PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 96 (110) (2014), 193–210 DOI: 10.2298/PIM1410193M

GENERALIZED COHERENT PAIRS ON THE UNIT CIRCLE AND

SOBOLEV ORTHOGONAL POLYNOMIALS

Francisco Marcellán and Natalia C. Pinzón-Cortés

Abstract. A pair of regular Hermitian linear functionals (U, V) is said to be an (M, N)-coherent pair of order m on the unit circle if their corresponding sequences of monic orthogonal polynomials {ϕ_{n} (z)}_{n⩾0} and {ψ_{n} (z)}_{n⩾0} satisfy

∑_{i=0}^{M} a_{i,n} ϕ_{n+m−i}^{(m)} (z) = ∑_{j=0}^{N} b_{j,n} ψ_{n−j} (z), n ⩾ 0,

where M, N, m ⩾ 0, a_{i,n} and b_{j,n} , for 0 ⩽ i ⩽ M, 0 ⩽ j ⩽ N, n ⩾ 0, are complex numbers such that a_{M,n} ≠ 0, n ⩾ M, b_{N,n} ≠ 0, n ⩾ N, and a_{i,n} = b_{i,n} = 0, i > n. When m = 1, (U, V) is called a (M, N)-coherent pair on the unit circle.

We focus our attention on the Sobolev inner product

⟨p(z), q(z)⟩_{λ} = ⟨U, p(z)\overline{q}(1/z)⟩ + λ⟨V, p^{(m)} (z)\overline{q^{(m)}} (1/z)⟩, λ > 0, m ∈ ℤ^{+} ,

assuming that U and V is an (M, N)-coherent pair of order m on the unit circle. We generalize and extend several recent results of the framework of Sobolev orthogonal polynomials and their connections with coherent pairs. Besides, we analyze the cases (M, N) = (1, 1) and (M, N) = (1, 0) in detail. In particular, we illustrate the situation when U is the Lebesgue linear functional and V is the Bernstein–Szegő linear functional. Finally, a matrix interpretation of (M, N)-coherence is given.

1. Introduction

In the theory of orthogonal polynomials with respect to measures supported on the real line the notion of coherent pair (in our terminology, (1, 0)-coherent pair of order 1) arose in the framework of Sobolev orthogonal polynomials and it

2010 Mathematics Subject Classification: 13B05; 42C05. Key words and phrases: coherent pairs, Sobolev inner products, structure relations, Hermit-

ian linear functionals, orthogonal polynomials on the unit circle, Sobolev orthogonal polynomials, Lebesgue linear functional, Bernstein–Szegő linear functional, Hessenberg matrices.

The work of the authors has been supported by Dirección General de Investigación, Desarrollo e Innovación, Ministerio de Economía y Competitividad of Spain, grant MTM2012-36732-C03-01.

To Professor Giuseppe Mastroianni on the occasion of his retirement.

was introduced by Iserles, Koch, Nørsett and Sanz-Serna [4]. From this pioneering contribution, coherent pairs and Sobolev orthogonal polynomials have been widely studied and extended in recent decades (for a historical summary, see e.g., the introductory sections in the recent papers [6] and [8] as well as [9]. Lately, de Jesus, Marcellán, Petronilho, and Pinzón-Cortés [5] generalized all those works focussing the attention on the Sobolev inner product

⟨p(x), r(x)⟩_{λ} = ∫_{ℝ} p(x) r(x) dμ_{0} + λ ∫_{ℝ} p^{(m)} (x) r^{(m)} (x) dμ_{1}, λ > 0, m ∈ ℤ^{+} ,

where p(x) and r(x) are polynomials with real coefficients. The positive definite Borel measures μ_{0} and μ_{1} supported on the real line are said to be a (M, N)-coherent pair of order m if their corresponding sequences of monic orthogonal polynomials (SMOP) {P_{n}(x)}_{n⩾0} and {Q_{n}(x)}_{n⩾0} , satisfy an algebraic relation

(1.1)  ∑_{i=0}^{M} a_{i,n} P_{n+m−i}^{(m)} (x) = ∑_{i=0}^{N} b_{i,n} Q_{n−i} (x), n ⩾ 0,

where M, N are fixed non-negative integers, {a_{i,n} }_{n⩾0} , {b_{i,n} }_{n⩾0} ⊂ ℂ, a_{M,n} ≠ 0 if n ⩾ M, b_{N,n} ≠ 0 if n ⩾ N, a_{i,n} = b_{i,n} = 0 when i > n.

Similarly, in the framework of orthogonal polynomials of a discrete variable, Álvarez-Nodarse, Petronilho, Pinzón-Cortés, and Sevinik-Adıgüzel [1] analyzed the case when μ_{0} and μ_{1} are discrete measures supported either on a uniform lattice or on a q-lattice, and they are an (M, N)-D_{ν}-coherent pair of order m, ν = ω ∈ ℂ∖{0} or ν = q ∈ ℂ ∖ {0, 1}, when in (1.1) instead of the standard derivative operator you consider either D_{ω} p(x) = \frac{p(x+ω)−p(x)}{ω} or D_{q}p(x) = \frac{p(qx)−p(x)}{(q−1)x} , the difference and the q-difference operator, respectively.

In this work we will deal with the analysis of coherent pairs in the framework of the theory of orthogonal polynomials on the unit circle. The structure of the manuscript is as follows. In Section 2, we will state the basic definitions, notations and results which will be useful in the forthcoming sections. In Section 3, we will introduce the Sobolev inner product

(1.2) ⟨p(z), q(z)⟩_{λ} = ⟨U, p(z) \overline{q}(1/z)⟩+λ⟨V, p^{(m)} (z) \overline{q^{(m)}}(1/z)⟩, λ > 0, m ∈ ℤ^{+} ,

and we will study its corresponding sequence of monic orthogonal polynomials when the regular Hermitian linear functionals U and V form an (M, N)-coherent pair of order m. In this way, we will highlight the cases (M, N) = (1, 1) and (M, N) = (1, 0). These results generalize those given by Branquinho, Foulquié-Moreno, Marcellán, and Rebocho in [2]. Finally, as an example, we will consider the cases when U is the Lebesgue linear functional and V is the Bernstein-Szegő linear functional. In Section 4, we will analyze (M, N)-coherent pairs of hermitian linear functionals on the unit circle from a matrix point of view, showing an interesting relation between the Hessenberg matrices associated with the regular Hermitian linear functionals that constitute such an (M, N)-coherent pair of order 1 (see [7] for the case of the linear functionals on the real line). As a special case, we will study the case when U is the linear functional associated with the Lebesgue measure on the unit circle. Furthermore, when (U, V) is an (M, N)-coherent pair of order m on the unit circle, we will give a matrix representation of the multiplication operator by z in terms of the basis of Sobolev polynomials orthogonal with respect to (1.2), involving a matrix similar to the Hessenberg matrix associated with U.

2. Preliminaries

Let us consider the unit circle 𝕋 = {z ∈ ℂ : |z| = 1}, the linear space of Laurent polynomials with complex coefficients Λ = span_{ℂ}{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 ⟨p(z), q(z)⟩_{U} = ⟨U, p(z)\overline{q}(1/z)⟩, where p, q ∈ ℙ, the linear space of polynomials with complex coefficients, and ⟨U, f(z)⟩ denotes the image of f(z) ∈ Λ by the linear functional U. Notice that

⟨p(z), q(z)⟩_{U} = ⟨p(z)\overline{q}(1/z), 1⟩_{U} = ⟨1, \overline{p}(1/z)q(z)⟩_{U} = ⟨\overline{q}(1/z), \overline{p}(1/z)⟩_{U}, p, q ∈ ℙ.

When c_{−n} = \overline{c}_{n}, n ⩾ 0, the linear functional U is said to be Hermitian. In this way, an Hermitian linear functional U is said to be quasi-definite or regular (resp. positive definite) if det(T_{n}) ≠ 0 (resp. > 0) for n ⩾ 0, where T_{n} = [c_{j−k} ]_{k,j=0}^{n} , n ⩾ 0. Notice that in this case U is an Hermitian bilinear form. Besides, an Hermitian linear functional U is regular if and only if there exists a (unique) sequence of monic orthogonal polynomials on the unit circle (OPUC, in short) with respect to such a linear functional, i.e., if there exists a sequence of monic polynomials {ϕ_{n}(z)}_{n⩾0} such that deg(ϕ_{n}(z)) = n, n ⩾ 0, and ⟨ϕ_{m}(z), ϕ_{n}(z)⟩_{U} = κ_{n}δ_{m,n} , κ_{n} ≠ 0. This sequence of monic OPUC {ϕ_{n}(z)}_{n⩾0} satisfies the forward Szegő recurrence relation

(2.1)  ϕ_{n}(z) = zϕ_{n−1} (z) + α_{n}ϕ_{n−1}^{∗} (z), n ⩾ 1, ϕ_{0}(z) = 1,

where ϕ_{n}^{∗}(z) denotes the reversed polynomial of ϕ_{n}(z) defined by ϕ_{n}^{∗}(z)=z^{n} \overline{ϕ}_{n}(1/z), n ⩾ 0, and α_{n} = ϕ_{n}(0), n ⩾ 0, are said to be the Verblunsky coefficients of U. The sequence of monic OPUC {ϕ_{n}(z)}_{n⩾0} is completely determined by the sequence of Verblunsky coefficients {α_{n}}_{n⩾0} which satisfy |α_{n}| ≠ 1 (resp. < 1) for n ⩾ 1 when U is regular (resp. positive definite) [10, 11].

