(1,1)-Coherent Pairs on the Unit Circle

Abstract

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 {𝜙_𝑛(𝑥)}_𝑛≥0 and {𝜓_𝑛(𝑥)}_𝑛≥0 satisfy 𝜙_𝑛^[1](𝑧)+𝑎_𝑛𝜙^[1]_𝑛−1(𝑧)=𝜓_𝑛(𝑧)+𝑏_𝑛𝜓_𝑛−1(𝑧), 𝑎_𝑛≠0, 𝑛≥1, where 𝜙_𝑛^[1](𝑧)=𝜙^′_𝑛+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) {𝑃_𝑛(𝑥)}_𝑛≥0 and {𝑄_𝑛(𝑥)}_𝑛≥0 satisfy 𝑃^′_𝑛+1(𝑥)𝑛+1+𝑎_𝑛𝑃^′_𝑛(𝑥)𝑛=𝑄_𝑛(𝑥)+𝑏_𝑛𝑄_𝑛−1(𝑥),𝑎_𝑛≠0,𝑛≥1. (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 𝑏_𝑛=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 {𝑆_𝑛(𝑥;𝜆)}_𝑛≥0 associated with the Sobolev inner product 𝑝(𝑥),𝑟(𝑥)_𝜆=_ℝ𝑝(𝑥)𝑟(𝑥)𝑑𝜇₀+𝜆_ℝ𝑝^′(𝑥)𝑟^′(𝑥)𝑑𝜇₁,𝜆>0,𝑝,𝑟∈ℙ, (2) and the SMOP {𝑃_𝑛(𝑥)}_𝑛≥0 associated with the positive Borel measure 𝜇₀ in the real line as follows: 𝑆_𝑛+1(𝑥;𝜆)+𝑐_𝑛(𝜆)𝑆_𝑛(𝑥;𝜆)=𝑃_𝑛+1(𝑥)+𝑛+1𝑛𝑎_𝑛𝑃_𝑛(𝑥),𝑛≥1, (3) where {𝑐_𝑛(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 𝑏_𝑛=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 {𝑃_𝑛(𝑥)}_𝑛≥0 and {𝑄_𝑛(𝑥)}_𝑛≥0 with respect to the regular linear functionals 𝒰 and 𝒱 are related by ^𝑀_𝑖=0𝑎_{𝑛−𝑖,𝑛,𝑚}𝑃^(𝑚)_{𝑛+𝑚−𝑖}(𝑥)=^𝑁_𝑖=0𝑏_{𝑛−𝑖,𝑛,𝑘}𝑄^(𝑘)_{𝑛+𝑘−𝑖}(𝑥),𝑛≥0, (4) where 𝑀, 𝑁, 𝑚, 𝑘∈ℤ⁺∪{0} and the real numbers 𝑎_{𝑛−𝑖,𝑛,𝑚},𝑏_{𝑛−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 ^{max{𝑀,𝑁}}_𝑗=0𝑐_{𝑛−𝑗,𝑛,𝑚}(𝜆)𝑆_{𝑛−𝑗+𝑚,𝑚}()=𝑥;𝜆_𝑀𝑗=0𝑎_{𝑛−𝑗,𝑛,𝑚}𝑃_{𝑛−𝑗+𝑚}(𝑥),𝑛≥0, (5) holds, where 𝑐_{𝑛−𝑗,𝑛,𝑚}(𝜆), 0<𝑗≤max{𝑀,𝑁}, 𝑛≥0, are rational functions in 𝜆 such that 𝑐_{𝑛−𝑗,𝑛,𝑚}(𝜆)=0 for 𝑛<𝑗≤max{𝑀,𝑁}, and {𝑆_𝑛,𝑚(𝑥;𝜆)}_𝑛≥0 is the Sobolev SMOP with respect to the inner product ⟨𝑝(𝑥),𝑟(𝑥)⟩_𝜆,𝑚=_ℝ𝑝(𝑥)𝑟(𝑥)𝑑𝜇₀+𝜆_ℝ𝑝^(𝑚)(𝑥)𝑟^(𝑚)(𝑥)𝑑𝜇₁,𝜆>0,𝑚∈ℤ⁺, (6)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 𝐷_𝜔 as well as the 𝑞-derivative operator 𝐷_𝑞 defined by 𝐷_𝜔𝑝(𝑥)=𝑝(𝑥+𝜔)−𝑝(𝑥)𝜔{0},𝐷,𝜔∈ℂ⧵_𝑞𝑝(𝑥)=𝑝𝑞𝑥−𝑝(𝑥)𝑞−1𝑥for𝐷𝑥≠0,_𝑞𝑝(0)=𝑝^′(0),𝑞∈ℂ⧵{0,1}, (7) 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)-𝐷_𝜔-coherent pairs and (1,1)-𝐷_𝑞-coherent pairs. Finally, Álvarez-Nodarse et al. [13] analyzed the more general case, (𝑀,𝑁)-𝐷_𝜔-coherent pairs of order (𝑚,𝑘) and (𝑀,𝑁)-𝐷_𝑞-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 {𝑃_𝑛(𝑧)}_𝑛≥0 is semiclassical and {𝑄_𝑛(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 {𝑃_𝑛(𝑧)}_𝑛≥0 and {𝑄_𝑛(𝑧)}_𝑛≥0 satisfy, for 𝑛≥0, ^𝑀1_𝑗=0𝛼_𝑛,𝑗𝑃^′_{𝑛+1+𝑀1−𝑗}(𝑧)𝑛+1+𝑀₁+−𝑗^𝑀2_𝑗=0𝜂_𝑛,𝑗𝑃^∗_{𝑛+𝑀2−𝑗}(𝑧)^′=^𝑁1_𝑗=0𝛽_𝑛,𝑗𝑄_{𝑛+𝑁1−𝑗}(𝑧)+^𝑁2_𝑗=0𝛾_𝑛,𝑗𝑄^∗_{𝑛+𝑁2−𝑗}(𝑧), (8) with 𝑁₁=𝑀₁, max{𝑀₂,𝑁₂}<𝑁₁, and some extra conditions, then {𝑃_𝑛(𝑧)}_𝑛≥0 and {𝑄_𝑛(𝑧)}_𝑛≥0 are semiclassical sequences of OPUC. Moreover, when 𝑃_𝑛(𝑧)=𝑄_𝑛(𝑧) for all 𝑛 and under some extra conditions, (8) is a necessary condition for the semiclassical character of {𝑃_𝑛(𝑧)}_𝑛≥0. Finally, they analyzed the (0,1)-coherence case (𝑃^′_𝑛+1(𝑧))/(𝑛+1)=𝑄_𝑛(𝑧)+𝑏_𝑛𝑄_𝑛−1(𝑧), 𝑏_𝑛≠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{𝑧^𝑛∶𝑛∈ℤ}, and a linear functional 𝒰∶Λ→ℂ. We can associate with 𝒰 a sequence of moments {𝑐_𝑛}_𝑛∈ℤ defined by 𝑐_𝑛=⟨𝒰,𝑧^𝑛⟩, 𝑛∈ℤ, and a bilinear form as follows: 𝑝(𝑧),𝑞(𝑧)=𝒰,𝑝(𝑧)𝑞1𝑧, (9) where 𝑝, 𝑞∈ℙ, the linear space of polynomials with complex coefficients. Its Gram matrix with respect to {𝑧^𝑛}_𝑛≥0 is an infinite Toeplitz matrix (𝑐_𝑗−𝑘)_{𝑗,𝑘≥0} with leading principal minors given by Δ_𝑛=det((𝑐_𝑗−𝑘)^𝑛_𝑗,𝑘=0), 𝑛∈ℤ⁺∪{0}.

The linear functional 𝒰 is said to be Hermitian if 𝑐_−𝑛=𝑐_𝑛, quasidefinite or regular if Δ_𝑛≠0 for all 𝑛∈ℤ⁺∪{0}, and positive definite if Δ_𝑛>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) {𝜙_𝑛(𝑧)}_𝑛≥0; this is, it satisfies that deg(𝜙_𝑛(𝑧))=𝑛 and ⟨𝜙_𝑚(𝑧),𝜙_𝑛(𝑧)⟩=𝜅_𝑛𝛿_𝑚,𝑛, with 𝜅_𝑛≠0, for 𝑛, 𝑚∈ℤ⁺∪{0}. Every monic OPUC 𝜙_𝑛(𝑧) has an explicit representation, the so-called Heine’s formula, as follows:𝜙_𝑛(𝑧)=1Δ_𝑛−1𝑐₀𝑐₁⋯𝑐_𝑛𝑐⋮⋮⋮⋮_{−(𝑛−1)}𝑐_{−(𝑛−2)}⋯𝑐₁1𝑧⋯𝑧^𝑛,𝑛≥1,𝜙₀(𝑧)=1. (10) Besides, they satisfy the forward and backward Szegő recurrence relations𝜙_𝑛(𝑧)=𝑧𝜙_𝑛−1(𝑧)+𝛼_𝑛𝜙^∗_𝑛−1(𝑧),𝜙_𝑛(𝑧)=1−𝛼_𝑛|²𝑧𝜙_𝑛−1(𝑧)+𝛼_𝑛𝜙^∗_𝑛(𝑧),𝑛≥1,𝜙₀(𝑧)=1, (11) where 𝛼_𝑛=𝜙_𝑛(0), 𝑛≥1, are said to be the Verblunsky (reflection, Schur, Szegő, or Geronimus) coefficients and 𝜙^∗_𝑛(𝑧)=𝑧^𝑛𝜙_𝑛(1/𝑧), 𝑛∈ℤ⁺∪{0}, is called the reversed polynomial of 𝜙_𝑛(𝑧). Conversely, if {𝜙_𝑛(𝑧)}_𝑛≥0 is a sequence of monic polynomials which satisfies (11) and |𝛼_𝑛|≠1 for 𝑛≥1, then {𝜙_𝑛(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]) |𝛼_𝑛|≠1 (resp., |𝛼_𝑛|<1), for 𝑛≥1.

