Orthogonal Polynomials Associated with Complementary Chain Sequences
Using the minimal parameter sequence of a given chain sequence, we introduce the concept of complementary chain sequences, which we view as perturbations of chain sequences. Using the relation between these complementary chain sequences and the corresponding Verblunsky coefficients, the para-orthogo...
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nasplib_isofts_kiev_ua-123456789-1478412025-02-23T17:45:15Z Orthogonal Polynomials Associated with Complementary Chain Sequences Behera, K.K. Sri Ranga, A. Swaminathan, A. Using the minimal parameter sequence of a given chain sequence, we introduce the concept of complementary chain sequences, which we view as perturbations of chain sequences. Using the relation between these complementary chain sequences and the corresponding Verblunsky coefficients, the para-orthogonal polynomials and the associated Szegő polynomials are analyzed. Two illustrations, one involving Gaussian hypergeometric functions and the other involving Carathéodory functions are also provided. A connection between these two illustrations by means of complementary chain sequences is also observed. This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html. nts The authors wish to thank the anonymous referees for their constructive criticism that resulted in significant improvement of the content leading to the final version. The work of the second author was supported by funds from CNPq, Brazil (grants 475502/2013-2 and 305073/2014-1) and FAPESP, Brazil (grant 2009/13832-9). 2016 Article Orthogonal Polynomials Associated with Complementary Chain Sequences / K.K. Behera, A. Sri Ranga, A. Swaminathan // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 42C05; 33C45; 30B70 DOI:10.3842/SIGMA.2016.075 https://nasplib.isofts.kiev.ua/handle/123456789/147841 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України |
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Using the minimal parameter sequence of a given chain sequence, we introduce the concept of complementary chain sequences, which we view as perturbations of chain sequences. Using the relation between these complementary chain sequences and the corresponding Verblunsky coefficients, the para-orthogonal polynomials and the associated Szegő polynomials are analyzed. Two illustrations, one involving Gaussian hypergeometric functions and the other involving Carathéodory functions are also provided. A connection between these two illustrations by means of complementary chain sequences is also observed. |
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Behera, K.K. Sri Ranga, A. Swaminathan, A. |
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Behera, K.K. Sri Ranga, A. Swaminathan, A. Orthogonal Polynomials Associated with Complementary Chain Sequences Symmetry, Integrability and Geometry: Methods and Applications |
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Behera, K.K. Sri Ranga, A. Swaminathan, A. |
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Behera, K.K. |
| title |
Orthogonal Polynomials Associated with Complementary Chain Sequences |
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Orthogonal Polynomials Associated with Complementary Chain Sequences |
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Orthogonal Polynomials Associated with Complementary Chain Sequences |
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Orthogonal Polynomials Associated with Complementary Chain Sequences |
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Orthogonal Polynomials Associated with Complementary Chain Sequences |
| title_sort |
orthogonal polynomials associated with complementary chain sequences |
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Інститут математики НАН України |
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2016 |
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Orthogonal Polynomials Associated with Complementary Chain Sequences / K.K. Behera, A. Sri Ranga, A. Swaminathan // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 27 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT beherakk orthogonalpolynomialsassociatedwithcomplementarychainsequences AT srirangaa orthogonalpolynomialsassociatedwithcomplementarychainsequences AT swaminathana orthogonalpolynomialsassociatedwithcomplementarychainsequences |
| first_indexed |
2025-11-24T04:39:15Z |
| last_indexed |
2025-11-24T04:39:15Z |
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1849645235398770688 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 075, 17 pages
Orthogonal Polynomials Associated
with Complementary Chain Sequences?
Kiran Kumar BEHERA †, A. SRI RANGA ‡ and A. SWAMINATHAN †
† Department of Mathematics, Indian Institute of Technology Roorkee,
Uttarakhand-247667, India
E-mail: krn.behera@gmail.com, mathswami@gmail.com
‡ Departamento de Matemática Aplicada, IBILCE, UNESP-Univ. Estadual Paulista,
15054-000, São José do Rio Preto, SP, Brazil
E-mail: ranga@ibilce.unesp.br
Received March 17, 2016, in final form July 22, 2016; Published online July 27, 2016
http://dx.doi.org/10.3842/SIGMA.2016.075
Abstract. Using the minimal parameter sequence of a given chain sequence, we intro-
duce the concept of complementary chain sequences, which we view as perturbations of
chain sequences. Using the relation between these complementary chain sequences and the
corresponding Verblunsky coefficients, the para-orthogonal polynomials and the associated
Szegő polynomials are analyzed. Two illustrations, one involving Gaussian hypergeometric
functions and the other involving Carathéodory functions are also provided. A connection
between these two illustrations by means of complementary chain sequences is also observed.
Key words: chain sequences; orthogonal polynomials; recurrence relation; Verblunsky
coefficients; continued fractions; Carathéodory functions; hypergeometric functions
2010 Mathematics Subject Classification: 42C05; 33C45; 30B70
1 Preliminaries on Szegő polynomials
The Szegő polynomials {Φn}, also referred to as orthogonal polynomials on the unit circle
(OPUC), enjoy the orthogonality property∫
∂D
(z̄)jΦn(z)dµ(z) =
∫
∂D
(z)−jΦn(z)dµ(z) = 0 for j = 0, 1, . . . , n− 1, n ≥ 1.
Here µ(z) = µ(eiθ) is a nontrivial measure defined on the unit circle ∂D = {z = eiθ : 0 ≤ θ ≤ 2π}.
Denoting the orthonormal Szegő polynomials by φn(z) = χnΦn(z), we also have the equivalent
definition∫
∂D
φn(z)φm(z)dµ(z) = δm,n.
Further, defining the moments µn =
∫
∂D e
−inθdµ(θ), n = 0,±1, . . . , where µ−n = µ̄n, we have∫
∂D
(z̄)nΦn(z)dµ(z) =
∆n
∆n−1
6= 0, n = 0, 1, . . . .
Here ∆n = det{µi−j}ni,j=0 are the associated Toeplitz matrices with ∆−1 = 1.
