AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 307974 10.1155/2013/307974 307974 Research Article ( 1,1 ) -Coherent Pairs on the Unit Circle Garza Luis 1 Marcellán Francisco 2 Pinzón-Cortés Natalia C. 2 Cao Jinde 1 Facultad de Ciencias Universidad de Colima Bernal Díaz del Castillo 340 28045 Colima, COL Mexico ucol.mx 2 Departamento de Matemáticas Universidad Carlos III de Madrid Avenida de la Universidad 30 28911 Leganés Spain uc3m.es 2013 10 11 2013 2013 23 07 2013 19 09 2013 2013 Copyright © 2013 Luis Garza et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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)}n0 and {ψn(x)}n0 satisfy ϕn(z)+anϕn-1(z)=ψn(z)+bnψn-1(z), an0, n1, where ϕn(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 is a (1,1)-coherent pair if and only if their corresponding sequences of monic orthogonal polynomials (SMOP) {Pn(x)}n0 and {Qn(x)}n0 satisfy (1)Pn+1(x)n+1+anPn(x)n=Qn(x)+bnQn-1(x),an0,n1. This concept is a generalization of the notion of coherent pair, for us (1,0)-coherent pair, introduced by Iserles et al. in , where bn=0, for every n1.

In the work by Delgado and Marcellán , 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)}n0 associated with the Sobolev inner product (2)p(x),r(x)λ=p(x)r(x)dμ0+λp(x)r(x)dμ1,λ>0,p,r, and the SMOP {Pn(x)}n0 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+1n  anPn(x),n1, where {cn(λ)}n1 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  for the (1,0)-coherence case (when bn=0, n1), 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  (see also ), the so-called (M,N)-coherent pairs of order (m,k), where the derivatives of order m and k of two SMOP {Pn(x)}n0 and {Qn(x)}n0 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),n0, where M, N, m, k+{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 mk, 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),n0, holds, where cn-j,n,m(λ), 0<jmax{M,N}, n0, are rational functions in λ such that cn-j,n,m(λ)=0 for n<jmax{M,N}, and {Sn,m(x;λ)}n0 is the Sobolev SMOP with respect to the inner product (6)p(x),r(x)λ,m=p(x)r(x)dμ0+λp(m)(x)r(m)(x)dμ1,λ>0,m+,p, r. 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.  and of Marcellán and Pinzón-Cortés .

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 . 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{0},(Dqp)(x)=p(qx)-p(x)(q-1)xfor  x0,(Dqp)(0)=p(0),q{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 ω0 or q1, 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.  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 .

Furthermore, Branquinho et al. in  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)}n0 is semiclassical and {Qn(z)}n0 is quasiorthogonal of order at most 6 with respect to the functional [zA(z)+(1/z)A¯(1/z)]𝒰, A. 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  obtained that if the sequences {Pn(z)}n0 and {Qn(z)}n0 satisfy, for n0, (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)}n0 and {Qn(z)}n0 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)}n0. Finally, they analyzed the (0,1)-coherence case (Pn+1(z))/(n+1)=Qn(z)+bnQn-1(z), bn0, n1, 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:|z|=1}, the linear space of Laurent polynomials with complex coefficients Λ=span{zn:n}, and a linear functional 𝒰:Λ. We can associate with 𝒰 a sequence of moments {cn}n defined by cn=𝒰,zn, n, and a bilinear form as follows: (9)p(z),q(z)=𝒰,p(z)q¯(1z), where p, q, the linear space of polynomials with complex coefficients. Its Gram matrix with respect to {zn}n0 is an infinite Toeplitz matrix (cj-k)j,k0 with leading principal minors given by Δn=det((cj-k)j,k=0n), n+{0}.

The linear functional 𝒰 is said to be Hermitian if c-n=c¯n, quasidefinite or regular if Δn0 for all n+{0}, and positive definite if Δn>0 for all n+{0}. We will denote by the set of Hermitian linear functionals defined on Λ.