On the other hand, for every positive definite Hermitian linear functional U we have an integral representation of the associated inner product as

⟨p(z), q(z)⟩_{U} = ⟨U, p(z) \overline{q}(1/z)⟩ = \frac{1}{2π} ∫_{0}^{2π} p(z) \overline{q}(1/z) dμ(θ), z = e^{iθ}, p, q ∈ ℙ,

where μ is a nontrivial probability measure supported on an infinite subset of 𝕋 [10, 11]. In this way, the Bernstein–Szegő linear functional with parameter −C is a well known and elementary positive definite linear functional associated with the measure dμ(θ) = \frac{1−|C|^{2}}{|1+Ce^{iθ} |^{2}} \frac{dθ}{2π}, C ∈ ℂ, |C| < 1, such that the corresponding monic OPUC and moments are ϕ_{0}(z) = 1, ϕ_{n}(z) = z^{n−1} (z + C), n ⩾ 1, and c_{n} = (−C)^{n}, n ⩾ 0, respectively. Notice that for C = 0, you recover the Lebesgue linear functional.

From (2.1), it is easy to check that the multiplication operator by z in terms of the (monic orthogonal) basis {ϕ_{n}(z)}_{n⩾0} is given by

zϕ_{n}(z) = ϕ_{n+1} (z) − α_{n+1} ⟨ϕ_{n}(z), ϕ_{n}(z)⟩_{U} ∑_{j=0}^{n} \frac{\overline{α}_{j}}{⟨ϕ_{j} (z), ϕ_{j} (z)⟩_{U}} ϕ_{j} (z), n ⩾ 0,

which can be written in a matrix form as

(2.2)  zΦ(z) = H_{ϕ}Φ(z),

where Φ(z) is an infinite vector and H_{ϕ} is an infinite lower Hessenberg matrix given by Φ(z) = [ϕ_{0}(z), ϕ_{1}(z), ⋯]^{T}, and

H_{ϕ} = [h_{k,j} ]_{k,j⩾0} = \{ \begin{matrix}1, & if j = k + 1, \\ −\frac{⟨ϕ_{n} (z),ϕ_{n} (z)⟩_{U}}{⟨ϕ_{j} (z),ϕ_{j} (z)⟩_{U}} α_{n+1} \overline{α}_{j} , & if j ⩽ k, \\ 0, & if j > k + 1. \end{matrix}

On the other hand, for fixed m, n ⩾ 0, ϕ_{n}^{[m]}(z) = ϕ_{n+m}^{(m)} (z)/(n + 1)_{m} will denote the monic polynomial of degree n, that is the mth derivative of the monic polynomial of degree n + m, ϕ_{n+m} (z). Here (n + 1)_{m} is the Pochhammer symbol defined by (a)_{n} = a(a + 1) ⋯ (a + n − 1), n ⩾ 1, and (a)_{0} = 1.

3. Sobolev orthogonal polynomials and (M, N) -coherent pairs of order m on the unit circle

Let us consider the Sobolev inner product

⟨p(z), q(z)⟩_{λ} =∫_{𝕋}p(z) \overline{q}(1/z) dμ_{0}(z)+λ∫_{𝕋}p^{(m)} (z) \overline{q^{(m)}}(1/z) dμ_{1}(z), λ > 0, m ∈ ℤ^{+} ,

where p, q ∈ ℙ, μ_{0}, μ_{1} are nontrivial probability measures supported on an infinite subset of the unit circle 𝕋. These measures are associated with positive definite Hermitian linear functionals U and V, respectively, defined on Λ, and whose sequences of monic OPUC are {ϕ_{n}(z)}_{n⩾0} and {ψ_{n}(z)}_{n⩾0} , respectively. Thus, the Sobolev inner product can also be written as

(3.1) ⟨p(z), q(z)⟩_{λ} = ⟨U, p(z) \overline{q}(1/z)⟩+ λ⟨V, p^{(m)} (z) \overline{q^{(m)}}(1/z)⟩

= ⟨p(z), q(z)⟩_{U} + λ⟨p^{(m)} (z), q^{(m)} (z)⟩_{V}, p, q ∈ ℙ, λ > 0, m ∈ ℤ^{+} ,

where ⟨ · , · ⟩_{U} and ⟨ · , · ⟩_{V} are the inner products associated with U and V, respectively. Moreover, using this notation, we can also consider the case when U and V are regular Hermitian linear functionals.

On the other hand, the sequence of monic Sobolev OPUC with respect to the inner product ⟨·, ·⟩_{λ} will be denoted by {S_{n}(z; λ)}_{n⩾0} .

Every monic Sobolev OPUC S_{n}(z; λ), n ⩾ 0, can be written as

S_{n}(z; λ) = \frac{1}{det([w_{k,j} ]_{k,j=0}^{n−1} )}|\begin{matrix}w_{0,0} & ⋯ & w_{0,n−1} & w_{0,n} \\ ⋮ & ... & ⋮ & ⋮ \\ w_{n−1,0} & ⋯ & w_{n−1,n−1} & w_{n−1,n} \\ 1 & ⋯ & z^{n−1} & z^{n} \end{matrix} | , n ⩾ 1,

S_{0}(z; λ) = 1, where w_{k,j} = ⟨z^{k}, z^{j} ⟩_{λ} = u_{k−j} + λ(k − m + 1)_{m}(j − m + 1)_{m}v_{k−j} , k, j ⩾ 0. Thus, every coefficient of S_{n}(z; λ) is a rational function of λ such that its numerator and denominator are polynomials of the same degree. Hence, there exist the monic limit polynomials

(3.2)  T_{n}(z) = lim_{λ→∞} S_{n}(z; λ), n ⩾ 0.

In this way,

(3.3) ⟨T_{n}(z), z^{j} ⟩_{U} = 0, j < min{n, m}, and ⟨T_{n}^{(m)} (z), z^{j} ⟩_{V} = 0, j < n − m.

As a consequence,

(3.4) T_{n}(z) =∑_{j=0}^{n} \frac{⟨T_{n}(z), ϕ_{j} (z)⟩_{U}}{⟨ϕ_{j} (z), ϕ_{j} (z)⟩_{U}} ϕ_{j} (z) =∑_{j=0}^{n−m} \frac{⟨T_{n}(z), ϕ_{j+m} (z)⟩_{U}}{⟨ϕ_{j+m} (z), ϕ_{j+m} (z)⟩_{U}} ϕ_{j+m} (z),

for n ⩾ m, as well as

(3.5)  \frac{T_{n+m}^{(m)} (z)}{(n + 1)_{m}} = ∑_{j=0}^{n} \frac{⟨T_{n+m}^{(m)} (z)/(n + 1)_{m} , ψ_{j} (z)⟩_{V}}{⟨ψ_{j} (z), ψ_{j} (z)⟩_{V}} ψ_{j} (z) = ψ_{n}(z),

for n ⩾ 0. Thus,

ψ_{n}(z) = ϕ_{n}^{[m]}(z) + ∑_{j=0}^{n−1} \frac{(j + 1)_{m}}{(n + 1)_{m}} \frac{⟨T_{n+m} (z), ϕ_{j+m} (z)⟩_{U}}{⟨ϕ_{j+m} (z), ϕ_{j+m} (z)⟩_{U}} ϕ_{j}^{[m]}(z), n ⩾ 0.

From (3.3), for n ⩾ 0 we get

T_{n}(z) = ∑_{j=0}^{n} \frac{⟨T_{n}(z), S_{j} (z; λ)⟩_{λ}S_{j} (z; λ)}{⟨S_{j} (z; λ), S_{j} (z; λ)⟩_{λ}} = S_{n}(z; λ)+ ∑_{j=m}^{n−1} \frac{⟨T_{n}(z), S_{j} (z; λ)⟩_{U}S_{j} (z; λ)}{⟨S_{j} (z; λ), S_{j} (z; λ)⟩_{λ}} ,

and together with (3.4), for n ⩾ m,

S_{n}(z; λ) + ∑_{j=m}^{n−1} \frac{⟨T_{n}(z), S_{j} (z; λ)⟩_{U}}{⟨S_{j} (z; λ), S_{j} (z; λ)⟩_{λ}} S_{j} (z; λ) = ϕ_{n}(z) + ∑_{j=m}^{n−1} \frac{⟨T_{n}(z), ϕ_{j} (z)⟩_{U}}{⟨ϕ_{j} (z), ϕ_{j} (z)⟩_{U}} ϕ_{j} (z).