The monic Szegő polynomials satisfy the first order recurrence relations
Φn(z) = zΦn−1(z)− ᾱn−1Φ∗n−1(z), Φ∗n(z) = −αn−1zΦn−1(z) + Φ∗n−1(z), n ≥ 1,
?This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applica-
tions. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html
mailto:krn.behera@gmail.com
mailto:mathswami@gmail.com
mailto:ranga@ibilce.unesp.br
http://dx.doi.org/10.3842/SIGMA.2016.075
http://www.emis.de/journals/SIGMA/OPSFA2015.html
2 K.K. Behera, A. Sri Ranga and A. Swaminathan
where Φ∗n(z) = znΦn(1/z̄). The complex numbers αn−1 = −Φn(0) are called the Verblunsky
coefficients [22]. The Verblunsky coefficients completely characterize the Szegő polynomials
in the sense that any sequence {αn−1}∞n=1 lying within the unit circle gives rise to a unique
probability measure µ(z) which leads to a unique sequence of Szegő polynomials. The above
result, called the Verblunsky theorem in [22], is the analogue of Favard’s theorem on the real
line. Conversely, algorithms exist in the literature that extracts these coefficients from any
given Szegő system of orthogonal polynomials. Notable among them are the Schur algorithm,
the Levinson algorithm and their modified versions given in [7, 14, 26].
The Szegő polynomials also satisfy the three term recurrence relation
Φn+1(z) =
(
Φn+1(0)
Φn(0)
+ z
)
Φn(z)− (1− |Φn(0)|2)Φn+1(0)
Φn(0)
zΦn−1(z), n ≥ 1, (1.1)
with Φ0(z) = 1 and Φ1(z) = z + Φ1(0). Note that if Φn(0) = 0, n ≥ 1, then the three term
recurrence relation ceases to exist. In such a case, Φn(z) = zn, which is given as the free case in
[22, p. 85]. Denoting
ηn+1 =
Φn+1(0)
Φn(0)
and ρn+1 =
(
1− |Φn(0)|2
)
Φn+1(0)
Φn(0)
, n ≥ 1,
the following expressions are easily obtained from (1.1):
Φn+1(0) =
Φn(0)
1− |Φn(0)|2
∫
∂D zΦn(z)dµ(z)∫
∂D zΦn−1(z)dµ(z)
, 1− |Φn(0)|2 =
∫
∂D z
−nΦn(z)dµ(z)∫
∂D z
−(n−1)Φn−1(z)dµ(z)
and
χ−2n =
ρ2ρ3 · · · ρn+1
η2η3 · · · ηn+1
µ0 = µ0
(
1− |Φ1(0)|2
)(
1− |Φ2(0)|2
)
· · ·
(
1− |Φn(0)|2
)
. (1.2)
For early developments on the subject, we refer to the monographs [8, 10, 25]. For a compendium
of modern research in the area as well as historical notes, we refer to [22, 23].
In order to develop a quadrature formula on the unit circle, Jones et al. [15] introduced
the para-orthogonal polynomials which vanish only on the unit circle and, for z, ωn ∈ C with
|ωn| = 1, have the representation
Xn(z, ωn) = Φn(z) + ωnΦ∗n(z), n ≥ 1.
The para-orthogonal polynomials satisfy the properties
〈Xn, zm〉 = 0, m = 1, 2, . . . , n− 1, 〈Xn, 1〉 6= 0, 〈Xn, zn〉 6= 0,
which are termed as deficiency in the orthogonality of these para-orthogonal polynomials. In
recent years, these para-orthogonal polynomials have been linked to kernel polynomials Kn(z, ω),
see [6, 11, 27]. The kernel polynomials Kn(z, ω) satisfy the Christoffel–Darboux formula
Kn(z, ω) =
n∑
k=0
φk(z)φk(ω) =
φ∗n+1(z)φ
∗
n+1(ω)− φn+1(z)φn+1(ω)
1− zω̄
.
Denoting τn(ω) = Φn(ω)/Φ∗n(ω), for n ≥ 1, the monic kernel polynomials related to the Szegő
polynomials are given by
Pn(ω; z) =
zΦn(z)− ωτn(ω)Φ∗n(z)
z − ω
, n ≥ 1, (1.3)
Orthogonal Polynomials Associated with Complementary Chain Sequences 3
and are shown in [6] to satisfy a three term recurrence relation of the form
Pn+1(ω; z) = [z + bn+1(ω)]Pn(ω; z)− an+1(ω)zPn−1(ω; z), n ≥ 1,
where
bn(ω) =
τn(ω)
τn−1(ω)
, an+1 = [1 + τn(ω)αn−1]
[
1− ωτn(ω)αn
]
ω, n ≥ 1.
The polynomials Pn(ω; z) are τn(w)-invariant sequences of polynomials which can be easily
verified from (1.3). Note that a sequence of polynomials {Yn} is called τn-invariant if [15]
Y∗n(z) = τnYn(z), n ≥ 1.
An important concept that is used in the sequel is the theory of chain sequences. We give a brief
introduction to chain sequences and then illustrate the role played by them in the theory of
OPUC.
A sequence {dn}∞n=1 which satisfies
dn = (1− gn−1)gn, n ≥ 1,
is called a positive chain sequence [5] (see also [12, Section 7.2]). Here {gn}∞n=0, called the
parameter sequence is such that 0 ≤ g0 < 1, 0 < gn < 1 for n ≥ 1. This is a stronger
condition than the one used in [26], in which dn is also allowed to be zero. The parameter
sequence {gn}∞n=0 is called a minimal parameter sequence and denoted by {mn}∞n=0 if m0 = 0.
Every chain sequence has a minimal parameter sequence [5, pp. 91–92]. Further, for a fixed
chain sequence {dn}n≥1, let G be the set of all parameter sequences {gk} of {dn}n≥1. Let the
sequence {Mn}∞n=0 be defined by
Mn = inf{gn, for each n, {gk} ∈ G}, n ≥ 0,
where inf is infimum of the set. Then, {Mn} is called the maximal parameter sequence of {dn}.
The role of chain sequences in the study of orthogonal polynomials on the real line is well
known. Similarly, a positive chain sequence {dn} appears in the three term recurrence relation for
the polynomials Rn(z), that turn out to be scaled versions of the (kernel) polynomials Pn(1, z),
namely,
Rn+1(z) = [(1 + icn+1)z + (1− icn+1)]Rn(z)− 4dn+1zRn−1(z), n ≥ 1, (1.4)
with R0(z) = 1 and R1(z) = (1 + ic1)z + (1− ic1). It is indeed shown in [6], that
Rn(z) =
n−1∏
j=0
[1− τjαj ]
n−1∏
j=0
[1− Re(τjαj)]
Pn(1; z),
where
τj = τj(1) =
j∏
k=1
1− ick
1 + ick
, j ≥ 1,
on condition that
cn =
− Im(τn−1αn−1)
1− Re(τn−1αn−1)
and dn+1 = (1− gn)gn+1, n ≥ 1,
4 K.K. Behera, A. Sri Ranga and A. Swaminathan
is a chain sequence with parameter sequence
gn =
1
2
|1− τn−1αn−1|2
[1− Re(τn−1αn−1)]
, n ≥ 1.