𝒰 is regular if and only if there exists a (unique) sequence of monic orthogonal polynomials on the unit circle (OPUC) {ϕn(z)}n0; this is, it satisfies that deg(ϕn(z))=n and ϕm(z),ϕn(z)=κnδm,n, with κn0, for n, m+{0}. Every monic OPUC ϕn(z) has an explicit representation, the so-called Heine’s formula, as follows:(10)ϕn(z)=1Δn-1|c0c1cnc-(n-1)c-(n-2)c11zzn|,n1,      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),n1,ϕ0(z)=1, where αn=ϕn(0), n1, are said to be the Verblunsky (reflection, Schur, Szegő, or Geronimus) coefficients and ϕn*(z)=znϕ¯n(1/z), n+{0}, is called the reversed polynomial of ϕn(z). Conversely, if {ϕn(z)}n0 is a sequence of monic polynomials which satisfies (11) and |αn|1 for n1, then {ϕn(z)}n0 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 ) |αn|1 (resp., |αn|<1), for n1.

A positive definite Hermitian linear functional 𝒰 has an integral representation (see ) (12)p(z),q(z)=𝒰,p(z)q¯(1z)=12π02πp(z)q¯(1z)dμ(θ),z=eiθ,  p,q, 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 ), an analytic function F(z), defined on 𝔻={z:|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)=12π02πeiθ+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 ), a Carathéodory function (13) admits the expansions (14)F(z)=c0+2n=1c-nzn,|z|<1,F(z)=-c0-2n=1cnz-n,|z|>1, where {cn}n0 are the moments of the measure associated with F(z).

To complete this section, we state the following definitions. Let {ϕn(z)}n0 be a sequence of monic OPUC with corresponding Verblunsky coefficients {αn}n1, and let N+{0}. The polynomials defined by (15)ϕn(N)(z)=zϕn-1(z)+αn+Nϕn-1*(z),n1,ϕ0(N)(z)=1, are called the associated polynomials of {ϕn(z)}n0 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}n1={γ1,,γN,α1,α2,}. Then the monic OPUC defined by the forward Szegő relation associated with {α~n}n1 are said to be the antiassociated polynomials of {ϕn(z)}n0 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)}n0 and {ψn(z)}n0, are related by (16)ϕn(z)+anϕn-1(z)=ψn(z)+bnψn-1(z),an0,n1, where ϕn(z)=(ϕn+1(z))/(n+1), for n. In such a case, the pair {ϕn(z)}n0 and {ψn(z)}n0 is also said to be a (1,1)-coherent pair. If bn=0 for every n1, then (𝒰,𝒱) is called a (1,0)-coherent pair.

Lemma 1.

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

a1b1 if and only if ϕn(z)ψn(z), for every n1.

For n1, one has (17)ϕn(z)=ψn(z)+(bn-an)ψn-1(z)+k=0n-2(-1)n-(k+1)anan-1ak+2(bk+1-ak+1)ψk(z),(18)ψn(z)=ϕn(z)+(an-bn)ϕn-1(z)+k=0n-2(-1)n-(k+1)bnbn-1bk+2(ak+1-bk+1)ϕk(z).

Proof.

From (16) it is easy to check that a1=b1 if and only if there exists N, N1, such that ϕN(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(z)=(-1)n(a1-b1)j=2naj𝒱,1,n1, 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+{0}. Besides, the reversed polynomials are ϕn*(z)=1,  n+{0}, and its Verblunsky coefficients are αn=ϕn(0)=0, for n1. Furthermore, its moments are cn=δn,0, for n+{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 and |C|<1. Its corresponding monic OPUC are ϕn(z)=zn-1(z+C) for n1 and ϕ0(z)=1. Its reversed polynomials are ϕn*(z)=1+C¯z, for n1, and its Verblunsky coefficients are αn=ϕn(0)=0, for n2 and α1=C. Besides, its moments are cn=(-C)n for n+{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.

