

On the other hand, from (26) we obtain (22) and (23). Besides, from the forward Szegő relation and (26), we can obtain another expression for ψn+1(z), n≥0. By comparing the coefficients of zn, we get an+1-bn+1=an-bn-bn+1|βn|2, for n≥1. Hence, since an+1=an and |βn-1|≠1, for n≥2, (21) follows.

We are interested in the cases where 𝒱 is also a positive definite linear functional. Notice that, aside from the trivial case when a1=b1, all of the coherence coefficients are determined from the values of a1, b1, and b2 (or, equivalently, a1, b1, and a2). Not every choice of these parameters will yield a positive definite linear functional 𝒱. For instance, if |b2|=1 and |a1-b1|=|β1|=2, then we can see from (22) that |bn|=1, n⩾3, and |βn|=2, n⩾2. However, it is possible to choose the values of a1, b1, and b2 in order to get a positive definite linear functional 𝒱, or at least its rational spectral transformation. We have the following cases.

Proposition 4.

Let (𝒰,𝒱) be a (1,1)-coherent pair on the unit circle such that their corresponding monic OPUC satisfy (16), and let 𝒰 be the linear functional associated with the Lebesgue measure. Assume that 𝒱 is normalized (i.e., v0=1). Then, one has the following.

Let |b1-a1|<1. If a2=a1-b1 (i.e., b2=0), then bn=0 and an=a1-b1 for every n≥2. Besides, 𝒱 is the linear functional associated with the Bernstein-Szegő measure with parameter b1-a1. Furthermore, if bN=0 for some N≥2,
then b2=0.

If a1, b1, a2∈ℝ and either 0<a1-b1<a2<1 or -1<a2<a1-b1<0 holds, then the Carathéodory function associated with 𝒱 is (29)F𝒱=-b1-a1a2FB(z)+b1-a1+a2a2,where FB(z) is the Carathéodory function associated with the Bernstein-Szegő measure with parameter -a2. As a consequence, the orthogonality measure associated with 𝒱 is (30)dμ2=-b1-a1a21-|a2|2|1+a2eiθ|2dθ2π+b1-a1+a2a2dθ2π.

For any values of a1, b1, the value of b2 can be chosen in such a way that 𝒱 is the linear functional associated with a rational spectral transformation of a Nevai class measure.

Proof.

(i) Notice that a1≠b1 because a2≠0. We first prove that if bN=0 for some N≥2, then bn=0 for n≥2. Assume that for some N≥2, bN=0. From (21), (22), and (23) it follows that bn=0=βn and ψn(z)=zn-1(z+a2) for n≥N. Besides, another expression for ψN(z) is ψN(z)=zψN-1(z)+βNψN-1*(z)=zψN-1(z), where ψN-1(z) is given by (23). Thus, the comparison of the coefficients of zN-1 in both expressions of ψN(z) yields a2=a2-bN-1, and thus, bN-1=0. Following the same argument for bN-1,…,b2, we conclude that bn=0 for n=2,…,N-1 and a2=a1-b1. Therefore, bn=0=βn for n≥2, β1=a1-b1=a2, and ψn(z)=zn-1(z+a1-b1) for n≥1. As a consequence, from (21) and (20), it follows that an+1=a1-b1 and vn=(b1-a1)n, n≥0. Finally, since |β1|=|b1-a1|<1, then 𝒱 is the linear functional associated with the Bernstein-Szegő measure.

(ii) From (20), the Carathéodory function associated with 𝒱 is F𝒱=1+2∑k≥1(b1-a1)(-a2)k-1zk. Since |a2|<1, then (see [19]) the Bernstein-Szegő polynomials of parameter -a2 have moments cn=(-a2)n and are orthogonal with respect to the measure ((1-|a2|2)/|1+a2eiθ|2)(dθ/2π), and their associated Carathéodory function is FB(z)=1-2a2∑k≥1(-a2)k-1zk. Therefore, (29) holds. In other words (see [23]), F𝒱 can be obtained by applying a rescaling to the moments of FB(z), followed by a perturbation of its first moment (i.e., a diagonal perturbation of the corresponding Toeplitz matrix). Thus, the orthogonality measure associated with 𝒱 is given by (30).

(iii) From (21), given β1=a1-b1, we have b3=b2/(1-|β2|2)=b2/(1-|b2β1|2), so we can choose |b2| small enough so that β2 is sufficiently close to 0. Thus, b3 will also be close to 0, and since
(31)βn=-bnβn-1,n≥2,bn=bn-11-|βn-1|2,n≥3,{|bn|}n⩾2 will be an increasing sequence and, as a consequence, {|βn|}n⩾2 will be a decreasing sequence. Besides, b2 can be chosen so that |bn| converges to a constant b, 0<b<1, and therefore the product ∏k=2n-1|1-|βk|2| will also converge to |b2|/b. This shows that βn→0, and thus {βn}n⩾2 defines a Nevai measure μ. As a consequence, since 𝒱 has {βn}n⩾1 as Verblunsky coefficients, 𝒱 can be expressed as an antiassociated perturbation of order 1 (see [24]) applied to the measure μ.