Finally, from (3.1), we obtain that ⟨ϕ_{n}(z), z^{j} ⟩_{λ} = 0, for j < n < m, and hence S_{n}(z; λ) = ϕ_{n}(z) for n < m, (since the uniqueness of the sequence of monic OPUC).

So, we have proved the following result.

Proposition 3.1. Given the Sobolev inner product (3.1), it follows that

ψ_{n}(z) = ϕ_{n}^{[m]}(z) + ∑_{j=0}^{n−1} \frac{(j + 1)_{m}}{(n + 1)_{m}} \frac{⟨T_{n+m} (z), ϕ_{j+m} (z)⟩_{U}}{⟨ϕ_{j+m} (z), ϕ_{j+m} (z)⟩_{U}} ϕ_{j}^{[m]}(z), n ⩾ 0,

S_{n}(z; λ) + ∑_{j=m}^{n−1} \frac{⟨T_{n}(z), S_{j} (z; λ)⟩_{U}}{⟨S_{j} (z; λ), S_{j} (z; λ)⟩_{λ}} S_{j} (z; λ) = ϕ_{n}(z) + ∑_{j=m}^{n−1} \frac{⟨T_{n}(z), ϕ_{j} (z)⟩_{U}}{⟨ϕ_{j} (z), ϕ_{j} (z)⟩_{U}} ϕ_{j} (z), (3.6)

n ⩾ m, and S_{n}(z; λ) = ϕ_{n}(z), n ⩽ m, where the monic polynomials T_{n}(z), n ⩾ 0, are given by (3.2).

Now, we are interested in the case when the regular Hermitian linear functionals U and V form a (M, N)-coherent pair of order m on the unit circle, which means that their corresponding sequences of monic OPUC {ϕ_{n}(z)}_{n⩾0} and {ψ_{n}(z)}_{n⩾0} satisfy the following algebraic relation

(3.7)  ϕ_{n}^{[m]}(z) + ∑_{i=1}^{M} a_{i,n} ϕ_{n−i}^{[m]} (z) = ψ_{n}(z) + ∑_{i=1}^{N} b_{i,n} ψ_{n−i} (z), n ⩾ 0,

where M, N are nonnegative integers, and the sequences {a_{i,n} }_{n⩾0} , {b_{i,n} }_{n⩾0} ⊂ ℂ are such that a_{M,n} ≠ 0, n ⩾ M, b_{N,n} ≠ 0, n ⩾ N, and a_{i,n} = b_{i,n} = 0, i > n.

In this case, what does (3.6) become? Our aim now is to answer this question. If we substitute (3.5) in (3.7) and then we integrate m times the resulting equation, (3.7) becomes, for n ⩾ 0,

\frac{ϕ_{n+m} (z)}{(n + 1)_{m}} + ∑_{i=1}^{M} a_{i,n} \frac{ϕ_{n−i+m} (z)}{(n − i + 1)_{m}} = \frac{T_{n+m} (z)}{(n + 1)_{m}} + ∑_{i=1}^{N} b_{i,n} \frac{T_{n−i+m} (z)}{(n − i + 1)_{m}} + ∑_{j=0}^{m−1} κ_{n,j} z^{j} .

Consequently, if we apply ⟨· , z^{k}⟩_{U} for k = 0, 1, …, m − 1, to both sides in previous equation, and using (3.3), it follows that ∑ _{j=0}^{m−1} κ_{n,j} u_{j−k} = 0, k = 0, …, m − 1 , which is a linear system with a unique solution since det([u_{j−k} ]_{k,j=0}^{m−1} ) ≠ 0. Therefore, κ_{n,j} = 0, j = 0, …, m − 1, n ⩾ 0. Thus,

(3.8) \frac{ϕ_{n+m} (z)}{(n + 1)_{m}} +∑_{i=1}^{M} a_{i,n} \frac{ϕ_{n−i+m} (z)}{(n − i + 1)_{m}} = \frac{T_{n+m} (z)}{(n + 1)_{m}}+∑_{i=1}^{N} b_{i,n} \frac{T_{n−i+m} (z)}{(n − i + 1)_{m}} , n ⩾ 0.

Furthermore, (3.9)

\frac{T_{n+m} (z)}{(n + 1)_{m}}+∑_{i=1}^{N} b_{i,n} \frac{T_{n−i+m} (z)}{(n − i + 1)_{m}} = \frac{S_{n+m} (z; λ)}{(n + 1)_{m}} +∑_{j=1}^{n+m} \frac{c_{j,n,λ}}{(n + 1)_{m}} S_{n−j+m} (z; λ), n ⩾ 0,

where from (3.8) and (3.5), the coefficients c_{j,n,λ} , 1 ⩽ j ⩽ n + m, n ⩾ 0, are

⟨S_{n−j+m} (z; λ), S_{n−j+m} (z; λ)⟩_{λ} \frac{c_{j,n,λ}}{(n + 1)_{m}}

= ∑_{i=1}^{M} \frac{a_{i,n} ⟨ϕ_{n−i+m} (z), S_{n−j+m} (z; λ)⟩_{U}}{(n − i + 1)_{m}} + λ ∑_{i=1}^{N} b_{i,n} ⟨ψ_{n−i} (z), S_{n−j+m}^{(m)} (z; λ)⟩_{V},

and, as a consequence, c_{j,n,λ} = 0 for j > i or j > K = max{M, N} or j > n. So, from (3.8) and (3.9) we get, for n ⩾ 0,

ϕ_{n+m} (z) + ∑_{i=1}^{M} \frac{(n + 1)_{m}}{(n − i + 1)_{m}} a_{i,n} ϕ_{n−i+m} (z) = S_{n+m} (z; λ) + ∑_{j=1}^{K} c_{j,n,λ} S_{n−j+m} (z; λ).

Besides, from the previous proposition S_{n}(z; λ) = ϕ_{n}(z) for n ⩽ m.

Conversely, if the previous equation holds, we can apply ⟨ · , p(z)⟩_{λ} on both sides of this equation, for any p ∈ ℙ_{n−K+m−1} . As a consequence,

λ ⟨( ϕ_{n+m}^{(m)} (z) + ∑_{i=1}^{M} \frac{(n + 1)_{m}}{(n − i + 1)_{m}} a_{i,n} ϕ_{n−i+m}^{(m)} (z)) , p^{(m)} (z)⟩_{V} = 0.

In other words,

⟨( ϕ_{n}^{[m]}(z) + ∑_{i=1}^{M} a_{i,n} ϕ_{n−i}^{[m]} (z)) , q(z)⟩_{V} = 0, q ∈ ℙ_{n−K−1} .

Since

ϕ_{n}^{[m]}(z) + ∑_{i=1}^{M} a_{i,n} ϕ_{n−i}^{[m]} (z) = ψ_{n}(z) + ∑_{j=1}^{n} b_{j,n} ψ_{n−j} (z), n ⩾ 0,

then b_{j,n} = 0, n − j ⩽ n − K − 1, which is a (M, K)-coherence relation of order m.

Summarizing, we generalize the relation between Sobolev orthogonal polynomials and (1, 0)-coherent pairs on the unit circle stated in [2].

Theorem 3.1. If (U, V) is an (M, N)-coherent pair of order m of Hermitian linear functionals on the unit circle given by (3.7), then S_{n}(z; λ) = ϕ_{n}(z), n ⩽ m, and

(3.10) ϕ_{n+m} (z) + ∑_{i=1}^{M} \frac{(n + 1)_{m}}{(n − i + 1)_{m}} a_{i,n} ϕ_{n−i+m} (z)

= S_{n+m} (z; λ) + ∑_{j=1}^{K} c_{j,n,λ} S_{n−j+m} (z; λ), n ⩾ 0,

where K = max{M, N}, c_{j,n,λ} = 0 for n < j ⩽ K, and

(3.11) c_{j,n,λ} = \frac{(n + 1)_{m}}{⟨S_{n−j+m} (z; λ), S_{n−j+m} (z; λ)⟩_{λ}} [ ∑_{i=j}^{M} \frac{a_{i,n} ⟨ϕ_{n−i+m} (z), S_{n−j+m} (z; λ)⟩_{U}}{(n − i + 1)_{m}}

+ λ ∑_{i=j}^{N} b_{i,n} ⟨ψ_{n−i} (z), S_{n−j+m}^{(m)} (z; λ)⟩_{V}], 1 ⩽ j ⩽ K.