It is also not difficult to verify that in this case Rn(z) has rn,n =
n∏
k=1
(1 + ick) as the leading
coefficient and rn,0 = r̄n,n =
n∏
k=1
(1− ick) as the constant term.
It is known that {Rn(z)} can be used to obtain a sequence of OPUC [2, 3, 6], with respect
to the measure µ(z) and having the shifted sequence {αn−1}∞n=1 as the Verblunsky coefficients.
A further interesting fact is that the above parameter sequence {gn+1}∞n=0 is such that g1 =
(1− ε)M1, (0 ≤ ε < 1), where {Mn+1}∞n=0 is the maximal parameter sequence of {dn+1}∞n=1 and
that ε, is the size of the pure point at z = 1 in the probability measure µ(z) associated with the
Verblunsky coefficients {αn−1}∞n=1. This means, if the measure does not have a pure point at
z = 1 then {gn+1}∞n=0 is the maximal parameter sequence of {dn+1}∞n=1.
Consider now the Uvarov transformation of the measure µ(z), [6, p. 11],∫
∂D
f(z)dµ(t)(z) =
(1− t)
1− ε
∫
∂D
f(z)dµ(z) +
t− ε
1− ε
f(1),
so that µ(t)(z) has a jump t, 0 ≤ t < 1, at z = 1. These measures µ(t)(z) are associated with
the positive chain sequence {dn}∞n=1 obtained from {dn+1}∞n=1 by including the additional term
d1 = (1− t)M1. We also denote the generalized sequence of Verblunsky coefficients associated
with µ(t)(z) by
{
α
(t)
n−1
}∞
n=1
.
Note 1.1. Since the measure µ(t)(z) has a parameter ‘t’, the notations for the polynomials Rn(z)
and the sequences {cn}, {dn} should have involved a ′t′. However, it has been proved [6, p. 7],
that the kernel polynomials Pn(1; z) and hence Rn(z) as well as the sequences {cn} and {dn} are
independent of ‘t’ and so their notations are devoid of ‘t’. But the minimal parameters depend
on d1 and this has been reflected in the notation m
(t)
n .
As shown in [6, Theorem 1.1], µ(t)(z) can also be given by∫
∂D
f(z)dµ(t)(z) = (1− t)
∫
∂D
f(z)dµ(0)(z) + tf(1). (1.5)
As is obvious from the notation, µ(0)(z) are the measures arising when t = 0. This is the case
when d1 = M1, so that both the minimal and maximal parameter sequences coincide. This
equality can also be interpreted as the measure having zero jump.
Further, the Verblunsky coefficients α
(t)
n−1 have the representation
α
(t)
n−1 = τn
[
1− 2m
(t)
n − icn
1 + icn
]
, n ≥ 1. (1.6)
where
{
m
(t)
n
}
is the minimal parameter sequence of the positive chain sequence {dn}∞n=1. The
Szegő polynomials corresponding to (1.6) are [3, Theorem 5.2]
Φ(t)
n (z) =
Rn(z)− 2
(
1−m(t)
n
)
Rn−1(z)
n∏
k=1
(1 + ick)
, n ≥ 1. (1.7)
Orthogonal Polynomials Associated with Complementary Chain Sequences 5
It can be verified from (1.4) that if ck = 0, k ≥ 0, αn−1, n ≥ 1, are all real. The Rn(z) are
then the singular predictor polynomials of the second kind given in [7]. Indeed, if cn = 0, n ≥ 1,
it can be easily shown from (1.7) that
(z − 1)Rn(z) = zΦ(t)
n (z)−
(
Φ(t)
n
)∗
(z).
We would like to mention here that the Szegő polynomials, Verblunsky coefficients and the re-
lated measure have also been obtained for the para-orthogonal polynomials that are an extension
of singular predictor polynomials of first kind. See [2] for the details of these extensions and
also for a survey of recent developments in the theory connecting chain sequences and OPUC.
The purpose of the present manuscript is to introduce a particular perturbation in the chain
sequence {dn}, called the complementary chain sequence, and study its effect on the Verblunsky
coefficients of the corresponding Szegő polynomials. The motivation for this follows from the fact
that (1.6) guarantees an explicit relation between the Verblunsky coefficients and the minimal
parameter sequence
{
m
(t)
n
}
of {dn}.
This manuscript is organized as follows. In Section 2 the concept of complementary chain
sequences using the minimal parameter sequences is introduced. Using this concept, perturba-
tions of Verblunsky coefficients are studied. As an illustration of this concept, in Section 3, the
Szegő polynomials which characterizes the positive Perron–Carathéodory (PPC) fractions from
a particular chain sequence are constructed. An interplay by these PPC fractions in finding
a relation between this chain sequence, its complementary chain sequence and their respective
Carathéodry functions is obtained in this section. In Section 4, another illustration of character-
izing the Szegő polynomials using Gaussian hypergeometric functions is provided. For particular
values, using complementary chain sequences, the corresponding Verblunsky coefficients of these
Szegő polynomials are also shown to be perturbed Verblunsky coefficients obtained earlier.
2 Complementary chain sequences
As is obvious from the definition of chain sequences, the minimal and maximal parameter
sequences are uniquely defined for any given chain sequence. Also, the chain sequence for which
the minimal and maximal parameter sequences coincide, that is, M0 = 0, has its own importance
as illustrated in the previous section. Such a chain sequence is said to determine its parameters
uniquely and is referred to as a single parameter positive chain sequence (SPPCS) [2]. By Wall’s
criteria for maximal parameter sequence [26, p. 82], this is equivalent to
∞∑
n=1
m1
1−m1
· m2
1−m2
· m3
1−m3
· · · mn
1−mn
=∞. (2.1)
Thus, introducing a perturbation in the minimal parameters mn will lead to a uniquely defined
change in the chain sequence.
Definition 2.1. Suppose {dn}∞n=1 is a chain sequence with {mn}∞n=0 as its minimal parameter
sequence. Let {kn}∞n=0 be another sequence given by k0 = 0 and kn = 1−mn for n ≥ 1. Then
the chain sequence {an}∞n=1 having {kn}∞n=0 as its minimal parameter sequence is called the
complementary chain sequence of {dn}.
Such chain sequences enjoy interesting relations like [26, equation (75.3)]
√
1 + z
1 + d1z
1+
d2z
1+
d3z
1+
...
·
√
1 + z
1 + a1z
1+
a2z
1+
a3z
1+
...
= 1.