If a1=b1, then 𝒱 is also the linear functional associated with the Lebesgue measure, and an=bn for n1.

If a1b1 and |ψn(0)|=|βn|1,  n1, then (20)vn=(-a2)n-1(b1-a1)v0,      n1,(21)a2=b2(1-|β1|2)+β1,      an=a2,bn=bn-11-|βn-1|2=b2k=2n-1(1-|βk|2),      n3,(22)β1=a1-b1,βn=(-1)n-1bnb2β1=-bnβn-1,n2,(23)ψ1(z)=z+a1-b1,      andfor  n2,ψn(z)=zn+(a2-bn)zn-1+k=1n-2(-1)n-k-1bnbk+2(a2-bk+1)zk+βn, where {vn}n0 is the sequence of moments associated with 𝒱.

Proof.

Since ϕn(z)=zn for n+{0}, then (16) becomes (24)zn+anzn-1=ψn(z)+bnψn-1(z),an0,n1. Thus, applying the linear functional 𝒱 on the previous expression, we get (25)vn=-anvn-1=(-1)n-1ana2(b1-a1)v0,n2,v1=(b1-a1)v0.

(i) If a1=b1, then from (25) we have vn=0 for n1. Thus, ψn(z)=zn for n1, and, as a consequence, from (24) we obtain an=bn for every n1.

(ii) From (18), we have (26)ψn(z)=zn+(an-bn)zn-1+k=0n-2(-1)n-(k+1)bnbn-1bk+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-1bk+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,n2. Since an0, n1, and a1b1, (25) yields vn0 for n1. Thus, from (28), we conclude that an+1=an for n2 or, equivalently, an+1=a2 for n2. 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), n0. By comparing the coefficients of zn, we get an+1-bn+1=an-bn-bn+1|βn|2, for n1. Hence, since an+1=an and |βn-1|1, for n2, (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, n3, and |βn|=2, n2. 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 n2. Besides, 𝒱 is the linear functional associated with the Bernstein-Szegő measure with parameter b1-a1. Furthermore, if bN=0 for some N2, then b2=0.

If a1, b1, a2 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 a1b1 because a20. We first prove that if bN=0 for some N2, then bn=0 for n2. Assume that for some N2, bN=0. From (21), (22), and (23) it follows that bn=0=βn and ψn(z)=zn-1(z+a2) for nN. 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 n2, β1=a1-b1=a2, and ψn(z)=zn-1(z+a1-b1) for n1. As a consequence, from (21) and (20), it follows that an+1=a1-b1 and vn=(b1-a1)n, n0. 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+2k1(b1-a1)(-a2)k-1zk. Since |a2|<1, then (see ) 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)=1-2a2k1(-a2)k-1zk. Therefore, (29) holds. In other words (see ), 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,n2,bn=bn-11-|βn-1|2,n3,{|bn|}n2 will be an increasing sequence and, as a consequence, {|βn|}n2 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 βn0, and thus {βn}n2 defines a Nevai measure μ. As a consequence, since 𝒱 has {βn}n1 as Verblunsky coefficients, 𝒱 can be expressed as an antiassociated perturbation of order 1 (see ) 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+1Ckj=2n-kaj+1n+1Cn]v0,n1, where j=k1k2aj=1 whenever k2<k1, and the sequence of monic OPUC {ψn(z)}n0 is given by ψ0(z)=1, ψ1(z)=z+(a1-b1)+(1/2)C, and, for n2, (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-1bk+3×[bk+2(ak+1-bk+1)-k+1k+2C(ak+2-bk+2)]zk. Furthermore, |βn|=|ψn(0)|1, n1, and (34)β1=(a1-b1)+12C,β2=-[b2β1-12Ca2],.andfor  n3,βn=(-1)n-1bnbn-1b3[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,.n2.

Proof.

