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 , a_n ≠ 0, n ≥ 1, where . 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

()

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

()

and the SMOP {P_n(x)} _n≥0 associated with the positive Borel measure μ₀ in the real line as follows:

()

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

()

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

()

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, 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

()

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 , 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,

()

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

, 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 {zⁿ : 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, zⁿ〉, n ∈ ℤ, and a bilinear form as follows:

()

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 (c_j−k) _j,k≥0 with leading principal minors given by

, n ∈ ℤ⁺ ∪ {0}.

The linear functional 𝒰 is said to be Hermitian if , 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:

()

Besides, they satisfy the forward and backward Szegő recurrence relations

()

where α_n = ϕ_n(0), n ≥ 1, are said to be the Verblunsky (reflection, Schur, Szegő, or Geronimus) coefficients and

, 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])

()

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

()

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

()

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

()

are called the associated polynomials of {ϕ_n(z)} _n≥0 of order N. Similarly, given a finite set of complex numbers

, with |γ_n | ≠ 1, n = 1,2, …, N, let us define the new Verblunsky coefficients

. Then the monic OPUC defined by the forward Szegő relation associated with

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

()

where

, 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 , for every n ≥ 1.
(ii)
For n ≥ 1, one has
()

()

Proof. From (16) it is easy to check that a₁ = b₁ if and only if there exists N ∈ ℕ, N ≥ 1, such that . 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

()

where

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) = zⁿ, for n ∈ ℤ⁺ ∪ {0}. Besides, the reversed polynomials are , 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) = zⁿ⁻¹(z + C) for n ≥ 1 and ϕ₀(z) = 1. Its reversed polynomials are , 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
()

()

()

()
where {v_n} _n≥0 is the sequence of moments associated with 𝒱.

Proof. Since for n ∈ ℤ⁺ ∪ {0}, then (16) becomes

()

Thus, applying the linear functional 𝒱 on the previous expression, we get

()

(i) If a₁ = b₁, then from (25) we have v_n = 0 for n ≥ 1. Thus, ψ_n(z) = zⁿ for n ≥ 1, and, as a consequence, from (24) we obtain a_n = b_n for every n ≥ 1.

(ii) From (18), we have

()

Multiplying (26) by z⁻¹ and applying 𝒱, we obtain

()

Thus, multiplying this equation by b_n+1 and adding it to the previous equation for n + 1, we get

()

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 zⁿ, 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 , then we can see from (22) that |b_n | = 1, n⩾3, and , 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
()
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
()
(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) = zⁿ⁻¹(z + a₂) for n ≥ N. Besides, another expression for ψ_N(z) is , 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) = zⁿ⁻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 , 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

()

{|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

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

()

where

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,

()

Furthermore, |β_n | = |ψ_n(0)| ≠ 1, n ≥ 1, and

()

Proof. Since , for n ≥ 0, then, from (19), we get

()

where

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) = zⁿ⁻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) = 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.
- (3)
  If (1/2)Ca₂ ≠ b₂β₁ and b_n ≠ 0, for n ≥ 3, then
  ()
  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,

()

If we multiply this equation by b_n+1 and we add it to the previous equation for n + 1, then we obtain

()

Hence, from (39) and (36), it follows that

()

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

()

Thus, from (39) and (41), we obtain, for n ≥ 2,

()

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

()

and (42) is, for n ≥ 2,

()

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,

()

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) = zⁿ⁻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 and

()

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

()

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

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.

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

Abstract

1. Introduction

2. Preliminaries

3. (1,1)-Coherent Pairs on the Unit Circle

4. The Lebesgue Linear Functional

5. The Bernstein-Szegő Linear Functional

Acknowledgments

References

References

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

Abstract

1. Introduction

2. Preliminaries

3. (1,1)-Coherent Pairs on the Unit Circle

4. The Lebesgue Linear Functional

5. The Bernstein-Szegő Linear Functional

Acknowledgments

References

References

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