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

and Applied Analysis 3 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 functionalU is said to beHermitian 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 byH 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 n δ 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:

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 {  (; )} ≥0 associated with the Sobolev inner product where {  ()} ≥1 are rational functions in  > 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   = 0,  ≥ 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 (, )-coherent pairs of order (, ), where the derivatives of order  and  of two SMOP {  ()} ≥0 and {  ()} ≥0 with respect to the regular linear functionals U and V are related by where , , ,  ∈ Z + ∪ {0} and the real numbers  −,, ,  −,, satisfy some natural conditions.They showed that the regular linear functionals U and V are related by a rational factor, and, when  ̸ = , those linear functionals are semiclassical.Besides, they proved that if ( 0 ,  1 ) is a (, )coherent pair of order (, 0) of positive Borel measures on the real line, then holds, where  −,, (), 0 <  ≤ max{, },  ≥ 0, are rational functions in  such that  −,, () = 0 for  <  ≤ max{, }, and { , (; )} ≥0 is the Sobolev SMOP with respect to the inner product ,  ∈ P. Also, they showed that (, max{, })-coherence of order (, 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][9][10].They used the difference operator   as well as the -derivative operator   defined by instead of the usual derivative operator .In this way, they obtained similar results to those by Meijer and similar classification as a limit case when either  → 0 or  → 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)-  -coherent pairs and (1, 1)-  -coherent pairs.Finally, Álvarez-Nodarse et al. [13] analyzed the more general case, (, )-  -coherent pairs of order (, ) and (, )- coherent pairs of order (, ), 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 {  ()} ≥0 is semiclassical and {  ()} ≥0 is quasiorthogonal of order at most 6 with respect to the functional [() + (1/)(1/)]U,  ∈ P. Besides, they analyzed the cases when either U or V is the Lebesgue measure or U is 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.

Preliminaries
Let us consider the unit circle T = { ∈ C : || = 1}, the linear space of Laurent polynomials with complex coefficients Λ = span{  :  ∈ Z}, and a linear functional U : Λ → C. We can associate with U a sequence of moments {  } ∈Z defined by   = ⟨U,   ⟩,  ∈ Z, and a bilinear form as follows: where ,  ∈ P, the linear space of polynomials with complex coefficients.Its Gram matrix with respect to {  } ≥0 is an infinite Toeplitz matrix ( − ) ,≥0 with leading principal minors given by Δ  = det(( − )  ,=0 ),  ∈ Z + ∪ {0}.The linear functional U is said to be Hermitian if  − =   , quasidefinite or regular if Δ  ̸ = 0 for all  ∈ Z + ∪ {0}, and positive definite if Δ  > 0 for all  ∈ Z + ∪ {0}.We will denote by H the set of Hermitian linear functionals defined on Λ.
If U is a Hermitian regular (resp., positive definite) linear functional, then (see [16][17][18]) A positive definite Hermitian linear functional U has an integral representation (see [19]) where  is a nontrivial probability measure supported on an infinite subset of T. A measure  belongs to the Nevai class (see [20,21] On the other hand (see [19]), an analytic function (), defined on D = { ∈ C : || < 1}, is said to be a Carathéodory function if and only if (0) = 1 and Re () > 0 on D. If  is a probability measure on T, then is a Carathéodory function.Conversely, the Herglotz representation theorem claims that every Carathéodory function () has a representation given by ( 13) for a unique probability measure  on T.Besides (see [22]), a Carathéodory function ( 13) admits the expansions where {  } ≥0 are the moments of the measure associated with ().

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  1 =  1 , then V is also the linear functional associated with the Lebesgue measure, and where {V  } ≥0 is the sequence of moments associated with V.
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  +1 (),  ≥ 0. By comparing the coefficients of   , we get We are interested in the cases where V is also a positive definite linear functional.Notice that, aside from the trivial case when  1 =  1 , all of the coherence coefficients are determined from the values of  1 ,  1 , and  2 (or, equivalently,  1 ,  1 , and  2 ).Not every choice of these parameters will yield a positive definite linear functional V.For instance, if then we can see from ( 22) that |  | = 1,  ⩾ 3, and |  | = √ 2,  ⩾ 2. However, it is possible to choose the values of  1 ,  1 , and  2 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., V 0 = 1).Then, one has the following.
where   () is the Carathéodory function associated with the Bernstein-Szegő measure with parameter − 2 .As a consequence, the orthogonality measure associated with V is (iii) For any values of  1 ,  1 , the value of  2 can be chosen in such a way that V is the linear functional associated with a rational spectral transformation of a Nevai class measure.
(ii) From ( 20), the Carathéodory function associated with [19]) the Bernstein-Szegő polynomials of parameter − 2 have moments   = (− 2 )  and are orthogonal with respect to the measure Therefore, (29) holds.In other words (see [23]),  V can be obtained by applying a rescaling to the moments of   (), 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  1 =  1 −  1 , we have ), so we can choose | 2 | small enough so that  2 is sufficiently close to 0. Thus,  3 will also be close to 0, and since {|  |} ⩾2 will be an increasing sequence and, as a consequence, {|  |} ⩾2 will be a decreasing sequence.Besides,  2 can be chosen so that |  | converges to a constant , 0 <  < 1, and therefore the product will also converge to | 2 |/.This shows that   → 0, and thus {  } ⩾2 defines a Nevai measure .As a consequence, since V has {  } ⩾1 as Verblunsky coefficients, V can be expressed as an antiassociated perturbation of order 1 (see [24]) applied to the measure .

The Bernstein-Szegy Linear Functional
Now, we proceed to analyze the companion measure V when U is the Bernstein-Szegő linear functional defined as above.
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  1 ,  1 ,  2 ,  2 , and  3 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.(1) If V is normalized (i.e., V 0 = 1) and   = 0 for some  ≥ 3, then  = 0; this is, U is the Lebesgue linear functional.As a consequence, and  3 can be chosen so that V is the linear functional associated with an antiassociated perturbation of order 2 applied to a Nevai measure.