Since ϕn(z)=zn+(n/(n+1))Czn-1, for n0, then, from (19), we get (36)vn=-nn+1Cvn-1+(-1)n(a1-b1)×j=2najv0,n1, 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)}n0 is a sequence of monic OPUC, then it follows that |βn|1, n1.

On the other hand, from the forward Szegő relation and (33), we can get another expression of ψn(z), for n2. 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 n1.

Let a1b1.

If 𝒱 is normalized (i.e., v0=1) and bN=0 for some N3, 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 n1. 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 n1; 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)Ca2b2β1 and bn0, for n3, then (37)bn=bn-11-|βn-1|2=b3k=3n-1(1-|βk-1|2),n4,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 n2, (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-1bk+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,n2. 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,n0.

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,n2. Thus, from (39) and (41), we obtain, for n2, (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 n0, and, as a consequence, (40) becomes (43)0=(-1)n1(n+1)(n+2)(n+3)×Cn+1[an+3+n+1n+4C]v0,n0, and (42) is, for n2, (44)0=(-1)n-11n(n+1)Cn-1×[1n+2C-1n-1an+1]v0,n2. Then, if C0, from (43) and (44) it follows that an=-((n-2)/(n+1))C, for n3, and an=((n-2)/(n+1))C, for n3, 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 a1b1.

(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 N3 (and therefore βN=0), then C=0. Indeed, if bN=0 for some N3, then from (33) for n=N+1,  N2, it follows that βN+1=0, for N3. Furthermore, from the forward Szegő relation and (33) for n=N, we obtain an expression of ψN+1(z), for N3. Hence, comparing the coefficients of this expression and (33) for n=N+1, we obtain, for N3, (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 aN0, 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 n2. Therefore, from the forward Szegő relation it follows that ψn(z)=zn-1(z+β1) for n1.

(ii)(3) From the forward Szegő relation and (33) we obtain an expression of ψn(z), for n3. If we compare the coefficients of z of this expression and (33), we get β2[b4-b3]=b4b3β2k=33b¯k|βk-1|2 and (48)bn-1b4β2[bn-b3]=bnbn-1b3β2k=3n-1b¯k|βk-1|2,n5. Thus, if (1/2)Ca2b2β1, then from (34) it follows that β20, and, as a consequence, if b4,,bn-1, n5, are nonzero, then from (48) we get (49)bn=b31-b3k=3n-1b¯k|βk-1|2,      n4. Besides, from (34), |βn|=|bnβn-1| for n3, and if b30, then by induction on n we can prove that bn=bn-1/(1-|βn-1|2), for n4, 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|}n3 will be an increasing sequence, and hence {|βn|}n3 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}n1 are the Verblunsky coefficients of 𝒱, this linear functional 𝒱 is an antiassociated perturbation of order 2 (see ) 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.