A positive definite Hermitian linear functional 𝒰 has an integral representation (see [19]) 𝑝(𝑧),𝑞(𝑧)=𝒰,𝑝(𝑧)𝑞1𝑧=12𝜋₀^2𝜋𝑝(𝑧)𝑞1𝑧𝑑𝜇(𝜃),𝑧=𝑒^𝑖𝜃,𝑝,𝑞∈ℙ, (12) 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 lim_𝑛→∞|𝜙_𝑛(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 𝐹(𝑧)=12𝜋₀^2𝜋𝑒^𝑖𝜃+𝑧𝑒^𝑖𝜃(𝜃)−𝑧𝑑𝜇 (13) 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 𝐹(𝑧)=𝑐₀+2^∞_𝑛=1𝑐_−𝑛𝑧^𝑛,|𝑧|𝐹(𝑧)<1,=−𝑐₀−2^∞_𝑛=1𝑐_𝑛𝑧^−𝑛,|𝑧|>1, (14) where {𝑐_𝑛}_𝑛≥0 are the moments of the measure associated with 𝐹(𝑧).

To complete this section, we state the following definitions. Let {𝜙_𝑛(𝑧)}_𝑛≥0 be a sequence of monic OPUC with corresponding Verblunsky coefficients {𝛼_𝑛}_𝑛≥1, and let 𝑁∈ℤ⁺∪{0}. The polynomials defined by 𝜙_𝑛^(𝑁)(𝑧)=𝑧𝜙_𝑛−1(𝑧)+𝛼_𝑛+𝑁𝜙^∗_𝑛−1(𝑧),𝑛≥1,𝜙₀^(𝑁)(𝑧)=1, (15) are called the associated polynomials of {𝜙_𝑛(𝑧)}_𝑛≥0 of order 𝑁. Similarly, given a finite set of complex numbers {𝛾_𝑛}^𝑁_𝑛=1, with |𝛾_𝑛|≠1,  𝑛=1,2,…,𝑁, let us define the new Verblunsky coefficients {𝛼_𝑛}_𝑛≥1={𝛾₁,…,𝛾_𝑁,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 {𝜙_𝑛(𝑧)}_𝑛≥0 of order 𝑁.

3. (1,1)-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, {𝜙_𝑛(𝑧)}_𝑛≥0 and {𝜓_𝑛(𝑧)}_𝑛≥0, are related by 𝜙_𝑛^[1](𝑧)+𝑎_𝑛𝜙^[1]_𝑛−1(𝑧)=𝜓_𝑛(𝑧)+𝑏_𝑛𝜓_𝑛−1(𝑧),𝑎_𝑛≠0,𝑛≥1, (16) where 𝜙_𝑛^[1](𝑧)=(𝜙^′_𝑛+1(𝑧))/(𝑛+1), for 𝑛∈ℕ. In such a case, the pair {𝜙_𝑛(𝑧)}_𝑛≥0 and {𝜓_𝑛(𝑧)}_𝑛≥0 is also said to be a (1,1)-coherent pair. If 𝑏_𝑛=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.(i)𝑎₁≠𝑏₁ if and only if 𝜙_𝑛^[1](𝑧)≠𝜓_𝑛(𝑧), for every 𝑛≥1. (ii)For 𝑛≥1, one has 𝜙_𝑛^[1](𝑧)=𝜓_𝑛(𝑧)+𝑏_𝑛−𝑎_𝑛𝜓_𝑛−1(𝑧)+^𝑛−2_𝑘=0(−1)^{𝑛−(𝑘+1)}𝑎_𝑛𝑎_𝑛−1⋯𝑎_𝑘+2𝑏_𝑘+1−𝑎_𝑘+1𝜓_𝑘(𝑧),𝜓_𝑛(𝑧)=𝜙_𝑛^[1](𝑧)+𝑎_𝑛−𝑏_𝑛𝜙^[1]_𝑛−1(𝑧)+^𝑛−2_𝑘=0(−1)^{𝑛−(𝑘+1)}𝑏_𝑛𝑏_𝑛−1⋯𝑏_𝑘+2𝑎_𝑘+1−𝑏_𝑘+1𝜙_𝑘^[1](𝑧). (17)(18)