Conversely, if there exist sequences of complex numbers {a_{i,n} }_{n⩾0} , 1 ⩽ i ⩽ M, and {c_{j,n,λ} }_{n⩾0} , 1 ⩽ j ⩽ K, such that (3.10) holds with a_{i,n} = 0 if n − i + m < 0, and c_{j,n,λ} = 0 if n − j + m < 0, then

ϕ_{n}^{[m]}(z) + ∑_{i=1}^{M} a_{i,n} ϕ_{n−i}^{[m]} (z) = ψ_{n}(z) + ∑_{j=1}^{K} b_{j,n} ψ_{n−j} (z), n ⩾ 0,

i.e., (U, V) is a (M, K)-coherent pair of order m on the unit circle (assuming b_{K,n} ≠ 0, n ⩾ K), where b_{j,n} = 0 for n < j ⩽ K, and

b_{j,n} = \frac{⟨(ϕ_{n}^{[m]}(z) + ∑ _{i=1}^{M} a_{i,n} ϕ_{n−i}^{[m]} (z)), ψ_{n−j} (z)⟩_{V}}{⟨ψ_{n−j} (z), ψ_{n−j} (z)⟩_{V}} , 1 ⩽ j ⩽ min\{K, n\}, n ⩾ 0.

Remark 3.1. For j = K and n ⩾ K (3.11) reads

c_{K,n,λ} = \frac{(n + 1)_{m}}{⟨S_{n−K+m} (z; λ), S_{n−K+m} (z; λ)⟩_{λ}} [ \frac{a_{M,n}}{(n − M + 1)_{m}} ⟨ϕ_{n−M+m} (z),

ϕ_{n−M+m} (z)⟩_{U}δ_{M,K} + λ(n − N + 1)_{m}b_{N,n} ⟨ψ_{n−N} (z), ψ_{n−N} (z)⟩_{V}δ_{N,K} ].

Thus, for every n ⩾ K we will claim that

• if M > N and a_{M,n} ≠ 0, then c_{K,n,λ} ≠ 0,

• if M < N and b_{N,n} ≠ 0, then c_{K,n,λ} ≠ 0,

• if M = N(= K) and a_{M,n} b_{N,n} ≠ 0, then c_{K,n,λ} ≠ 0 holds if and only if a_{K,n} ⟨ϕ_{n−K+m} (z), ϕ_{n−K+m} (z)⟩_{U}+λ(n−K +1)_{m}^{2}b_{K,n} ⟨ψ_{n−K} (z), ψ_{n−K} (z)⟩_{V} ≠ 0.

Applying Theorem 3.1, we obtain the monic Sobolev OPUC S_{n}(z; λ), n ⩾ 0, as well as the coefficients c_{j,n,λ} , 1 ⩽ j ⩽ K, n ⩾ 0, assuming that (U, V) is an (M, N)-coherent pair of order m of Hermitian linear functionals on the unit circle. Nevertheless, we can compute the sequences {⟨S_{n}(z; λ), S_{n}(z; λ)⟩_{λ}}_{n⩾0} and {c_{j,n,λ} }_{n⩾0} , 1 ⩽ j ⩽ K, without knowing the explicit expressions of the monic Sobolev OPUC S_{n}(z; λ), n ⩾ 0. Of course, if we also want to obtain these monic Sobolev OPUC polynomials, then we can use (3.10).

Indeed, let (3.12)

s_{n} = ⟨S_{n}(z; λ), S_{n}(z; λ)⟩_{λ}, ã_{i,n} = \frac{(n + 1)_{m}}{(n − i + 1)_{m}} a_{i,n} , b̃_{i,n} = (n + 1)_{m}b_{i,n} , n ⩾ 0,

where ã_{i,n} = b̃_{i,n} = 0 for i > n, ã_{0,n} = 1 and b̃_{0,n} = (n + 1)_{m}, for n ⩾ 0.

Since (3.10) and (3.11) hold setting c_{0,n,λ} = 1 for n ⩾ 0, from (3.7) and (3.10), it follows that (3.11) becomes, for n ⩾ j and 0 ⩽ j ⩽ K,

s_{n−j+m} c_{j,n,λ} = ∑_{i=j}^{M} ∑_{ℓ=0}^{M} ã_{i,n} \overline{ã}_{ℓ,n−j} ⟨ϕ_{n−i+m} (z), ϕ_{n−j−ℓ+m} (z)⟩_{U}

− ∑_{i=j}^{M} ∑_{ℓ=1}^{K} ã_{i,n} \overline{c}_{ℓ,n−j,λ} ⟨ϕ_{n−i+m} (z), S_{n−j−ℓ+m} (z; λ)⟩_{U}

+ λ ∑_{i=j}^{N} ∑_{ℓ=0}^{N} b̃_{i,n} \overline{b̃}_{ℓ,n−j} ⟨ψ_{n−i} (z), ψ_{n−j−ℓ} (z)⟩_{V}

− λ ∑_{i=j}^{N} ∑_{ℓ=1}^{K} b̃_{i,n} \overline{c}_{ℓ,n−j,λ} ⟨ψ_{n−i} (z), S_{n−j−ℓ+m}^{(m)} (z; λ)⟩_{V},

where ⟨ϕ_{n−i+m} (z), S_{n−j−ℓ+m} (z; λ)⟩_{U} = 0, and ⟨ψ_{n−i} (z), S_{n−j−ℓ+m}^{(m)} (z; λ)⟩ = 0, for i < j + ℓ or j + ℓ > K (⩾ M, N). Consequently, for n ⩾ j and 0 ⩽ j ⩽ K, we get

s_{n−j+m} c_{j,n,λ} = ∑_{i=j}^{M} ã_{i,n} \overline{ã}_{i−j,n−j} ⟨ϕ_{n−i+m} (z), ϕ_{n−i+m} (z)⟩_{U}

+ λ ∑_{i=j}^{N} b̃_{i,n} \overline{b̃}_{i−j,n−j} ⟨ψ_{n−i} (z), ψ_{n−i} (z)⟩_{V}

− ∑_{ℓ=1}^{K−j} \overline{c}_{ℓ,n−j,λ} ∑_{i=j+ℓ}^{M} ã_{i,n} ⟨ϕ_{n−i+m} (z), S_{n−j−ℓ+m} (z; λ)⟩_{U}

− λ ∑_{ℓ=1}^{K−j} \overline{c}_{ℓ,n−j,λ} ∑_{i=j+ℓ}^{N} b̃_{i,n}⟨ψ_{n−i} (z), S_{n−j−ℓ+m}^{(m)} (z; λ)⟩_{V}.

Notice that, according to (3.11), the sum of the last two terms is equal to − ∑ _{ℓ=1}^{K−j} \overline{c}_{ℓ,n−j,λ} s_{n−j−ℓ+m} c_{j+ℓ,n,λ} . Therefore, replacing n by n + j, we have got the recurrence relation stated in the following theorem.

Theorem 3.2. If (U, V) is an (M, N)-coherent pair of order m on the unit circle given by (3.7), then

(3.13) s_{n+m} c_{j,n+j,λ} = ζ_{j,n,λ} − ∑_{ℓ=1}^{K−j} \overline{c}_{ℓ,n,λ} c_{j+ℓ,n+j,λ} s_{n−ℓ+m} , 0 ⩽ j ⩽ K, n ⩾ 0,

s_{n} = ⟨ϕ_{n}(z), ϕ_{n}(z)⟩_{U}, n < m, c_{j,n,λ} = 0, n < j ⩽ K, c_{0,n,λ} = 1, n ⩾ 0, where

ζ_{j,n,λ} = ∑_{i=j}^{M} ã_{i,n+j} \overline{ã}_{i−j,n} ⟨ϕ_{n+j−i+m} (z), ϕ_{n+j−i+m} (z)⟩_{U}

+ λ ∑_{i=j}^{N} b̃_{i,n+j} \overline{b̃}_{i−j,n} ⟨ψ_{n+j−i} (z), ψ_{n+j−i} (z)⟩_{V}, 0 ⩽ j ⩽ K,

with s_{n}, ã_{i,n} and b̃_{i,n}, n ⩾ 0, given by (3.12), and K = max{M, N}.

Remark 3.2. • For the recurrence relation appearing in (3.13), we can associate the following matrix with K + 1 rows and infinitely many columns

[ \begin{matrix}s_{m} & s_{m+1} & s_{m+2} & ⋯ \\ 0 & c_{1,1,λ} & c_{1,2,λ} & c_{1,3,λ} & ⋯ \\ 0 & 0 & c_{2,2,λ} & c_{2,3,λ} & c_{2,4,λ} & ⋯ \\ ⋮ & ... & ... & ... & ... & ... & ... \\ 0 & ⋯ & 0 & 0 & c_{K,K,λ} & c_{K,K+1,λ} & c_{K,K+2,λ} & ⋯ \end{matrix} ] ,

which indicates the order for the computation of the sequences {s_{m+n} }_{n⩾0} and {c_{j,n+j,λ} }_{n⩾0} , 1 ⩽ j ⩽ K, through its decreasing diagonals.