6 K.K. Behera, A. Sri Ranga and A. Swaminathan
They also satisfy
d1 − a1 = 1− 2k1 = 2m1 − 1
and
dn − an = 4mn−1 = −∇kn, n ≥ 2.
where 4 and ∇ are the forward and backward difference operators respectively. Further of
particular interest is the ratio of these two chain sequences given by
d1
a1
=
m1
1−m1
,
dn
an
=
kn−1
1− kn−1
mn
1−mn
, n ≥ 2.
This implies
mn
1−mn
=
dn
an
mn−1
1−mn−1
= · · · = dndn−1 · · · d1
anan−1 · · · a1
, n ≥ 1. (2.2)
Substituting (2.2) in (2.1), we have the following lemma.
Lemma 2.2. Let {dn}∞n=1 and {an}∞n=1 be two complementary chain sequences of each other.
Then {dn}∞n=1 will be a SPPCS if and only if
∞∑
n=1
n∏
j=1
d1d2 · · · dj
a1a2 · · · aj
=∞.
Remark 2.3. The above lemma is useful while considering a chain sequence and its complemen-
tary chain sequence without using the information on the corresponding minimal parameters.
Lemma 2.4. Let {dn}∞n=1 and {an}∞n=1 be two complementary chain sequences of each other.
If {dn}∞n=1 is not a SPPCS, then {an}∞n=1 is a SPPCS.
Proof. If {dn}∞n=1 is not a SPPCS then its minimal parameter sequence {mn}∞n=0 is such that
∞∑
n=1
n∏
j=1
mj
1−mj
<∞.
Hence, lim
n→∞
n∏
j=1
mj/(1−mj) = 0, and we have
∞∑
n=1
n∏
j=1
kj
1− kj
=
∞∑
n=1
n∏
j=1
1−mj
mj
=∞.
Thus, concluding the proof of the lemma. �
Lemma 2.5. Let {dn}∞n=1 be a chain sequence and {an}∞n=1 be its complementary chain sequence
with minimal parameter sequences {mn}∞n=0 and {kn}∞n=0 respectively.
– If 0 < mn < 1/2, n ≥ 1, then an is a SPPCS.
– If 1/2 < mn < 1, n ≥ 1, then dn is a SPPCS.
Proof. Observe that if 0 < mn < 1/2, kn/(1− kn) > 1 for all n ≥ 1. Similarly, 1/2 < mn < 1
implies mn/(1−mn) > 1 for all n ≥ 1. The results now follow from (2.1). �
Orthogonal Polynomials Associated with Complementary Chain Sequences 7
It is known that [26, p. 79] if dn ≥ 1/4, n ≥ 1, every parameter sequence {gn}, in particular
the minimal parameter sequence {mn} of {dn} is non-decreasing. For the special case when
dn = 1/4, n ≥ 1, mn → 1/2 as n → ∞. This implies 0 < mn < 1/2, n ≥ 1. By Lemma 2.5,
{an} is a SPPCS. In other words, the chain sequence complementary to the constant chain
sequence {1/4} determines its parameters gn uniquely, which are further given by
g0 = 0, gn =
n+ 2
2(n+ 1)
, n ≥ 1.
Moreover, if dn ≥ 1/4, there exist some n ∈ N such that an < 1/4 ≤ dn. Indeed,
dn = (1−mn−1)mn ≥ mn−1(1−mn) = an, n ≥ 2,
with the sign of the difference of d1 and a1 depending on whether m1 ∈ (0, 1/2) or (1/2, 1). If
an ∈ (1/4, 1) for n ≥ 1, kn has to be non-decreasing. This is a contradiction as kn = 1−mn for
n ≥ 1.
The effect of complementary chain sequences in studying perturbation of Verblunsky coeffi-
cients given by (1.6) has interesting consequences. In this context, we give the following result.
Theorem 2.6. Let {cn}∞n=1 and {dn+1}∞n=1 be, respectively, the real sequence and positive chain
sequence as given in (1.4). Let
{
m
(t)
n
}∞
n=0
be the minimal parameter sequence of the augmented
positive chain sequence {dn}∞n=1, where d1 = (1−t)M1 and {Mn+1}∞n=0 is the maximal parameter
sequence of {dn+1}∞n=1. Let
{
k
(t)
n
}∞
n=0
be the minimal parameter sequence of the positive chain
sequence {an}∞n=1 obtained as complementary to {dn}∞n=1. Set τn = 1−icn
1+icn
τn−1,
α
(t)
n−1 = τn
[
1− 2m
(t)
n − icn
1 + icn
]
and β
(t)
n−1 = τn
[
1− 2k
(t)
n − icn
1 + icn
]
,
for n ≥ 1, with τ0 = 1. Let µ(t)(z) and ν(t)(z) be, respectively, the probability measures having
α
(t)
n−1 and β
(t)
n−1 as the corresponding Verblunsky coefficients. Then the following can be stated:
1. For 0 < t < 1, the measure µ(t)(z) has a pure point of size t at z = 1, while ν(t)(z) does
not.
2. β
(t)
n−1 = −τnτn−1α(t)
n−1, n ≥ 1.
3. For n ≥ 1, if cn = (−1)nc, c ∈ R, β
(t)
n−1 = −1−ic
1+icα
(t)
n−1, n ≥ 1.
4. If cn = 0, n ≥ 1 then the Verblunsky coefficients, which are real, are such that β
(t)
n−1 =
−α(t)
n−1, n ≥ 1.
Proof. First we observe that α
(t)
n−1 are the generalized Verblunsky coefficients of the mea-
sure µ(t)(z) as given by (1.5). Consequently, for 0 < t < 1 the probability measure µ(t)(z) has
a pure point of size t at z = 1. Since d1 = (1 − t)Mn, choosing M0 = t > 0, the sequence
{t,M1,M2,M3, . . .} is the maximal parameter sequence of {dn}∞n=1. Since t > 0, {dn}∞n=1 is
a non SPPCS and hence, by Lemma 2.4 the sequence {an}∞n=1 is a SPPCS so that
{
k
(t)
n
}∞
n=0
is also its maximal parameter sequence. Thus, by results established in [6], the measure ν(t)(z)
does not have a pure point at z = 1. This proves the first part of the theorem.
Now to prove the second part, we first have
β
(t)
n−1 = τn
[
1− 2k
(t)
n − icn
1 + icn
]
= τn
[
−1 + 2m
(t)
n − icn
1 + icn
]
.
8 K.K. Behera, A. Sri Ranga and A. Swaminathan
By conjugation of the expression for α
(t)
n−1, we have
−α(t)
n−1 = τn
[
−1 + 2m
(t)
n − icn
1− icn
]
,
which leads to the second part of the theorem.