Proof. From (16) it is easy to check that 𝑎₁=𝑏₁ if and only if there exists 𝑁∈ℕ, 𝑁≥1, such that 𝜙_𝑁^[1](𝑧)=𝜓_𝑁(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 𝒱,𝜙_𝑛^[1](𝑧)=(−1)^𝑛𝑎₁−𝑏₁^𝑛_𝑗=2𝑎_𝑗⟨𝒱,1⟩,𝑛≥1, (19) where ∏^𝑘2_𝑗=𝑘1𝑎_𝑗=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 𝜙_𝑛(𝑧)=𝑧^𝑛, for 𝑛∈ℤ⁺∪{0}. Besides, the reversed polynomials are 𝜙^∗_𝑛(𝑧)=1,  𝑛∈ℤ⁺∪{0}, and its Verblunsky coefficients are 𝛼_𝑛=𝜙_𝑛(0)=0, for 𝑛≥1. Furthermore, its moments are 𝑐_𝑛=𝛿_𝑛,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+𝐶𝑒^𝑖𝜃|²)(𝑑𝜃/2𝜋), with 𝐶∈ℂ and |𝐶|<1. Its corresponding monic OPUC are 𝜙_𝑛(𝑧)=𝑧^𝑛−1(𝑧+𝐶) for 𝑛≥1 and 𝜙₀(𝑧)=1. Its reversed polynomials are 𝜙^∗_𝑛(𝑧)=1+𝐶𝑧, for 𝑛≥1, and its Verblunsky coefficients are 𝛼_𝑛=𝜙_𝑛(0)=0, for 𝑛≥2 and 𝛼₁=𝐶. Besides, its moments are 𝑐_𝑛=(−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 Functional

Theorem 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. (i)If 𝑎₁=𝑏₁, then 𝒱 is also the linear functional associated with the Lebesgue measure, and 𝑎_𝑛=𝑏_𝑛 for 𝑛≥1. (ii)If 𝑎₁≠𝑏₁ and |𝜓_𝑛(0)|=|𝛽_𝑛|≠1,  𝑛≥1, then 𝑣_𝑛=−𝑎₂^𝑛−1𝑏₁−𝑎₁𝑣₀,𝑛≥1,𝑎₂=𝑏₂1−𝛽₁²+𝛽₁,𝑎_𝑛=𝑎₂,𝑏_𝑛=𝑏_𝑛−11−𝛽_𝑛−1²=𝑏₂∏^𝑛−1_𝑘=21−𝛽_𝑘²,𝑛≥3,𝛽₁=𝑎₁−𝑏₁,𝛽_𝑛=(−1)^𝑛−1𝑏_𝑛⋯𝑏₂𝛽₁=−𝑏_𝑛𝛽_𝑛−1,𝑛≥2,𝜓₁(𝑧)=𝑧+𝑎₁−𝑏₁𝜓,𝑎𝑛𝑑𝑓𝑜𝑟𝑛≥2,_𝑛(𝑧)=𝑧^𝑛+𝑎₂−𝑏_𝑛𝑧^𝑛−1+^𝑛−2_𝑘=1(−1)^{𝑛−𝑘−1}𝑏_𝑛⋯𝑏_𝑘+2𝑎₂−𝑏_𝑘+1𝑧^𝑘+𝛽_𝑛, (20)(21)(22)(23) where {𝑣_𝑛}_𝑛≥0 is the sequence of moments associated with 𝒱.

Proof. Since 𝜙_𝑛^[1](𝑧)=𝑧^𝑛 for 𝑛∈ℤ⁺∪{0}, then (16) becomes 𝑧^𝑛+𝑎_𝑛𝑧^𝑛−1=𝜓_𝑛(𝑧)+𝑏_𝑛𝜓_𝑛−1(𝑧),𝑎_𝑛≠0,𝑛≥1. (24) Thus, applying the linear functional 𝒱 on the previous expression, we get 𝑣_𝑛=−𝑎_𝑛𝑣_𝑛−1=(−1)^𝑛−1𝑎_𝑛⋯𝑎₂𝑏₁−𝑎₁𝑣₀𝑣,𝑛≥2,₁=𝑏₁−𝑎₁𝑣₀. (25)
(i) If 𝑎₁=𝑏₁, then from (25) we have 𝑣_𝑛=0 for 𝑛≥1. Thus, 𝜓_𝑛(𝑧)=𝑧^𝑛 for 𝑛≥1, and, as a consequence, from (24) we obtain 𝑎_𝑛=𝑏_𝑛 for every 𝑛≥1.
