Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 307974, 8 pages
http://dx.doi.org/10.1155/2013/307974
Research Article
(1, 1)-Coherent Pairs on the Unit Circle
Luis Garza,
1
Francisco Marcellán,
2
and Natalia C. Pinzón-Cortés
2
1
Facultad de Ciencias, Universidad de Colima, Bernal D´ ıaz del Castillo 340, 28045 Colima, COL, Mexico
2
Departamento de Matem´ aticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Legan´ es, Spain
Correspondence should be addressed to Luis Garza; garzaleg@gmail.com
Received 23 July 2013; Accepted 19 September 2013
Academic Editor: Jinde Cao
Copyright © 2013 Luis Garza et al. Tis 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 {
()}
≥0
and {
()}
≥0
satisfy
[1]
()+
[1]
−1
() =
()+
−1
(),
̸ =0, ≥1,
where
[1]
() =
+1
()/( + 1). In this contribution, we consider the cases when U is the linear functional associated with the
Lebesgue and Bernstein-Szeg˝ o measures, respectively, and we obtain a classifcation 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 coefcients P is a (1, 1)-coherent
pair if and only if their corresponding sequences of monic
orthogonal polynomials (SMOP) {
()}
≥0
and {
()}
≥0
satisfy
+1
()
+1
+
()
=
() +
−1
(),
̸ = 0, ≥ 1.
(1)
Tis concept is a generalization of the notion of coherent pair,
for us (1, 0)-coherent pair, introduced by Iserles et al. in [1],
where
=0, for every ≥1.
In the work by Delgado and Marcell´ an [2], the notion
of a generalized coherent pair of measures, in short, (1, 1)-
coherent pair of measures, arose as a necessary and sufcient
condition for the existence of an algebraic relation between
the SMOP {
(; )}
≥0
associated with the Sobolev inner
product
⟨(),()⟩
=∫
R
()()
0
+∫
R
()
()
1
, > 0, , ∈ P,
(2)
and the SMOP {
()}
≥0
associated with the positive Borel
measure
0
in the real line as follows:
+1
(;)+
()
(; )
=
+1
() +
+1
(), ≥ 1,
(3)
where {
()}
≥1
are rational functions in >0. Besides,
they obtained the classifcation 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. Tis 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.
Te 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
∑
=0
−,,
()
+−
()
=
∑
=0
−,,
()
+−
(), ≥ 0,
(4)