Clearly with cn = (−1)nc, n ≥ 1 we have τ2n = 1 and τ2n+1 = 1−ic
1+ic . Thus, the third part of
the theorem is established.
The last part follows by taking τnτn−1 = 1, n ≥ 1. This is only possible if cn = 0, n ≥ 1. �
The perturbation of the Verblunsky coefficients in case of OPUC and of the recurrence
coefficients in case of the real line play an important role in the spectral theory of orthogonal
polynomials. The reader is referred to [9] and [18] for some details. For a recent work in this
direction, we refer to [4].
The last two parts of Theorem 2.6 are important cases of Aleksandrov transformation and,
in the case of last part gives rise to second kind polynomials for the measure µ(t) [22]. In
this particular case, the recurrence relation (1.4) assumes a very simple form, similar to that
considered in [7].
In the next section, starting with particular minimal parameter sequences and assuming
cn = 0, n ≥ 1, we construct the para-orthogonal polynomials and the related Szegő polynomials
to illustrate our results.
3 An illustration involving Carathéodory functions
In a series of papers [13, 14, 15], Jones et al. during their investigation of the connection between
Szegő polynomials and continued fractions introduced the following
δ0 −
2δ0
1 +
1
δ̄1z +
(
1− |δ1|2
)
z
δ1 +
1
δ̄2z +
(
1− |δ2|2
)
z
δ2z +
· · · . (3.1)
These are called Hermitian Perron–Carathéodory fractions or HPC-fractions and are also used
to solve the trigonometric moment problem. They are completely determined by δn ∈ C, where
δ0 6= 0 and |δn| 6= 1 for n ≥ 1. Under the stronger conditions δ0 > 0 and |δn| < 1, for n ≥ 1,
(3.1) is called a positive PC fraction (PPC-fractions). Let Pn(z) and Qn(z) be respectively
the numerator and denominator of the nth approximant of a PPC-fraction where Qn(z) is
a polynomial of degree n and Pn(z) of degree at most n. Then [15, Theorems 3.1 and 3.2] Φn(z)
are precisely the odd ordered denominators Q2n+1(z) and Φ∗n(z) the even ordered denomina-
tors Q2n(z). The δ′ns are then given by δn = Φn(0) and are called the Schur parameters or
the reflection coefficients. This gives the following equivalent set of recurrence relations for the
Szegő polynomials:
Φ∗n(z) = δ̄nzΦn−1z + Φ∗n−1(z),
Φn(z) = δnΦ∗n(z) +
(
1− |δn|2
)
zΦn−1(z), n ≥ 1.
Further, if (3.1) is a positive PC-fraction, there exists a pair of formal power series
L0 = µ0 + 2
∞∑
k=1
µkz
k, L∞ = −µ0 − 2
∞∑
k=1
µ−kz
−k,
where µk are the moments as defined earlier and such that
L0 − Λ0
(
P2n
Q2n
)
= O
(
zn+1
)
, L∞ − Λ∞
(
P2n+1
Q2n+1
)
= O
(
1
zn+1
)
.
Orthogonal Polynomials Associated with Complementary Chain Sequences 9
Here, Λ0(R(z)) and Λ∞(R(z)) are the Laurent series expansion of the rational function R(z)
about 0 and ∞ respectively. For details regarding correspondence of continued fractions to
power series, see [16, 17].
For |ζ| < 1, the polynomials
Ψn(z) =
∫
∂D
z + ζ
z − ζ
(Φn(z)− Φn(ζ))dµ(ζ), n ≥ 1,
are known in literature as the associated Szegő polynomials or polynomials of the second
kind [10]. They arise as the odd ordered numerators of (3.1). The function −Ψ∗n(z) is called
the polynomial associated with Φ∗n(z) and are the even ordered numerators in (3.1). It is also
known that for |z| < 1, there exists a function C(z) =
∫
∂D
ζ+z
ζ−zdµ(ζ) with Re C(z) > 0 such that
C(z)− Ψ∗n(z)
Φ∗n(z)
= O
(
zn+1
)
.
C(z) is called the Carathéodory function associated with the PPC-fraction (3.1) or with the Szegő
polynomials Φn(z) obtained from this PPC-fraction. The ratio Ψn(z)/Φn(z) also converges to
a function Ĉ(z) called the Carathéodory reciprocal of C(z) [14] and is defined by
C(z) = −Ĉ(1/z̄).
The convergence is uniform on compact subsets of |z| < 1 and |z| > 1 respectively. Also, L0 is
the Taylor series expansion of C(z) about 0 and L∞ is that of Ĉ(z) about ∞.
Consider the sequence {δn}∞n=1, which satisfies δ0 > 0, |δn| < 1 and
δn+1 − δn = δnδn+1, n ≥ 1. (3.2)
Our aim in this section is to use a chain sequence to construct the Szegő polynomials Φ
(t)
n (z),
having δn ∈ R and satisfying (3.2) as the Verblunsky coefficients. We will also use the com-
plementary chain sequence to get another sequence of Szegő polynomials Φ̃
(t)
n (z) which has −δn
as the Verblunsky coefficients. The associated Carathéodory function in each case is also given
and it is shown that there exists a relation between them.
We start with the sequence
{
m
(t)
n
}∞
n=0
, where m
(t)
0 = 0 and m
(t)
n = (1 − δn)/2, n ≥ 1.
These minimal parameters are obtained by first substituting ck = 0, k ≥ 1 in the Verblunsky
coefficients (1.6) and then equating them to δn. The corresponding chain sequence is
d1 =
1− δ1
2
and dn =
1
4
(1 + δn−1)(1− δn) =
1
4
(1− 2δn−1δn), n ≥ 2.
The following are two algebraic relations of δn which will be needed later and can be proved by
simple induction using (3.2).
δ1δ2 + δ2δ3 + δ3δ4 + · · ·+ δnδn+1 = δn+1 − δ1, n ∈ N.
and
δn =
δn+1
1 + δn+1
= · · · = δn+k
1 + kδn+k
, k ∈ N. (3.3)
Proposition 3.1. The monic polynomial
Rn(z) = 1 +
n∑
k=1
[1 + 2k(n− k)δ1δn]zk (3.4)
satisfies the recurrence relation
Rn+1(z) = (z + 1)Rn(z)− (1− 2δnδn+1)zRn−1(z), n ≥ 1,
with the initial conditions, R0(z) = 1 and R1(z) = z + 1.