(ii) From (18), we have 𝜓_𝑛(𝑧)=𝑧^𝑛+𝑎_𝑛−𝑏_𝑛𝑧^𝑛−1+^𝑛−2_𝑘=0(−1)^{𝑛−(𝑘+1)}𝑏_𝑛𝑏_𝑛−1⋯𝑏_𝑘+2𝑎_𝑘+1−𝑏_𝑘+1𝑧^𝑘. (26) Multiplying (26) by 𝑧⁻¹ and applying 𝒱, we obtain 0=𝑣_𝑛−1+𝑎_𝑛−𝑏_𝑛𝑣_𝑛−2+^𝑛−2_𝑘=0(−1)^{𝑛−(𝑘+1)}𝑏_𝑛𝑏_𝑛−1⋯𝑏_𝑘+2𝑎_𝑘+1−𝑏_𝑘+1𝑣_𝑘−1. (27) Thus, multiplying this equation by 𝑏_𝑛+1 and adding it to the previous equation for 𝑛+1, we get 0=𝑣_𝑛+𝑎_𝑛+1𝑣=_𝑛−1⁽²⁵⁾−𝑎_𝑛𝑣_𝑛−1+𝑎_𝑛+1𝑣_𝑛−1=𝑎_𝑛+1−𝑎_𝑛𝑣_𝑛−1,𝑛≥2. (28) Since 𝑎_𝑛≠0, 𝑛≥1, and 𝑎₁≠𝑏₁, (25) yields 𝑣_𝑛≠0 for 𝑛≥1. Thus, from (28), we conclude that 𝑎_𝑛+1=𝑎_𝑛 for 𝑛≥2 or, equivalently, 𝑎_𝑛+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 𝜓_𝑛+1(𝑧), 𝑛≥0. By comparing the coefficients of 𝑧^𝑛, we get 𝑎_𝑛+1−𝑏_𝑛+1=𝑎_𝑛−𝑏_𝑛−𝑏_𝑛+1|𝛽_𝑛|², for 𝑛≥1. Hence, since 𝑎_𝑛+1=𝑎_𝑛 and |𝛽_𝑛−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 |𝑏_𝑛|=1, 𝑛⩾3, and |𝛽_𝑛|=√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.(i)Let |𝑏₁−𝑎₁|<1. If 𝑎₂=𝑎₁−𝑏₁ (i.e., 𝑏₂=0), then 𝑏_𝑛=0 and 𝑎_𝑛=𝑎₁−𝑏₁ for every 𝑛≥2. Besides, 𝒱 is the linear functional associated with the Bernstein-Szegő measure with parameter 𝑏₁−𝑎₁. Furthermore, if 𝑏_𝑁=0 for some 𝑁≥2, then 𝑏₂=0. (ii)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 𝐹_𝒱𝑏=−¹−𝑎₁𝑎₂𝐹_𝐵(𝑧)+𝑏₁−𝑎₁+𝑎₂𝑎₂, (29)where 𝐹_𝐵(𝑧) is the Carathéodory function associated with the Bernstein-Szegő measure with parameter −𝑎₂. As a consequence, the orthogonality measure associated with 𝒱 is 𝑑𝜇₂𝑏=−¹−𝑎₁𝑎₂1−𝑎₂²1+𝑎₂𝑒^𝑖𝜃2𝑑𝜃+𝑏2𝜋₁−𝑎₁+𝑎₂𝑎₂𝑑𝜃.2𝜋 (30)(iii)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 𝑏_𝑁=0 for some 𝑁≥2, then 𝑏_𝑛=0 for 𝑛≥2. Assume that for some 𝑁≥2, 𝑏_𝑁=0. From (21), (22), and (23) it follows that 𝑏_𝑛=0=𝛽_𝑛 and 𝜓_𝑛(𝑧)=𝑧^𝑛−1(𝑧+𝑎₂) for 𝑛≥𝑁. Besides, another expression for 𝜓_𝑁(𝑧) is 𝜓_𝑁(𝑧)=𝑧𝜓_𝑁−1(𝑧)+𝛽_𝑁𝜓^∗_𝑁−1(𝑧)=𝑧𝜓_𝑁−1(𝑧), where 𝜓_𝑁−1(𝑧) is given by (23). Thus, the comparison of the coefficients of 𝑧^𝑁−1 in both expressions of 𝜓_𝑁(𝑧) yields 𝑎₂=𝑎₂−𝑏_𝑁−1, and thus, 𝑏_𝑁−1=0. Following the same argument for 𝑏_𝑁−1,…,𝑏₂, we conclude that 𝑏_𝑛=0 for 𝑛=2,…,𝑁−1 and 𝑎₂=𝑎₁−𝑏₁. Therefore, 