• For j = 0, (3.13) becomes the following non-homogeneous linear difference equation of order K satisfied by {s_{n}}_{n⩾0}

s_{n+m} + ∑_{ℓ=1}^{K} |c_{ℓ,n,λ} |^{2}s_{n−ℓ+m} = ζ_{0,n,λ} , n ⩾ 0,

s_{n} = ⟨ϕ_{n}(z), ϕ_{n}(z)⟩_{U}, n < m, and c_{j,n,λ} = 0, n < j ⩽ K, where for n ⩾ 0,

ζ_{0,n,λ} = ∑_{i=0}^{M} |ã_{i,n} |^{2}⟨ϕ_{n−i+m} (z), ϕ_{n−i+m} (z)⟩_{U} + λ ∑_{i=0}^{N} |b̃_{i,n}|^{2}⟨ψ_{n−i} (z), ψ_{n−i} (z)⟩_{V}.

The sequences {s_{n} = ⟨S_{n}(z; λ), S_{n}(z; λ)⟩_{λ}}_{n⩾0} and {c_{j,n,λ} }_{n⩾0} , 1 ⩽ j ⩽ K, fulfill some additional properties in the cases when U and V constitute a (1, 1)- coherent or (1, 0)-coherent pair of order m of Hermitian linear functionals on the unit circle. This will be the topic to be studied in the next two subsections.

3.1. (1, 1) -coherent pairs of order m on the unit circle. Let (U, V) be a (1, 1)-coherent pair of order m of regular Hermitian linear functionals on the unit circle such that

(3.14)  ϕ_{n}^{[m]}(z) + a_{1,n} ϕ_{n−1}^{[m]} (z) = ψ_{n}(z) + b_{1,n} ψ_{n−1} (z), n ⩾ 0,

with a_{1,0} = b_{1,0} = 0. Let {S_{n}(z; λ)}_{n⩾0} be the sequence of monic Sobolev OPUC with respect to inner product (3.1) and let s_{n} = ⟨S_{n}(z; λ), S_{n}(z; λ)⟩_{λ}, n ⩾ 0, denotes their square norm in terms of the Sobolev inner product. Then,

i. From Theorem 3.1, the monic Sobolev OPUC S_{n}(z; λ), n ⩾ 0, satisfy

ϕ_{m+n} (z) + \frac{n + m}{n} a_{1,n} ϕ_{m+n−1} (z) = S_{m+n} (z; λ) + c_{1,n,λ} S_{m+n−1} (z; λ), n ⩾ 0,

and S_{n}(z; λ) = ϕ_{n}(z), n ⩽ m, where

c_{1,n,λ} = \frac{1}{s_{m+n−1}} [ \frac{n + m}{n} a_{1,n} ⟨ϕ_{m+n−1} (z), ϕ_{m+n−1} (z)⟩_{U}
+ λ(n)_{m}(n + 1)_{m}b_{1,n} ⟨ψ_{n−1} (z), ψ_{n−1} (z)⟩_{V}].

In particular, c_{1,0,λ} = 0.

ii. From Theorem 3.2, the sequences {s_{n}}_{n⩾0} and {c_{1,n,λ} }_{n⩾0} satisfy

s_{n+m} c_{1,n+1,λ} = ζ_{1,n,λ} and s_{n+m} = ζ_{0,n,λ} − |c_{1,n,λ} |^{2}s_{n+m−1} , n ⩾ 0,

where

(3.15) \begin{aligned}ζ_{1,n,λ} & = \frac{n + m + 1}{n + 1} a_{1,n+1} ⟨ϕ_{m+n} (z), ϕ_{m+n} (z)⟩_{U} \\ & + λ(n + 1)_{m}(n + 2)_{m}b_{1,n+1} ⟨ψ_{n}(z), ψ_{n}(z)⟩_{V}, \\ ζ_{0,n,λ} & = ⟨ϕ_{n+m} (z), ϕ_{n+m} (z)⟩_{U} + \frac{(n + m)^{2}}{n^{2}} |a_{1,n} |^{2}⟨ϕ_{n+m−1} (z), ϕ_{n+m−1} (z)⟩_{U} \\ & + λ(n + 1)_{m}^{2}[⟨ψ_{n}(z), ψ_{n}(z)⟩_{V} + |b_{1,n} |^{2}⟨ψ_{n−1} (z), ψ_{n−1} (z)⟩_{V}]. \end{aligned}

Therefore, if ζ_{1,n,λ} ≠ 0 for n ⩾ 0, then s_{n} and c_{1,n,λ} , n ⩾ 0, are given by

(3.16) s_{m+n+1} = ζ_{0,n+1,λ} − \frac{|ζ_{1,n,λ} |^{2}}{\overline{s}_{m+n}}, \frac{ζ_{1,n,λ}}{c_{1,n+1,λ}} = ζ_{0,n,λ} − ζ_{1,n−1,λ} \overline{c}_{1,n,λ} , n ⩾ 0,

with c_{1,0,λ} = 0, s_{m} = ζ_{0,0,λ} , s_{n} = ⟨ϕ_{n}(z), ϕ_{n}(z)⟩_{U}, n < m. iii. Rewriting the equations in (3.16) we get c_{1,1,λ} = ζ_{1,0,λ} /ζ_{0,0,λ} , and

\overline{c}_{1,n,λ} = \frac{ζ_{0,n,λ}}{ζ_{1,n−1,λ}} − \frac{\frac{ζ_{1,n,λ}}{ζ_{1,n−1,λ}}}{c_{1,n+1,λ}} , n ⩾ 1, \overline{s}_{m+n} = \frac{|ζ_{1,n,λ} |^{2}}{ζ_{0,n+1,λ} − s_{m+n+1}} , n ⩾ 0.

As a consequence, when ζ_{1,n,λ} ≠ 0 for n ⩾ 0, every constant s_{m+n} , n ⩾ 0, and c_{1,n,λ} , n ⩾ 1, can be represented by continued fractions as

\overline{s}_{m+n} = \frac{|ζ_{1,n,λ} |^{2}}{[ζ_{0,n+1,λ}} − \frac{|ζ_{1,n+1,λ} |^{2}]}{[ζ_{0,n+2,λ}} − ⋯, n ⩾ 0,

\overline{c}_{1,n,λ} = \frac{ζ_{0,n,λ}}{ζ_{1,n−1,λ}} − \frac{\frac{ζ_{1,n,λ}}{ζ_{1,n−1,λ}}}{[ \frac{ζ_{0,n+1,λ}}{ζ_{1,n,λ}}} − \begin{matrix}\frac{ζ_{1,n+1,λ}}{ζ_{1,n,λ}} & ] \\ [ & \frac{ζ_{0,n+2,λ}}{ζ_{1,n+1,λ}} \end{matrix} − ⋯, n ⩾ 1.

iv. From the theory of continued fractions, it is possible to define a sequence {ϖ_{n,λ} }_{n⩾0} as follows

(3.17)  ϖ_{0,λ} = 1 and ϖ_{n+1,λ} = s_{m+n} ϖ_{n,λ} , n ⩾ 0.

So, when s_{n} ∈ ℝ, n ⩾ 0, (for instance, when U and V are positive definite Hermitian linear functionals), then the first equation in (3.16) becomes

ϖ_{n+2,λ} = ζ_{0,n+1,λ} ϖ_{n+1,λ} − |ζ_{1,n,λ} |^{2}ϖ_{n,λ} , n ⩾ 0, ϖ_{1,λ} = ζ_{0,0,λ} , ϖ_{0,λ} = 1.

As a consequence, since ζ_{1,n,λ} ≠ 0, n ⩾ 0, let us consider the sequence of monic polynomials {ϖ_{n}(x; λ)}_{n⩾0} such that ϖ_{n}(0; λ) = ϖ_{n,λ} , n ⩾ 0, satisfying the threeterm recurrence relation

(3.18)  \begin{matrix}ϖ_{n+1} (x; λ) = (x + ζ_{0,n,λ} )ϖ_{n}(x; λ) − |ζ_{1,n−1,λ} |^{2}ϖ_{n−1} (x; λ), n ⩾ 0, \\ ϖ_{0}(x; λ) = 1, ϖ_{−1} (x; λ) = 0. \end{matrix}

From Favard’s theorem, the sequence of monic polynomials {ϖ_{n}(x; λ)}_{n⩾0} is orthogonal with respect to some regular linear functional on ℙ, which will be positive definite when ζ_{0,n,λ} ∈ ℝ. Then, this sequence will be orthogonal with respect to some positive Borel measure supported on the real line. Therefore, the sequences {s_{n}}_{n⩾0} and {c_{1,n,λ} }_{n⩾0} fulfill

s_{m+n} = \frac{ϖ_{n+1} (0; λ)}{ϖ_{n}(0; λ)} , c_{1,n+1,λ} = ζ_{1,n,λ} \frac{ϖ_{n}(0; λ)}{ϖ_{n+1} (0; λ)} , n ⩾ 0.