10 K.K. Behera, A. Sri Ranga and A. Swaminathan
Proof. First, note that R1(z) given by (3.4) satisfies the initial condition. Suppose Rn(z) has
this form and satisfies the recurrence relation for n = 1, 2, . . . , j. We shall now show
Rj+1(z) + (1− 2δjδj+1)zRj−1(z) = (z + 1)Rj(z). (3.5)
Using (3.3), the coefficient of zk in the left-hand side of (3.5) is
1 + 2k(j − k + 1)δ1δj+1 + (1− 2δjδj+1)[1 + 2(k − 1)(j − k)δ1δj−1]
= 1 + 2
k(j − k + 1)
j
(δj+1 − δ1) + 1− 2(δj+1 − δj) + 2
(k − 1)(j − k)
j − 2
(δj−1 − δ1)
− 2 · 2(k − 1)(j − k)
j − 2
(δj−1 − δ1)(δj+1 − δj). (3.6)
It is easy to verify that the coefficients of δj+1 and δj−1 vanish in (3.6). The coefficient of δ1 is
−2k(j − k + 1)
j
− 2(k − 1)(j − k)
j − 2
− 2 · 2(k − 1)(j − k)
j(j − 2)
+
2 · 2(k − 1)(j − k)
(j − 1)(j − 2)
= −2k(j − k)
j − 1
− 2(k − 1)(j − k + 1)
j − 1
. (3.7)
Similarly, the coefficient of δj is
2 +
2 · 2(k − 1)(j − k)
j − 1
=
2k(j − k)
j − 1
+
2(k − 1)(j − k + 1)
j − 1
. (3.8)
Using (3.7) and (3.8) in (3.6), the coefficient of zk in the left-hand side of (3.5) is given by
[1 + 2(k − 1)(j − k + 1)δ1δj ] + [1 + 2k(j − k)δ1δj ],
which is nothing but the coefficient of zk in the right-hand side of (3.5). Hence, by induction
the proof is complete. �
We now obtain the Szegő polynomials Φ
(t)
n (z) from the para-orthogonal polynomials Rn(z)
given by (3.4). Using (1.7) and (3.4), it can be seen that the coefficient of zk, 1 ≤ k ≤ n − 1,
in Φ
(t)
n (z) is −δn(1− 2kδ1). Hence, the Szegő polynomials are given by
Φ(t)
n (z) = zn − δn
[
(1− 2(n− 1)δ1)z
n−1 + · · ·+ (1− 2δ1)z + 1
]
, n ≥ 1, (3.9)
with α
(t)
n−1 = −δn.
We now give the Carathéodory function associated with the parameters δn’s given by (3.2).
Consider
C(z) = 1− 2(1− σ)z
1 + (1− 2σ)z
=
1− z
1 + (1− 2σ)z
, |z| < 1,
where 0 < σ < 1. That C(z) corresponds to a PPC-fraction with the parameter γn, where
γn =
1
n+ σ
1−σ
, n ≥ 1. (3.10)
can be shown by applying the algorithm [14] which is similar to the Schur algorithm. With the
initial values C0(z) = (1− z)/(1 + (1− 2σ)z), γ0 = C0(0) = 1, define
C1(z) =
γ0 − C0(z)
γ0 + C0(z)
, γ1 = C′1(0).
Orthogonal Polynomials Associated with Complementary Chain Sequences 11
Then
C1(z) =
z
1 + σ
1−σ −
(
1− 1−2σ
1−σ
)
z
, and γ1 =
1
1 + σ
1−σ
.
Assume for k ≥ 1 the following
Ck(z) =
z
k + σ
1−σ −
(
k − 1−2σ
1−σ
)
z
, γk = C′k(0).
This is true for k = 1. Now define
Ck+1(z) =
γkz − Ck(z)
γkCk(z)− z
, n ≥ 1. (3.11)
It can be shown that
γk =
1− σ
k − (k − 1)σ
=
1
k + σ
1−σ
,
which is also true for k = 1. Simplifying (3.11), we obtain
Ck+1 =
z(
k + 1 + σ
1−σ
)
−
(
k + 1− 1−2σ
1−σ
)
z
,
from which γk+1 = 1
k+1+ σ
1−σ
. Hence by induction, (3.10) and because of the uniqueness of
the Carathéodory function that corresponds to a given PPC-fraction, the assertion follows.
Moreover, observe that δn = −γn satisfies (3.2) and so Φ
(t)
n (0) = 1
n+ σ
1−σ
.
From the power series expansion of C(z), we also obtain the moments as
µ0 = 1, µk = (−1)k(1− α)(1− 2α)k−1, k ≥ 1.
Using the fact that the Verblunsky coefficients are all real, from (1.2), we have
χ−2n =
n∏
k=1
(
1− δ2k
)
.
Further
δn =
1
n+ σ
1−σ
=
1− σ
n(1− σ) + σ
, n ≥ 1,
and we obtain
1− δ2n =
[n(1− σ) + σ − 1 + σ][n(1− σ) + σ + 1− σ]
[n(1− σ) + σ]2
=
[(n− 1)− (n− 2)σ][(n+ 1)− nσ]
[n− (n− 1)σ]2
,
which yields the fact that
χ−2n =
σ[(n+ 1)− nσ]
[n− (n− 1)σ]
.
12 K.K. Behera, A. Sri Ranga and A. Swaminathan
Rewriting the right-hand expression as σ
(
1 + 1−σ
n(1−σ)+σ
)
gives
χ−2n =
∥∥Φ(t)
n (z)
∥∥2 = σ(1 + δn),
which tends to σ > 0 as n→∞.
Consider now the parameter sequence
{
k
(t)
n
}∞
n=0
, defined by k
(t)
0 = 0 and k
(t)
n = 1 −m(t)
n =
(1 + δn)/2, n ≥ 1. From (3.2), it is easy to check that 1 + δn+1 = 1/(1− δn), n ≥ 1. In this case,
the constant sequence {1/4} becomes the complementary chain sequence so that equation (1.4)
assumes the form
R̃n+1(z) = [1 + z]R̃n(z)− zR̃n−1(z), n ≥ 1.
The polynomials satisfying the above recurrence relation are the palindromic polynomials zn +
λ(zn−1 + · · · + z) + 1. For λ = 1, the para-orthogonal polynomials are the partial sums of the
geometric series given by
R̃n(z) = 1 + z + z2 + · · ·+ zn =
1− zn+1
1− z
, n ≥ 1.