𝑏_𝑛=0=𝛽_𝑛 for 𝑛≥2, 𝛽₁=𝑎₁−𝑏₁=𝑎₂, and 𝜓_𝑛(𝑧)=𝑧^𝑛−1(𝑧+𝑎₁−𝑏₁) for 𝑛≥1. As a consequence, from (21) and (20), it follows that 𝑎_𝑛+1=𝑎₁−𝑏₁ and 𝑣_𝑛=(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 𝐹_𝒱=1+2∑_𝑘≥1(𝑏₁−𝑎₁)(−𝑎₂)^𝑘−1𝑧^𝑘. Since |𝑎₂|<1, then (see [19]) the Bernstein-Szegő polynomials of parameter −𝑎₂ have moments 𝑐_𝑛=(−𝑎₂)^𝑛 and are orthogonal with respect to the measure ((1−|𝑎₂|²)/|1+𝑎₂𝑒^𝑖𝜃|²)(𝑑𝜃/2𝜋), and their associated Carathéodory function is 𝐹_𝐵(𝑧)=1−2𝑎₂∑_𝑘≥1(−𝑎₂)^𝑘−1𝑧^𝑘. Therefore, (29) holds. In other words (see [23]), 𝐹_𝒱 can be obtained by applying a rescaling to the moments of 𝐹_𝐵(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 𝛽_𝑛=−𝑏_𝑛𝛽_𝑛−1𝑏,𝑛≥2,_𝑛=𝑏_𝑛−11−𝛽_𝑛−1²,𝑛≥3, (31){|𝑏_𝑛|}_𝑛⩾2 will be an increasing sequence and, as a consequence, {|𝛽_𝑛|}_𝑛⩾2 will be a decreasing sequence. Besides, 𝑏₂ can be chosen so that |𝑏_𝑛| converges to a constant 𝑏, 0<𝑏<1, and therefore the product ∏^𝑛−1_𝑘=2|1−|𝛽_𝑘|²| will also converge to |𝑏₂|/𝑏. This shows that 𝛽_𝑛→0, and thus {𝛽_𝑛}_𝑛⩾2 defines a Nevai measure 𝜇. As a consequence, since 𝒱 has {𝛽_𝑛}_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 𝑣_𝑛=(−1)^𝑛×𝑎₁−𝑏₁^𝑛−1_𝑘=0𝑛+1−𝑘𝐶𝑛+1^{𝑘𝑛−𝑘}_𝑗=2𝑎_𝑗+1𝐶𝑛+1^𝑛𝑣₀,𝑛≥1, (32) where ∏^𝑘2_𝑗=𝑘1𝑎_𝑗=1 whenever 𝑘₂<𝑘₁, and the sequence of monic OPUC {𝜓_𝑛(𝑧)}_𝑛≥0 is given by 𝜓₀(𝑧)=1, 𝜓₁(𝑧)=𝑧+(𝑎₁−𝑏₁)+(1/2)𝐶, and, for 𝑛≥2, 𝜓_𝑛(𝑧)=𝑧^𝑛+𝑎_𝑛−𝑏_𝑛+𝑛𝐶𝑧𝑛+1^𝑛−1−𝑏_𝑛𝑎_𝑛−1−𝑏_𝑛−1−𝑛−1𝑛𝐶𝑎_𝑛−𝑏_𝑛𝑧^𝑛−2+^𝑛−3_𝑘=0(−1)^{𝑛−(𝑘+1)}𝑏_𝑛𝑏_𝑛−1⋯𝑏_𝑘+3×𝑏_𝑘+2𝑎_𝑘+1−𝑏_𝑘+1−𝑘+1𝐶𝑎𝑘+2^𝑘+2−𝑏_𝑘+2𝑧^𝑘. (33) Furthermore, |𝛽_𝑛|=|𝜓_𝑛(0)|≠1, 𝑛≥1, and 𝛽₁=𝑎₁−𝑏₁+12𝐶,𝛽₂=−𝑏₂𝛽₁−12𝐶𝑎₂,𝛽𝑎𝑛𝑑𝑓𝑜𝑟𝑛≥3,_𝑛=(−1)^𝑛−1𝑏_𝑛𝑏_𝑛−1⋯𝑏₃𝑏₂𝑎₁−𝑏₁−12𝐶𝑎₂−𝑏₂=−𝑏_𝑛𝛽_𝑛−1,𝑎_𝑛+𝑏_𝑛𝛽_𝑛−1²𝑛−1=−𝑛+1𝐶+𝛽₁+12𝐶𝑎₂𝛽₁−^𝑛−1_𝑘=2𝑏_𝑘𝛽_𝑘−1²,𝑛≥2. (34)(35)

Proof. Since 𝜙_𝑛^[1](𝑧)=𝑧^𝑛+(𝑛/(𝑛+1))𝐶𝑧^𝑛−1, for 𝑛≥0, then, from (19), we get 𝑣_𝑛𝑛=−𝑛+1𝐶𝑣_𝑛−1+(−1)^𝑛𝑎₁−𝑏₁×^𝑛_𝑗=2𝑎_𝑗𝑣₀,𝑛≥1, (36) where ∏^𝑘2_𝑗=𝑘1𝑎_𝑗=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 {𝜓_𝑛(𝑧)}_𝑛≥0 is a sequence of monic OPUC, then it follows that |𝛽_𝑛|≠1, 𝑛≥1.