Remark 3.3. If c_{1,n,λ} ∈ ℝ and ζ_{1,n,λ} ≠ 0 for n ⩾ 0, then the same analysis done in the previous item iv holds for the second equation in (3.16). In fact, using

θ_{n+1,λ} = \frac{ζ_{1,n,λ} /ζ_{1,n−1,λ}}{c_{1,n+1,λ}} θ_{n,λ} , n ⩾ 1, θ_{1,λ} = \frac{ζ_{1,0,λ}}{c_{1,1,λ}} θ_{0,λ} , θ_{0,λ} = 1,

instead of (3.17), we get the existence of a SMOP {θ_{n}(x; λ)}_{n⩾0} , such that θ_{n}(0; λ)= θ_{n,λ} for n ⩾ 0, which satisfies the three-term recurrence relation

(3.19)  \begin{matrix}θ_{n+1} (x; λ) = ( x + \frac{ζ_{0,n,λ}}{ζ_{1,n−1,λ}} ) θ_{n}(x; λ) − \frac{ζ_{1,n−1,λ}}{ζ_{1,n−2,λ}} θ_{n−1} (x; λ), n ⩾ 2, \\ θ_{2}(x; λ) = ( x + \frac{ζ_{0,1,λ}}{ζ_{1,0,λ}} ) θ_{1}(x; λ) − ζ_{1,0,λ} θ_{0}(x; λ), \\ θ_{1}(x; λ) = x + ζ_{0,0,λ} , θ_{0}(x; λ) = 1. \end{matrix}

Additionally, if ζ_{0,n,λ} , ζ_{1,n,λ} ∈ ℝ and ζ_{1,n,λ} > 0 for n ⩾ 0, then {θ_{n}(x; λ)}_{n⩾0} is orthogonal with respect to some positive Borel measure supported on the real line.

Finally, in this case, the sequences {s_{n}}_{n⩾0} and {c_{1,n,λ} }_{n⩾0} satisfy c_{1,1,λ} = ζ_{1,0,λ} \frac{θ_{0} (0;λ)}{θ_{1} (0;λ)} and

c_{1,n+1,λ} = \frac{ζ_{1,n,λ}}{ζ_{1,n−1,λ}} \frac{θ_{n}(0; λ)}{θ_{n+1} (0; λ)}, s_{m+n} = ζ_{1,n−1,λ} \frac{θ_{n+1} (0; λ)}{θ_{n}(0; λ)}, n ⩾ 1.

Example 3.1. In [3, Section 4] it was proved that if (U, V) is a (1, 1)-coherent pair of order m of Hermitian linear functionals on the unit circle given by (3.14), such that U is the Lebesgue linear functional, V is normalized, i.e., v_{0} = 1, |b_{1,1} − a_{1,1}| < 1, and, either a_{1,2} = a_{1,1} − b_{1,1} (i.e., b_{1,2} = 0) or b_{1,N} = 0 for some N ⩾ 2, then a_{1,n} = a_{1,1} − b_{1,1} as well as b_{1,n} = 0 for n ⩾ 2, and V is the Bernstein–Szegő linear functional with parameter −a_{1,2} = b_{1,1} −a_{1,1}. In this way, ⟨ϕ_{n}(z), ϕ_{n}(z)⟩_{U} = ⟨z^{n}, z^{n}⟩_{U} = 1, n ⩾ 0, ⟨ψ_{n}(z), ψ_{n}(z)⟩_{V} = ⟨z^{n}+a_{1,2}z^{n−1} , z^{n}+a_{1,2}z^{n−1} ⟩_{V} = 1−|a_{1,2}|^{2}, n ⩾ 1, and ⟨ψ_{0}(z), ψ_{0}(z)⟩_{V} = 1. Thus, from (3.15), the coefficients ζ_{0,n,λ} and |ζ_{1,n,λ} |^{2}, n ⩾ 0, appearing in the three-term recurrence relation (3.18) satisfied by the associated SMOP {ϖ_{n}(x; λ)}_{n⩾0} , are ζ_{0,0,λ} = 1 + λ(m!)^{2} ∈ ℝ,

ζ_{0,1,λ} = 1 + (m + 1)^{2}|a_{1,1}|^{2} + λ[(m + 1)!]^{2}(1 − |a_{1,2}|^{2} + |b_{1,1}|^{2}) ∈ ℝ,

ζ_{0,n,λ} = 1 + \frac{(n + m)^{2}}{n^{2}} |a_{1,2}|^{2} + λ(n + 1)_{m}^{2}(1 − |a_{1,2}|^{2}) ∈ ℝ, n ⩾ 2,

ζ_{1,0,λ} = (m + 1)a_{1,1} + λ(m!)^{2}(m + 1)b_{1,1}, ζ_{1,n,λ} = \frac{n + m + 1}{n + 1} a_{1,2}, n ⩾ 1,

where a_{1,2} = a_{1,1} − b_{1,1}. Additionally, from previous equations, the coefficients of the three-term recurrence relation (3.19) satisfied by the SMOP {θ_{n}(x; λ)}_{n⩾0} are

\frac{ζ_{0,n,λ}}{ζ_{1,n−1,λ}} = \frac{1 + \frac{(n+m)^{2}}{n^{2}} |a_{1,2}|^{2} + λ(n + 1)_{m}^{2}(1 − |a_{1,2}|^{2})}{\frac{n+m}{n} a_{1,2}} , n ⩾ 2,

\frac{ζ_{0,1,λ}}{ζ_{1,0,λ}} = \frac{1 + (m + 1)^{2}|a_{1,1}|^{2} + λ[(m + 1)!]^{2}(1 − |a_{1,2}|^{2} + |b_{1,1}|^{2})}{(m + 1)a_{1,1} + λ(m!)^{2}(m + 1)b_{1,1}} ,

\frac{ζ_{1,n,λ}}{ζ_{1,n−1,λ}} = \frac{n(n + m + 1)}{(n + 1)(n + m)} , n ⩾ 2, \frac{ζ_{1,1,λ}}{ζ_{1,0,λ}} = \frac{\frac{m+2}{2} a_{1,2}}{(m + 1)a_{1,1} + λ(m!)^{2}(m + 1)b_{1,1}} .

3.2. (1, 0) -coherent pairs of order m on the unit circle. When the regular Hermitian linear functionals U and V form a (1, 0)-coherent pair of order m on the unit circle, i.e., (3.14) reads

ϕ_{n}^{[m]}(z) + a_{1,n} ϕ_{n−1}^{[m]} (z) = ψ_{n}(z), n ⩾ 0, a_{1,0} = 0, a_{1,n} ≠ 0, n ⩾ 1,

all the results obtained in the previous subsection hold taking b_{1,n} = 0, for n ⩾ 0. In particular, the coefficients ζ_{1,n,λ} and ζ_{0,n,λ} , n ⩾ 0, become

ζ_{1,n,λ} = \frac{n + m + 1}{n + 1} a_{1,n+1} ⟨ϕ_{m+n} (z), ϕ_{m+n} (z)⟩_{U},
ζ_{0,n,λ} = ⟨ϕ_{n+m} (z), ϕ_{n+m} (z)⟩_{U} + \frac{(n + m)^{2}}{n^{2}} |a_{1,n} |^{2}⟨ϕ_{n+m−1} (z), ϕ_{n+m−1} (z)⟩_{U}
+ λ(n + 1)_{m}^{2}⟨ψ_{n}(z), ψ_{n}(z)⟩_{V},

i.e., constants and polynomials in the variable λ of degree 1. In this way, in the item iv, it is possible to show that ϖ_{n}(0; λ) is a polynomial in λ of degree n with leading coefficient ∏ _{j=0}^{n−1} (j + 1)_{m}^{2}⟨ψ_{j} (z), ψ_{j} (z)⟩_{V}, n ⩾ 1, using induction on n and (3.18). Consequently, since a_{1,n} ≠ 0 for n ⩾ 1, from (3.18), the monic polynomials ϖ̃_{n} (λ) = ϖ_{n}(0; λ)/(∏ _{j=0}^{n−1} (j + 1)_{m}^{2}⟨ψ_{j} (z), ψ_{j} (z)⟩_{V}), n ⩾ 1, are a SMOP in the variable λ because they satisfy the following three-term recurrence relation

(3.20)  ϖ̃_{n+1} (λ) = (λ + α_{n})ϖ̃_{n} (λ) − β_{n} ϖ̃_{n−1} (λ), n ⩾ 0, ϖ̃_{0} (λ) = 1,

where α_{0} = \frac{⟨ϕ_{m} (z),ϕ_{m} (z)⟩_{U}}{(1)_{m}^{2} ⟨ψ_{0} (z),ψ_{0} (z)⟩_{V}} , β_{0} = 0,

α_{n} = \frac{⟨ϕ_{n+m} (z), ϕ_{n+m} (z)⟩_{U} + \frac{(n+m)^{2}}{n^{2}} |a_{1,n} |^{2}⟨ϕ_{n+m−1} (z), ϕ_{n+m−1} (z)⟩_{U}}{(n + 1)_{m}^{2}⟨ψ_{n}(z), ψ_{n}(z)⟩_{V}} , n ⩾ 1,

β_{n} = \frac{|a_{1,n} |^{2}|⟨ϕ_{m+n−1} (z), ϕ_{m+n−1} (z)⟩_{U}|^{2}}{(n)_{m}^{4}⟨ψ_{n}(z), ψ_{n}(z)⟩_{V}⟨ψ_{n−1} (z), ψ_{n−1} (z)⟩_{V}} , n ⩾ 1.