Then (1.7) yields the Szegő polynomial
Φ̃(t)
n (z) = zn + δnz
n−1 + · · ·+ δnz + δn, n ≥ 1, (3.12)
with α
(t)
n−1 = −δn. The polynomials Φ̃
(t)
n (z) have been considered in [20] where it is proved that
Φ̃(t)
n (0) = δn = − 1
n+ σ
1−σ
, n ≥ 1. (3.13)
Further, the corresponding Carathéodory function is C̃(z) = 1+(1−2σ)z
1−z , |z| < 1, where 0 < σ < 1.
This is a special case when all the moments are equal to µ̃ = (1− σ). We summarize the above
facts as a theorem.
Theorem 3.2. Consider the real sequence {δn}∞n=0 satisfying δn − δn−1 = δn−1δn, n ≥ 1 under
the restrictions δ0 > 0 and |δn| < 1, n ≥ 1. If C(z) is a Carathéodory function whose PPC-
fraction can be obtained from the minimal parameter sequence {mn}, where 2mn = 1−δn, n ≥ 1,
then 1−mn gives the PPC-fraction corresponding to the Carathéodory function 1/C(z).
Note that an equivalent statement using Schur parameters is given in [21]. Further, let µ(t)(z)
be the probability measure associated with the positive chain sequence {dn}∞n=1. Since its
complementary chain sequence {1/4} is not a SPPCS, by Lemma (2.4) {dn}∞n=1 is a SPPCS and
hence µ(t)(z) has zero jump (t = 0) at z = 1. If ν(t)(z) is the measure associated with {1/4},
ν(t)(z) has a jump t = 1/2 at z = 1. Finally as shown in [20], ν(1/2)(θ) is of the form,
dν(1/2)(θ) = dν(1/2)s (θ) + (1− µ̃)d(θ),
where dν
(1/2)
s (θ) is a point measure with mass µ̃ at z=1 and mass zero elsewhere.
We end this illustration with two observations which we state as remarks.
Remark 3.3. Suppose the minimal parameters are given in terms of some variable ε. It follows
that the coefficients of the polynomial Rn(z) satisfying (1.4) with cn = 0 for n ≥ 1 will be given
in terms of ε. Since, it is clear that Rn(z) is palindromic for the chain sequence {dn} = {1/4},
Rn(z) can always be expressed as the sum of two polynomials, one of them being a palindromic
and the other one being such that it vanishes whenever ε is chosen so that dn = 1/4.
Remark 3.4. As n → ∞, both the minimal parameter sequences approach 1/2. From the
expressions (3.9) and (3.12) it is clear that for fixed z, Φ
(t)
n (z) and Φ̃
(t)
n (z) approach zn as n
becomes large. The polynomials zn are called the Szegő–Chebyshev polynomials and correspond
to the standard Lebesgue measure on the unit circle.
Orthogonal Polynomials Associated with Complementary Chain Sequences 13
4 An illustration using Gaussian hypergeometric functions
The Gaussian hypergeometric function, with the complex parameters a, b and c is defined by
the power series
F (a, b; c; z) =
∞∑
n=0
(a)n(b)n
(c)n(1)n
zn, |z| < 1,
where c 6= 0,−1,−2, . . . and (a)n is the Pochhammer symbol. With specialized values of the
parameters a, b and c, many elementary functions can be represented by the Gaussian hyper-
geometric functions or their ratios. If Re(c − a − b) > 0, the series converges for |z| = 1 to the
value given by
F (a, b; c; 1) =
∞∑
k=0
(a)k(b)k
(c)kk!
=
Γ(c)Γ(c− a− b)
Γ(c− a)Γ(c− b)
.
In case the series is terminating, we have the Chu–Vandermonde identity [1]
F (−n, b; c; 1) =
(c− b)n
(c)n
. (4.1)
Two hypergeometric functions F (a1, b1; c1; z) and F (a2, b2; c2, z) are said to be contiguous if
the difference between the corresponding parameters is at most unity. A linear combination
of two contiguous hypergeometric functions is again a hypergeometric function. Such relations
are called contiguous relations and have been used to explore many hidden properties of the
hypergeometric functions, for example by Gauss who found continued fraction expansions for
ratios of hypergeometric functions [19] and hence for the special functions that these ratios
represent. In some special cases, the contiguous relations can also be related to the recurrence
relations for orthogonal polynomials. Consider one such relation [1]
(c− a)F (a− 1, b; c; z) = (c− 2a− (b− a)z)F (a, b; c; z) + a(1− z)F (a+ 1, b; c; z),
which as shown in [24], can be transformed to the three term recurrence relation
%n+1(z) =
(
z +
c− b+ n
b+ n
)
%n(z)− n(c+ n− 1)
(b+ n− 1)(b+ n)
%n−1(z), n ≥ 1, (4.2)
satisfied by the monic polynomial
%n(z) =
(c)n
(b)n
F (−n, b; c; 1− z). (4.3)
It was also shown that for the specific values b = λ ∈ R and c = 2λ − 1, the polynomials (4.3)
are Szegő polynomials. We note that with b = λ+ 1, %n(z) given by (4.3) are called the circular
Jacobi polynomials [12, Example 8.2.5]. For other specialized values of b and c in (4.2), %n(z) also
becomes the para-orthogonal polynomial.
Let λ > −1/2 ∈ R. Taking b = λ+ 1 and c = 2λ+ 2, (4.2) reduces to
%n+1(z) = (z + 1)%n(z)− n(2λ+ n+ 1)
(λ+ n)(λ+ n+ 1)
z%n−1(z), n ≥ 1,
satisfied by
%n(z) = Rn(z) =
(2λ+ 2)n
(λ+ 1)n
F (−n, λ+ 1; 2λ+ 2; 1− z), n ≥ 1.
14 K.K. Behera, A. Sri Ranga and A. Swaminathan
Consider now the sequence {dn+1}∞n=1, where
dn+1 =
1
4
n(2λ+ n+ 1)
(λ+ n)(λ+ n+ 1)
, n ≥ 1.
As established in [2, Example 3], for λ > −1, the sequence {dn+1}∞n=1 is a positive chain sequence
and {mn}∞n=0, where
mn =
n
2(λ+ n+ 1)
, n ≥ 0,
is its minimal parameter sequence. When −1/2 ≥ λ > −1, {mn}∞n=0 is also the maximal
parameter sequence of {dn+1}∞n=1, which makes it a SPPCS. However, when λ > −1/2 then
{dn+1}∞n=1 is not a SPPCS and its maximal parameter sequence {Mn+1}∞n=0 is such that
Mn+1 =
2λ+ n+ 1
2(λ+ n+ 1)
, n ≥ 0.
The coefficients dn+1, n ≥ 1 are the same coefficients occurring in the recurrence formula for
ultraspherical (or Gegenbauer) polynomials.