On the other hand, from the forward Szegő relation and (33), we can get another expression of 𝜓_𝑛(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.(i)If 𝑎₁=𝑏₁, then 𝐶=0 and, therefore, 𝒰 and 𝒱 are Lebesgue linear functionals, and 𝑎_𝑛=𝑏_𝑛  for 𝑛≥1.(ii)Let 𝑎₁≠𝑏₁.(1) If 𝒱 is normalized (i.e., 𝑣₀=1) and 𝑏_𝑁=0 for some 𝑁≥3, then 𝐶=0; this is, 𝒰 is the Lebesgue linear functional. As a consequence, 𝑏_𝑛+1=0, 𝑎_𝑛+1=𝑎₁−𝑏₁, 𝜓_𝑛(𝑧)=𝑧^𝑛−1(𝑧+𝑎₁−𝑏₁), and 𝑣_𝑛=(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.(2) If (1/2)𝐶𝑎₂=𝑏₂𝛽₁, then 𝜓_𝑛(𝑧)=𝑧^𝑛−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.(3)If (1/2)𝐶𝑎₂≠𝑏₂𝛽₁ and 𝑏_𝑛≠0, for 𝑛≥3, then 𝑏_𝑛=𝑏_𝑛−11−𝛽_𝑛−1²=𝑏₃∏^𝑛−1_𝑘=31−𝛽_𝑘−1²,𝑛≥4, (37)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, 0=𝑣_𝑛−1+𝑎_𝑛−𝑏_𝑛+𝑛𝐶𝑣𝑛+1^𝑛−2−𝑏_𝑛𝑎_𝑛−1−𝑏_𝑛−1−𝑛−1𝑛𝐶𝑎_𝑛−𝑏_𝑛𝑣_𝑛−3+^𝑛−3_𝑘=0(−1)^{𝑛−(𝑘+1)}𝑏_𝑛𝑏_𝑛−1⋯𝑏_𝑘+3×𝑏_𝑘+2𝑎_𝑘+1−𝑏_𝑘+1−𝑘+1𝐶𝑎𝑘+2^𝑘+2−𝑏_𝑘+2𝑣_𝑘−1. (38) If we multiply this equation by 𝑏_𝑛+1 and we add it to the previous equation for 𝑛+1, then we obtain 0=𝑣_𝑛+𝑎_𝑛+1+𝑛+1𝐶𝑣𝑛+2^𝑛−1+𝑛𝑛+1𝐶𝑎_𝑛+1𝑣_𝑛−2,𝑛≥2. (39) Hence, from (39) and (36), it follows that 0=(−1)^𝑛+1𝑎₁−𝑏₁×^𝑛+1_𝑗=2𝑎_𝑗𝑣₀𝑎_𝑛+3−𝑎_𝑛+2+1()𝐶+1𝑛+3)(𝑛+4()𝑎𝑛+2)(𝑛+3^𝑛+3+𝑛+1()𝐶𝑛+2)(𝑛+3)(𝑛+4𝐶𝑣_𝑛,𝑛≥0. (40)
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 𝑣_𝑛+𝑎_𝑛+𝐶𝑛𝑣𝑛+1^𝑛−1+𝑎_𝑛𝐶(𝑛−1)𝑛𝑣_𝑛−2=0,𝑛≥2. (41) Thus, from (39) and (41), we obtain, for 𝑛≥2, 0=𝑎_𝑛+1−𝑎_𝑛+𝐶()𝑣𝑛+1)(𝑛+2^𝑛−1+𝑛𝑎_𝑛+1−()𝑎𝑛+1𝑛−1_𝑛𝑛𝐶𝑣_𝑛−2. (42)
Therefore, if 𝑎₁=𝑏₁, then from (32), the moments of 𝒱 are 𝑣_𝑛=(1/(𝑛+1))(−𝐶)^𝑛𝑣₀ for 𝑛≥0, and, as a consequence, (40) becomes 0=(−1)^𝑛1(𝑛+1)(𝑛+2)(𝑛+3)×𝐶^𝑛+1𝑎_𝑛+3+𝑛+1𝐶𝑣𝑛+4⁰,𝑛≥0, (43) and (42) is, for 𝑛≥2, 0=(−1)^𝑛−11𝑛()𝐶𝑛+1^𝑛−1×11𝑛+2𝐶−𝑎𝑛−1^𝑛+1𝑣₀,𝑛≥2. (44) Then, if 𝐶≠0, from (43) and (44) it follows that 𝑎_𝑛=−((𝑛−2)/(𝑛+1))𝐶, for 𝑛≥3, and 𝑎_𝑛=((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 𝑏_𝑁=0 for some 𝑁≥3 (and therefore 𝛽_𝑁=0), then 𝐶=0. Indeed, if 𝑏_𝑁=0 for some 𝑁≥3, then from (33) for 𝑛=𝑁+1,  𝑁≥2, it follows that 𝛽_𝑁+1=0, for 