5. The Bernstein-Szegő Linear Functional

Now, we proceed to analyze the companion measure 𝒱 when 𝒰 is the Bernstein-Szegő linear functional defined as above.

Theorem 5.

Let 𝒰 be the Bernstein-Szegő linear functional, and let (𝒰,𝒱) be a (1,1)-coherent pair on the unit circle given by (16). Then, the moments of 𝒱 are
(32)vn=(-1)n×[(a1-b1)∑k=0n-1n+1-kn+1Ck∏j=2n-kaj+1n+1Cn]v0,n≥1,
where ∏j=k1k2aj=1 whenever k2<k1, and the sequence of monic OPUC {ψn(z)}n≥0 is given by ψ0(z)=1, ψ1(z)=z+(a1-b1)+(1/2)C, and, for n≥2,
(33)ψn(z)=zn+[(an-bn)+nn+1C]zn-1-[bn(an-1-bn-1)-n-1nC(an-bn)]zn-2+∑k=0n-3(-1)n-(k+1)bnbn-1⋯bk+3×[bk+2(ak+1-bk+1)-k+1k+2C(ak+2-bk+2)]zk.
Furthermore, |βn|=|ψn(0)|≠1, n≥1, and
(34)β1=(a1-b1)+12C,β2=-[b2β1-12Ca2],.andforn≥3,βn=(-1)n-1bnbn-1⋯b3[b2(a1-b1)-12C(a2-b2)]=-bnβn-1,(35)an+bn[|βn-1|2-1]=-nn+1C+β1+12Ca2β¯1-∑k=2n-1bk|βk-1|2,.n≥2.

Proof.

Since ϕn[1](z)=zn+(n/(n+1))Czn-1, for n≥0, then, from (19), we get
(36)vn=-nn+1Cvn-1+(-1)n(a1-b1)×∏j=2najv0,n≥1,
where ∏j=k1k2aj=1 whenever k2<k1. From (36) and using induction on n, it is easy to verify that the moments of 𝒱 are given by (32). Besides, from (18) and (33), (34) holds. Furthermore, since {ψn(z)}n≥0 is a sequence of monic OPUC, then it follows that |βn|≠1, n≥1.

On the other hand, from the forward Szegő relation and (33), we can get another expression of ψn(z), for n≥2. Hence, comparing the coefficients of z and using (34), (35) follows.

As in the previous section, we are interested in the situations where 𝒱 is also a positive definite linear functional. Notice now that the values of a1, b1, a2, b2, and b3 determine all other coherence coefficients. We have the following cases.

Proposition 6.

Let 𝒰 be the Bernstein-Szegő linear functional, and let (𝒰,𝒱) be a (1,1)-coherent pair on the unit circle given by (16). Then, one has the following.

If a1=b1, then C=0 and, therefore, 𝒰 and 𝒱 are Lebesgue linear functionals, and an=bn for n≥1.

Let a1≠b1.

If 𝒱 is normalized (i.e., v0=1) and bN=0 for some N≥3, then C=0;
this is, 𝒰 is the Lebesgue linear functional. As a consequence, bn+1=0, an+1=a1-b1, ψn(z)=zn-1(z+a1-b1),
and vn=(b1-a1)n for every n≥1. In other words, for |b1-a1|<1, 𝒱 is the linear functional associated with the Bernstein-Szegő measure, with parameter b1-a1.

If (1/2)Ca2=b2β1, then ψn(z)=zn-1(z+a1-b1+(1/2)C) for n≥1;
this is, for |b1-a1-(1/2)C|<1, 𝒱 is the linear functional associated with the Bernstein-Szegő measure, with parameter b1-a1-(1/2)C.

If (1/2)Ca2≠b2β1 and bn≠0, for n≥3, then (37)bn=bn-11-|βn-1|2=b3∏k=3n-1(1-|βk-1|2),n≥4,and b3 can be chosen so that 𝒱 is the linear functional associated with an antiassociated perturbation of order 2 applied to a Nevai measure.

Proof.

(i) If we multiply (33) by z-1 and apply 𝒱, then we get, for n≥2,
(38)0=vn-1+[(an-bn)+nn+1C]vn-2-[bn(an-1-bn-1)-n-1nC(an-bn)]vn-3+∑k=0n-3(-1)n-(k+1)bnbn-1⋯bk+3×[bk+2(ak+1-bk+1)-k+1k+2C(ak+2-bk+2)]vk-1.
If we multiply this equation by bn+1 and we add it to the previous equation for n+1, then we obtain
(39)0=vn+[an+1+n+1n+2C]vn-1+nn+1Can+1vn-2,n≥2.
Hence, from (39) and (36), it follows that
(40)0=(-1)n+1(a1-b1)×∏j=2n+1ajv0[an+3-an+2+1(n+3)(n+4)C]+[1(n+2)(n+3)an+3+n+1(n+2)(n+3)(n+4)C]Cvn,n≥0.