As above, {ϖ̃_{n+1} (λ)}_{n⩾0} is the SMOP with respect to some positive Borel measure supported on the real line if α_{n} ∈ ℝ and β_{n+1} > 0 for n ⩾ 0 (for instance, when U and V are positive definite Hermitian linear functionals).

Finally, the sequences {s_{n}}_{n⩾0} and {c_{1,n,λ} }_{n⩾0} satisfy

s_{m+n} = κ_{n} \frac{ϖ̃_{n+1} (λ)}{ϖ̃_{n} (λ)} , n ⩾ 0, c_{1,n,λ} = κ̃_{n} \frac{ϖ̃_{n−1} (λ)}{ϖ̃_{n} (λ)} , n ⩾ 1,

with

κ_{n} = (n + 1)_{m}^{2}⟨ψ_{n}(z), ψ_{n}(z)⟩_{V}, κ̃_{n} = a_{1,n} \frac{(n + 1)_{m}}{(n)_{m}^{3}} \frac{⟨ϕ_{m+n−1} (z), ϕ_{m+n−1} (z)⟩_{U}}{⟨ψ_{n−1} (z), ψ_{n−1} (z)⟩_{V}} .

Example 3.2. In a similar way as in Example 3.1, taking into account that b_{1,n} = 0, n ⩾ 0, let us consider a (1, 0)-coherent pair of order m, (U, V), such that U is the Lebesgue linear functional and V is normalized, i.e., v_{0} = 1. If |a_{1,1}| < 1, then a_{1,n} = a_{1,1}, n ⩾ 2, and V is the Bernstein–Szegő linear functional with parameter −a_{1,1}. Hence, ⟨ϕ_{n}(z), ϕ_{n}(z)⟩_{U} = 1, n ⩾ 0, ⟨ψ_{n}(z), ψ_{n}(z)⟩_{V} = 1 − |a_{1,1}|^{2}, n ⩾ 1, ⟨ψ_{0}(z), ψ_{0}(z)⟩_{V} = 1. As a consequence, ζ_{0,0,λ} = 1 + λ(m!)^{2}, and for n ⩾ 1,

ζ_{1,n−1,λ} = \frac{(n + m)a_{1,1}}{n} , ζ_{0,n,λ} = 1 + \frac{(n + m)^{2}|a_{1,1}|^{2}}{n^{2}} + λ(n + 1)_{m}^{2}(1 − |a_{1,1}|^{2}).

Besides, α_{0} = \frac{1}{(1)_{m}^{2}} , β_{0} = 0, β_{1} = \frac{|a_{1,1} |^{2}}{(1)_{m}^{4} (1−|a_{1,1} |^{2} )} > 0,

α_{n} = \frac{1 + \frac{(n+m)^{2}}{n^{2}} |a_{1,1}|^{2}}{(n + 1)_{m}^{2}(1 − |a_{1,1}|^{2})} ∈ ℝ, n ⩾ 1, β_{n} = \frac{|a_{1,1}|^{2}}{(n)_{m}^{4}(1 − |a_{1,1}|^{2})^{2}} > 0, n ⩾ 2,

are the coefficients of the three-term recurrence relation (3.20) satisfied by the corresponding SMOP {ϖ̃_{n} (λ)}_{n⩾0} .

4. A matrix interpretation of (M, N) -coherence on the unit circle

Let U and V be two regular Hermitian linear functionals defined on Λ, and let {ϕ_{n}(z)}_{n⩾0} and {ψ_{n}(z)}_{n⩾0} be their corresponding sequences of monic OPUC. (U, V) is said to be a (M, N)-coherent pair on the unit circle if

(4.1)  \frac{ϕ′_{n+1} (z)}{n + 1} + ∑_{i=1}^{M} a_{i,n} \frac{ϕ′_{n−i+1} (z)}{n − i + 1} = ψ_{n}(z) + ∑_{i=1}^{N} b_{i,n} ψ_{n−i} (z), n ⩾ 0,

holds, where M, N are non-negative integers, a_{i,n} , b_{i,n} , n ⩾ 0 are complex numbers such that a_{M,n} ≠ 0 for n ⩾ M, b_{N,n} ≠ 0 for n ⩾ N, and a_{i,n} = b_{i,n} = 0 for i > n.

If we consider the infinite dimensional vectors

(4.2)  \begin{matrix}Φ(z) = [ϕ_{0}(z), ϕ_{1}(z), ⋯]^{T} , Φ_{1}(z) = [ϕ_{1}(z), ϕ_{2}(z), ⋯]^{T} \\ Ψ(z) = [ψ_{0}(z), ψ_{1}(z), ⋯]^{T} , \end{matrix}

then we can formulate the (M, N)-coherence relation (4.1) in a matrix form as

(4.3)  A_{1}Φ′_{1}(z) = BΨ(z),
(4.4)  AΦ′(z) = BΨ(z),

where A, A_{1}, and B are infinite matrices whose 0th rows (counting the rows from zero) are [1 1 0 ⋯], [1 0 ⋯], and [1 0 ⋯], respectively, and their corresponding nth rows, for n ⩾ 1, are

↓^{nth position}
[ \underbrace{0 ⋯ 0}_{n−M+1 zeros} \frac{a_{M,n}}{n−M+1} ⋯ \frac{a_{2,n}}{n−1}  \frac{a_{1,n}}{n}  \frac{1}{n+1} 0 ⋯]

[ \underbrace{0 ⋯ 0}_{n−M zeros} \frac{a_{M,n}}{n−M+1} ⋯ \frac{a_{1,n}}{n}  \frac{1}{n+1} 0 ⋯]

[ \underbrace{0 ⋯ 0}_{n−N zeros} b_{N,n} ⋯ b_{1,n} 1 0 ⋯]

respectively. Thus, A is a lower Hessenberg matrix with M + 1 nonzero diagonals, whose entries of its superdiagonal are \frac{1}{n+1} , n ⩾ 0, and the entries of its main diagonal are 1 and \frac{a_{1,n}}{n} , n ⩾ 1, and A_{1} and B are nonsingular lower triangular matrices with M + 1 and N + 1 nonzero diagonals, respectively, such that the entries of their main diagonals are \frac{1}{n+1} , n ⩾ 0, and 1’s, respectively.

Also, we can consider the multiplication operators by z in terms of the bases {ϕ_{n}(z)}_{n⩾0} and {ψ_{n}(z)}_{n⩾0} given in (2.2),

(4.5)  zΦ(z) = H_{ϕ}Φ(z) and zΨ(z) = H_{ψ} Ψ(z),

where H_{ϕ} and H_{ψ} are the infinite lower Hessenberg matrices associated with U and V, respectively. So, in this context, first we state a general result and, as a consequence, we apply it to coherent pairs .

Lemma 4.1. If the sequences of monic OPUC {ϕ_{n}(z)}_{n⩾0} , {ψ_{n}(z)}_{n⩾0} satisfy

(4.6)  Φ′(z) = MΨ(z),

where Φ(z) and Ψ(z) are given by (4.2) and M is an infinite matrix (such that its 0th row is zero since ϕ′_{0}(x) = 0), then

(4.7)  H_{ϕ}^{2}M − 2H_{ϕ}MH_{ψ} + MH_{ψ}^{2} = 0,

where H_{ϕ} and H_{ψ} are the Hessenberg matrices associated with {ϕ_{n}(z)}_{n⩾0} and {ψ_{n}(z)}_{n⩾0} , respectively.

Proof. We have that

MH_{ψ} Ψ(z) + Φ(z) =^{(4.6)}_{(4.5)} zΦ′(z) + Φ(z) = (zΦ(z))′ =^{(4.5)} H_{ϕ}Φ′e(z) =^{(4.6)} H_{ϕ}MΨ(z),

or, equivalently,

(4.8)  Φ(z) = (H_{ϕ}M − MH_{ψ} )Ψ(z).

Hence,

H_{ϕ}(H_{ϕ}M − MH_{ψ} )Ψ(z) =^{(4.8)} H_{ϕ}Φ(z) =^{(4.8)}_{(4.5)} (H_{ϕ}M − MH_{ψ} )H_{ψ} Ψ(z).  □

Proposition 4.1. If (U, V) is an (M, N)-coherent pair of regular Hermitian linear functionals on the unit circle given by (4.3), then

H_{ϕ}^{2} [ \begin{matrix}0 \\ A_{1}^{−1} B \end{matrix}] − 2H_{ϕ} [ \begin{matrix}0 \\ A_{1}^{−1} B \end{matrix}] H_{ψ} + [ \begin{matrix}0 \\ A_{1}^{−1} B \end{matrix}] H_{ψ}^{2} = 0,

where 0 = [0 0 ⋯] is the zero row.