Iserles A. Koch P. E. Nørsett S. P. Sanz-Serna J. M. On polynomials orthogonal with respect to certain Sobolev inner products Journal of Approximation Theory 1991 65 2 151 175 10.1016/0021-9045(91)90100-O MR1104157 ZBL0734.42016 Delgado A. M. Marcellán F. Companion linear functionals and Sobolev inner products: a case study Methods and Applications of Analysis 2004 11 2 237 266 MR2143522 10.4310/MAA.2004.v11.n2.a5 ZBL1087.42020 Meijer H. G. Determination of all coherent pairs Journal of Approximation Theory 1997 89 3 321 343 10.1006/jath.1996.3062 MR1451509 ZBL0880.42012 de Jesus M. N. Marcellán F. Petronilho J. Pinzón-Cortés N. C. ( M , N ) -coherent pairs of order (m,k) and Sobolev orthogonal polynomials Journal of Computational and Applied Mathematics 2014 256 16 35 10.1016/j.cam.2013.07.015 MR3095683 de Jesus M. N. Petronilho J. On linearly related sequences of derivatives of orthogonal polynomials Journal of Mathematical Analysis and Applications 2008 347 2 482 492 10.1016/j.jmaa.2008.06.017 MR2440344 ZBL1160.42011 de Jesus M. N. Petronilho J. Sobolev orthogonal polynomials and (M,N)-coherent pairs of measures Journal of Computational and Applied Mathematics 2013 237 1 83 101 10.1016/j.cam.2012.07.006 MR2966889 Marcellán F. Pinzón-Cortés N. C. Higher order coherent pairs Acta Applicandae Mathematicae 2012 121 105 135 10.1007/s10440-012-9696-0 MR2966968 ZBL1262.42009 Area I. Godoy E. Marcellán F. Classification of all Δ-coherent pairs Integral Transforms and Special Functions 2000 9 1 1 18 10.1080/10652460008819238 MR1785535 Area I. Godoy E. Marcellán F. q -coherent pairs and q-orthogonal polynomials Applied Mathematics and Computation 2002 128 2-3 191 216 10.1016/S0096-3003(01)00072-8 MR1891019 Area I. Godoy E. Marcellán F. Δ -coherent pairs and orthogonal polynomials of a discrete variable Integral Transforms and Special Functions 2003 14 1 31 57 10.1080/10652460304546 MR1949214 ZBL1047.42019 Marcellán F. Pinzón-Cortés N. C. (1,1)-Dω-coherent pairs Journal of Difference Equations and Applications 2013 10.1080/10236198.2013.778842 Marcellán F. Pinzón-Cortés N. C. ( 1 , 1 ) -q-coherent pairs Numerical Algorithms 2012 60 2 223 239 10.1007/s11075-012-9549-y MR2915777 ZBL1262.42009 Álvarez-Nodarse R. Petronilho J. Pinzón-Cortés N. C. Sevinik-Adgüzel R. On linearly related sequences of difference derivatives of discrete orthogonal polynomials in progress Branquinho A. Moreno A. F. Marcellán F. Rebocho M. N. Coherent pairs of linear functionals on the unit circle Journal of Approximation Theory 2008 153 1 122 137 10.1016/j.jat.2008.03.003 MR2432558 ZBL1149.42013 Branquinho A. Rebocho M. N. Structure relations for orthogonal polynomials on the unit circle Linear Algebra and Its Applications 2012 436 11 4296 4310 10.1016/j.laa.2012.01.034 MR2915284 ZBL1242.33011 Geronimus Ya. L. Orthogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval 1961 18 New York, NY, USA Consultants Bureau MR0133643 Geronimus Ya. L. Polynomials Orthogonal on a Circle and Their Applications 1962 3 Providence, RI, USA American Mathematical Society Translations Szegő G. Orthogonal Polynomials 1975 23 4th Providence, RI, USA American Mathematical Society American Mathematical Society Colloquium Publications Simon B. Orthogonal Polynomials on the Unit Circle 2005 54 Providence, RI, USA American Mathematical Society American Mathematical Society Colloquium Publications Máté A. Nevai P. G. Remarks on E. A. Rakhmanov's paper: “The asymptotic behavior of the ratio of orthogonal polynomials” Journal of Approximation Theory 1982 36 1 64 72 10.1016/0021-9045(82)90071-5 MR673857 Máté A. Nevai P. Totik V. Extensions of Szegő's theory of orthogonal polynomials. II Constructive Approximation 1987 3 1 51–72, 73–96 10.1007/BF01890553 MR892168 Peherstorfer F. Steinbauer R. Characterization of orthogonal polynomials with respect to a functional Journal of Computational and Applied Mathematics 1995 65 1–3 339 355 10.1016/0377-0427(95)00125-5 Castillo K. Garza L. Marcellán F. Linear spectral transformations, Hessenberg matrices, and orthogonal polynomials Rendiconti Circolo Matematico di Palermo 2010 2 supplement 82 3 26 Peherstorfer F. A special class of polynomials orthogonal on the unit circle including the associated polynomials Constructive Approximation 1996 12 2 161 185 10.1007/s003659900008 MR1393285 ZBL0854.42021