On the other hand, if we apply the linear functional 𝒱 to both sides of the (1,1)-coherence relation (16), we get v1+[a1+(C/2)]v0=b1v0 and
(41)vn+[an+Cnn+1]vn-1+anC(n-1)nvn-2=0,n≥2.
Thus, from (39) and (41), we obtain, for n≥2,
(42)0=[an+1-an+C(n+1)(n+2)]vn-1+[nan+1n+1-(n-1)ann]Cvn-2.

Therefore, if a1=b1, then from (32), the moments of 𝒱 are vn=(1/(n+1))(-C)nv0 for n≥0, and, as a consequence, (40) becomes
(43)0=(-1)n1(n+1)(n+2)(n+3)×Cn+1[an+3+n+1n+4C]v0,n≥0,
and (42) is, for n≥2,
(44)0=(-1)n-11n(n+1)Cn-1×[1n+2C-1n-1an+1]v0,n≥2.
Then, if C≠0, from (43) and (44) it follows that an=-((n-2)/(n+1))C, for n≥3, and an=((n-2)/(n+1))C, for n≥3, respectively, which is a contradiction. Thus, if a1=b1, then C=0; that is, 𝒰 is the Lebesgue linear functional, and in case the part i of Theorem 3 holds.

Now, let us assume a1≠b1.

(ii)(1) From part (i) of Proposition 4, it suffices to show that 𝒰 is the Lebesgue linear functional. Thus, let us prove that if bN=0 for some N≥3 (and therefore βN=0), then C=0. Indeed, if bN=0 for some N≥3, then from (33) for n=N+1, N≥2, it follows that βN+1=0, for N≥3. Furthermore, from the forward Szegő relation and (33) for n=N, we obtain an expression of ψN+1(z), for N≥3. Hence, comparing the coefficients of this expression and (33) for n=N+1, we obtain, for N≥3,
(45)(aN+1-bN+1)+N+1N+2C=aN+NN+1C,(46)-bN+1aN+NN+1C(aN+1-bN+1)=N-1NCaN,(47)bN+1N-1NCaN=0.
Since aN≠0, then from (47) it follows that either C=0 or bN+1=0. If C=0, then from (46) we get bN+1=0 and, as a consequence, from (45) we have aN+1=aN. If bN+1=0, then from (46) it follows that either C=0 (and thus, from (45), aN+1=aN) or aN+1=((N2-1)/N2)aN. If bN+1=0 and aN+1=((N2-1)/N2)aN, from (45) it follows that C=(((N+1)(N+2))/N2)aN. But if bN+1=0, we can follow a similar argument and conclude that C=((N+2)(N+3)/(N+1)2)aN+1, and since aN+1=((N2-1)/N2)aN, then we also have C=((N+2)(N+3)(N-1)/(N+1)N2)aN, which yields a contradiction. Therefore, C=0.

(ii)(2) If (1/2)Ca2=b2β1, then from (34) it follows that β2=0 and, as a consequence, βn=0 for every n≥2. Therefore, from the forward Szegő relation it follows that ψn(z)=zn-1(z+β1) for n≥1.

(ii)(3) From the forward Szegő relation and (33) we obtain an expression of ψn(z), for n≥3. If we compare the coefficients of z of this expression and (33), we get β2[b4-b3]=b4b3β2∑k=33b¯k|βk-1|2 and
(48)bn-1⋯b4β2[bn-b3]=bnbn-1⋯b3β2∑k=3n-1b¯k|βk-1|2,n≥5.
Thus, if (1/2)Ca2≠b2β1, then from (34) it follows that β2≠0, and, as a consequence, if b4,…,bn-1, n≥5, are nonzero, then from (48) we get
(49)bn=b31-b3∑k=3n-1b¯k|βk-1|2,n≥4.
Besides, from (34), |βn|=|bnβn-1| for n≥3, and if b3≠0, then by induction on n we can prove that bn=bn-1/(1-|βn-1|2), for n≥4, which is (37). Therefore, proceeding as in the proof of Proposition 4, we can choose |b3| small enough so that β3 is sufficiently close to 0. As a consequence, {|bn|}n⩾3 will be an increasing sequence, and hence {|βn|}n⩾3 will be a decreasing sequence. Also, we can choose b3 such that |bn| converges to a constant b, with 0<b<1. The infinite product ∏k=3n-1|1-|βk|2| will then converge to |b3|/b. Therefore, since {βn}n⩾1 are the Verblunsky coefficients of 𝒱, this linear functional 𝒱 is an antiassociated perturbation of order 2 (see [24]) applied to a Nevai measure μ.

Acknowledgments

The authors thank the referee the valuable comments. They greatly contributed to improve the contents of the paper. The work of Luis Garza was supported by Conacyt Grant no. 156668 and Beca Santander Iberoamérica para Jóvenes Profesores e Investigadores (Mexico). The work of Francisco Marcellán and Natalia C. Pinzón-Cortés has been supported by Dirección General de Investigación, Desarrollo e Innovación, Ministerio de Economía y Competitividad of Spain, Grant MTM2012-36732-C03-01.

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