Proof. The result is a straightforward consequence of Lemma 4.1 taking as M the matrix obtained from A_{1}^{−1} B by shifting the matrix one position downward, i.e., adding a zero row to top. □

Proposition 4.2. Let (U, V) be an (M, N)-coherent pair of regular Hermitian linear functionals on the unit circle given by (4.4), such that the matrix A is a nonsingular matrix (e.g., when M = 1 and N ⩾ 0, we have that a_{1,n} ≠ 0, n ⩾ 1, and, as a consequence, A is a nonsingular upper bidiagonal matrix). Let consider the matrices M_{ϕ} and M_{ψ} , which are similar to the corresponding Hessenberg matrices of U and V, respectively, such that

(4.9)  M_{ϕ} = AH_{ϕ}A^{−1} and M_{ψ} = BH_{ψ} B^{−1} .

Then,

(M_{ϕ} − M_{ψ} )^{2} = [M_{ϕ}, M_{ψ} ],

where [M_{ϕ}, M_{ψ} ] is the commutator of M_{ϕ} and M_{ψ} defined by [M_{ϕ}, M_{ψ} ] = M_{ϕ}M_{ψ} − M_{ψ} M_{ϕ}.

Proof. From (4.4) and (4.7), H_{ϕ}^{2}A^{−1} B − 2H_{ϕ}A^{−1} BH_{ψ} + A^{−1} BH_{ψ}^{2} = 0 holds. Therefore, multiplying on the left by A and on the right by B^{−1} in both sides of the previous equation and taking into account (4.9), the previous equation becomes 0 = M_{ϕ}^{2} − 2M_{ϕ}M_{ψ} + M_{ψ}^{2} = (M_{ϕ} − M_{ψ} ) ^{2} − [M_{ϕ}, M_{ψ} ] . □

Remark 4.1. When U is the Lebesgue linear functional we get ϕ_{n}^{[m]}(z) = z^{n} = ϕ_{n}(z), m, n ⩾ 0. So, if (U, V) is an (M, N)-coherent pair of order m, m ⩾ 0, and U is the Lebesgue linear functional, then (3.7) becomes

ϕ_{n}(z) + ∑_{i=1}^{M} a_{i,n} ϕ_{n−i} (z) = ψ_{n}(z) + ∑_{i=1}^{N} b_{i,n} ψ_{n−i} (z), n ⩾ 0,

which can be written in a matrix form as

(4.10)  Â_{1}Φ(z) = BΨ(z),

where Φ(z), Ψ(z), B are given as in (4.4), and the infinite matrix Â_{1} is as B, i.e., it is a lower triangular matrix with M + 1 nonzero diagonals, whose entries of its main diagonal are all 1’s and its nth row for n ⩾ 1, counting the rows from zero, is

[ \underbrace{0 ⋯ 0}_{n−M zeros} a_{M,n} ⋯ a_{1,n} 1 0 ⋯]

Consequently, Â_{1} is nonsingular as B. Hence,

Â_{1}H_{ϕ} Â_{1}^{−1} BΨ(z) =^{(4.10 )} Â_{1}H_{ϕ}Φ(z) =^{(4.5)} zÂ_{1}Φ(z) =^{(4.10 )} zBΨ(z) =^{(4.5)} BH_{ψ} Ψ(z),

from which we can conclude that Â_{1}H_{ϕ} Â_{1}^{−1} = M̂_{ϕ} = M_{ψ} = BH_{ψ} B^{−1} , where H_{ϕ} and H_{ψ} are the Hessenberg matrices associated with U and V, respectively. Therefore, H_{ϕ} and H_{ψ} are similar matrices.

4.1. A matrix interpretation of Sobolev orthogonal polynomials and (M, N) -coherence of order m on the unit circle. Let us recall that there is a close relation between Sobolev orthogonal polynomials (M, N)-coherent pairs of order m of regular Hermitian linear functionals on the unit circle, m ⩾ 1. More precisely, in Theorem 3.1 we showed that the (M, N)-coherence relation of order m on the unit circle (3.7) satisfied by the sequences of monic OPUC associated with the regular Hermitian linear functionals U and V implies (3.10), i.e.,

ϕ_{n+m} (z) + ∑_{i=1}^{M} ã_{i,n} ϕ_{n−i+m} (z) = S_{n+m} (z; λ) + ∑_{j=1}^{K} c_{j,n,λ} S_{n−j+m} (z; λ), n ⩾ 0,

and S_{n}(z; λ) = ϕ_{n}(z), n ⩽ m, where K = max{M, N}, c_{j,n,λ} = 0, n < j ⩽ K, ã_{i,n} = \frac{(n+1)_{m}}{(n−i+1)_{m}} a_{i,n} , n ⩾ 0, ã_{i,n} = 0, i > n, ã_{0,n} = 1, n ⩾ 0. Here S_{n}(z; λ), n ⩾ 0, are the monic Sobolev polynomials on the unit circle orthogonal with respect to inner product (3.1). In this way, if we write these algebraic relations as

ÃΦ(z) = Cs(z; λ),

where Φ(z) = [ϕ_{0}(z), ϕ_{1}(z), ⋯]^{T}, s(z; λ) = [S_{0}(z; λ), S_{1}(z; λ), ⋯]^{T}, and

↓^{(n+m)th position}
[ \underbrace{0 ⋯ 0}_{n−M+m zeros} ã_{M,n} ⋯ ã_{1,n} 1 0 ⋯]

[ \underbrace{0 ⋯ 0}_{n−K+m zeros} c_{K,n,λ} ⋯ c_{1,n,λ} 1 0 ⋯]

are the (n + m)th rows, respectively, n ⩾ 0, and [0 ⋯0 1 0 ⋯] are the first m rows, counting from zero, of the lower triangular matrices Ã and C (notice that these matrices have M + 1 and K + 1 nonzero diagonals, respectively, and 1’s as entries of their main diagonal), then we obtain the following result.

Proposition 4.3. If (U, V) is an (M, N)-coherent pair of order m, m ⩾ 1, of regular Hermitian linear functionals on the unit circle, then the matrix representation of the multiplication operator by z in terms of {S_{n}(z; λ)}_{n⩾0} , the basis of monic Sobolev OPUC with respect to the inner product (3.1), is given by

zs(z; λ) = H_{S,ϕ,λ} s(z; λ),

where H_{S,ϕ,λ} is a lower Hessenberg matrix similar to the Hessenberg matrix associated with the linear functional U.

Proof. Since

zs(z; λ) =^{(4.1)} C^{−1} ÃzΦ(z) =^{(2.2)} C^{−1} ÃH_{ϕ}Φ(z) =^{(4.1)} C^{−1} ÃH_{ϕ} Ã^{−1} Cs(z; λ),

and if H_{S,ϕ,λ} = C^{−1} ÃH_{ϕ} Ã^{−1} C = [C^{−1} Ã]H_{ϕ}[C^{−1} Ã]^{−1} , then the proof is complete.

□

References

1. R. Álvarez-Nodarse, J. Petronilho, N. C. Pinzón-Cortés, R. Sevinik-Adıgüzel, On linearly related sequences of difference derivatives of discrete orthogonal polynomials, J. Comput. Appl. Math., in press

2. A. Branquinho, A. Foulquié-Moreno, F. Marcellán, M. N. Rebocho, Coherent pairs of linear functionals on the unit circle, J. Approx. Theory 153 (2008), 122–137.

3. L. Garza, F. Marcellán, N. C. Pinzón-Cortés, (1, 1)-Coherent pairs on the unit circle, Abstract Appl. Anal. (2013), Article ID 307974, 8 pages

4. A. Iserles, P. E. Koch, S. P. Nørsett, J. M. Sanz-Serna, On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Theory 65 (1991), 151–175.

5. M. N. de Jesus, F. Marcellán, J. Petronilho, N. C. Pinzón-Cortés, (M, N)-coherent pairs of order (m, k) and Sobolev orthogonal polynomials, J. Comput. Appl. Math. 256 (2014), 16–35.

6. M. N. de Jesus, J. Petronilho, Sobolev orthogonal polynomials and (m, n)-coherent pairs of measures, J. Comput. Appl. Math. 237 (2013), 83–101.

7. F. Marcellán, N. C. Pinzón-Cortés, (M, N)-oherent pairs of linear functionals and Jacobi matrices, Appl. Math. Comput. 232 (2014), 76–83.

8. , Higher order coherent pairs, Acta Appl. Math. 121 (2012), 105–135.

9. H. G. Meijer, A short history of orthogonal polynomials in a Sobolev space, I. The nondiscrete case, Nieuw Arch. Wisk. 14 (1996), 93–112.

10. B. Simon, Orthogonal Polynomials on the Unit Circle, 2 vols. Am. Math. Soc. Colloq. Publ. Series. 54, Am. Math. Soc., Providence, Rhode Island, 2005.

11. G. Szegő, Orthogonal Polynomials, (fourth edition) Am. Math. Soc. Colloq. Publ. Series 23, Am. Math. Soc., Providence, Rhode Island, 1975.

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