𝑁≥3. Furthermore, from the forward Szegő relation and (33) for 𝑛=𝑁, we obtain an expression of 𝜓_𝑁+1(𝑧), for 𝑁≥3. Hence, comparing the coefficients of this expression and (33) for 𝑛=𝑁+1, we obtain, for 𝑁≥3, 𝑎_𝑁+1−𝑏_𝑁+1+𝑁+1𝑁+2𝐶=𝑎_𝑁+𝑁𝑁+1𝐶,−𝑏_𝑁+1𝑎_𝑁+𝑁𝐶𝑎𝑁+1^𝑁+1−𝑏_𝑁+1=𝑁−1𝑁𝐶𝑎_𝑁,𝑏_𝑁+1𝑁−1𝑁𝐶𝑎_𝑁=0. (45)(46)(47) Since 𝑎_𝑁≠0, then from (47) it follows that either 𝐶=0 or 𝑏_𝑁+1=0. If 𝐶=0, then from (46) we get 𝑏_𝑁+1=0 and, as a consequence, from (45) we have 𝑎_𝑁+1=𝑎_𝑁. If 𝑏_𝑁+1=0, then from (46) it follows that either 𝐶=0 (and thus, from (45), 𝑎_𝑁+1=𝑎_𝑁) or 𝑎_𝑁+1=((𝑁²−1)/𝑁²)𝑎_𝑁. If 𝑏_𝑁+1=0 and 𝑎_𝑁+1=((𝑁²−1)/𝑁²)𝑎_𝑁, from (45) it follows that 𝐶=(((𝑁+1)(𝑁+2))/𝑁²)𝑎_𝑁. But if 𝑏_𝑁+1=0, we can follow a similar argument and conclude that 𝐶=((𝑁+2)(𝑁+3)/(𝑁+1)²)𝑎_𝑁+1, and since 𝑎_𝑁+1=((𝑁²−1)/𝑁²)𝑎_𝑁, then we also have 𝐶=((𝑁+2)(𝑁+3)(𝑁−1)/(𝑁+1)𝑁²)𝑎_𝑁, 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, 𝛽_𝑛=0 for every 𝑛≥2. Therefore, from the forward Szegő relation it follows that 𝜓_𝑛(𝑧)=𝑧^𝑛−1(𝑧+𝛽₁) for 𝑛≥1.
(ii)(3) From the forward Szegő relation and (33) we obtain an expression of 𝜓_𝑛(𝑧), for 𝑛≥3. If we compare the coefficients of 𝑧 of this expression and (33), we get 𝛽₂[𝑏₄−𝑏₃]=𝑏₄𝑏₃𝛽₂∑³_𝑘=3𝑏_𝑘|𝛽_𝑘−1|² and 𝑏_𝑛−1⋯𝑏₄𝛽₂𝑏_𝑛−𝑏₃=𝑏_𝑛𝑏_𝑛−1⋯𝑏₃𝛽₂^𝑛−1_𝑘=3𝑏_𝑘𝛽_𝑘−1²,𝑛≥5. (48) Thus, if (1/2)𝐶𝑎₂≠𝑏₂𝛽₁, then from (34) it follows that 𝛽₂≠0, and, as a consequence, if 𝑏₄,…,𝑏_𝑛−1, 𝑛≥5, are nonzero, then from (48) we get 𝑏_𝑛=𝑏₃1−𝑏₃∑^𝑛−1_𝑘=3𝑏_𝑘𝛽_𝑘−1²,𝑛≥4. (49) Besides, from (34), |𝛽_𝑛|=|𝑏_𝑛𝛽_𝑛−1| for 𝑛≥3, and if 𝑏₃≠0, then by induction on 𝑛 we can prove that 𝑏_𝑛=𝑏_𝑛−1/(1−|𝛽_𝑛−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, {|𝑏_𝑛|}_𝑛⩾3 will be an increasing sequence, and hence {|𝛽_𝑛|}_𝑛⩾3 will be a decreasing sequence. Also, we can choose 𝑏₃ such that |𝑏_𝑛| converges to a constant 𝑏, with 0<𝑏<1. The infinite product ∏^𝑛−1_𝑘=3|1−|𝛽_𝑘|²| will then converge to |𝑏₃|/𝑏. Therefore, since {𝛽_𝑛}_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.