Further, for λ > −1/2 and 0 ≤ t < 1, if
{
m
(t)
n
}∞
n=0
is the minimal parameter sequence of
the positive chain sequence {dn}∞n=1, obtained by adding d1 = (1 − t)M1 to {dn+1}∞n=1, then
from (1.7)
Φ(t)
n (z) = Rn(z)− 2
(
1−m(t)
n
)
Rn−1(z), n ≥ 1
and are the monic OPUC with respect to the measure µ(t)(z), where µ(t)(z) is as defined by (1.5).
To find µ(t)(z), we first find the measure µ(0)(z) arising when {dn}∞n=1 becomes a SPPCS (t = 0).
As shown in [24], the monic OPUC are given by
Φ(0)
n (z) = Rn(z)− 2(1−Mn)Rn−1(z) =
(2λ+ 1)n
(λ+ 1)n
F (−n, λ+ 1; 2λ+ 1; 1− z), n ≥ 1.
Using the identity (4.1), the Verblunsky coefficients are given by
α
(0)
n−1 = −Φ(0)
n (0) = − (λ)n
(λ+ 1)n
, n ≥ 1. (4.4)
The Verblunsky coefficients α
(0)
n−1 are associated with the non-trivial probability measure given
by [24]
dµ(0)
(
eiθ
)
= τ (λ) sin2λ(θ/2)dθ,
where
τ (λ) =
|Γ(1 + λ)|2
Γ(2λ+ 1)
4λ.
Hence∫
∂D
f(ζ)dµ(t)(ζ) = (1− t)τ (λ)
∫ 2π
0
f
(
eiθ
)
sin2λ(θ/2)dθ + tf(1).
Further characterization of Szegő polynomials is provided below as it is not possible to find
closed form expressions for the coefficients of the para-orthogonal polynomials and Szegő poly-
nomials. Since {Rn(z)}, depends on the parameter b (= λ+1), in what follows, we denote Rn(z)
by R
(b)
n (z). We also denote cn and dn by c
(b)
n and d
(b)
n respectively. Now, note that if
Q(b)
n (z) =
1
2(1− t)M1
∫
T
R
(b)
n (z)−R(b)
n (ζ)
z − ζ
(1− ζ)dµ(t)(ζ), n ≥ 0,
Orthogonal Polynomials Associated with Complementary Chain Sequences 15
then
{
Q
(b)
n (z)
}∞
n=0
satisfies
Q
(b)
n+1(z) =
[(
1 + ic
(b)
n+1
)
z +
(
1− ic(b)n+1
)]
Q(b)
n (z)− 4d
(b)
n+1zQ
(b)
n−1(z), n ≥ 1,
with Q
(b)
0 (z) = 0 and Q
(b)
1 (z) = 1. That is, the three term recurrence for
{
Q
(b)
n (z)
}∞
n=0
is the same
as for
{
R
(b)
n (z)
}∞
n=0
, with the difference being only on the initial conditions. The polynomials{
Q
(b)
n (z)
}
are generally called the numerator polynomials associated with
{
R
(b)
n (z)
}
. Further,
observe that the three term recurrence for
{
Q
(b)
n (z)
}∞
n=0
can also be given in the shifted form
Q
(b)
n+2(z) =
[(
1 + ic
(b)
n+2
)
z +
(
1− ic(b)n+2
)]
Q
(b)
n+1(z)− 4d
(b)
n+2zQ
(b)
n (z), n ≥ 1, (4.5)
with Q
(b)
1 (z) = 1 and Q
(b)
2 (z) =
(
1 + ic
(b)
2
)
z +
(
1− ic(b)2
)
.
Consider now the parameter sequence given by k
(t)
n = 1−m(0)
n = n/[2(λ+ n)] for n ≥ 1. For
sake of clarity, we would like to note that t need not be necessarily 0. It depends on whether
the resulting chain sequence for
{
k
(t)
n
}
, given by
a
(b)
1 =
1
2λ+ 2
and a
(b)
n+1 =
1
4
(n+ 1)(2λ+ n)
(λ+ n)(λ+ n+ 1)
, n ≥ 1, (4.6)
is a SPPCS or not.
Let ν(t)(z) be the measure associated with the Verblunsky coefficients
{
β
(t)
n−1
}∞
n=1
given by
β
(t)
n−1 = τn
[
1− 2k
(t)
n − ic(b)n
1 + ic
(b)
n
]
, n ≥ 1.
Following Theorem 2.6, the corresponding OPUC are
Φ̃(t)
n (z) =
R̃
(b)
n (z)− 2
(
1− k(t)n
)
R̃
(b)
n−1(z)
n∏
k=1
(
1 + ic
(b)
k
) , n ≥ 1,
where the polynomials R̃
(b)
n are given by
R̃
(b)
n+1(z) =
[(
1 + ic
(b)
n+1
)
z +
(
1− ic(b)n+1
)]
R̃(b)
n (z)− 4a
(b)
n+1zR̃
(b)
n−1(z), n ≥ 1, (4.7)
with R̃
(b)
0 (z) = 1 and R̃
(b)
1 (z) =
(
1 + ic
(b)
1
)
z +
(
1 − ic(b)1
)
. Observing that c
(b)
n = c
(b−1)
n+1 , a
(b)
n+1 =
d
(b−1)
n+2 , n ≥ 1, we have from (4.5) and (4.7)
R̃(b)
n (z) = Q
(b−1)
n+1 (z), n ≥ 0,
and thus
Φ̃(t)
n (z) =
Q
(b−1)
n+1 (z)− 2
(
1− k(t)n
)
Q
(b−1)
n (z)
n∏
k=1
(
1 + ic
(b−1)
k+1
) , n ≥ 1.
That is, if R
(b)
n (z) generates the OPUC Φ
(t)
n (z), Q
(b−1)
n (z), which are the numerator polynomials
for R
(b−1)
n (z) generates the OPUC Φ̃
(t)
n (z) associated with the complementary chain sequences.
We note that, in the present case too, c
(b)
n (= cn) = 0, n ≥ 1 and so by Theorem 2.6 β
(t)
n−1 = −α(0)
n−1
for n ≥ 1. Hence dν(t)(z) are the Aleksandrov measures associated with dµ(0)(z) [22].
16 K.K. Behera, A. Sri Ranga and A. Swaminathan
Further, we note that such Szegő polynomials result from perturbations of the Verblunsky
coefficients obtained in Section 3. Indeed, for σ = λ/(1 +λ), {λδn} corresponds to the Verblun-
sky coefficients given by (4.4), wheras by Verblunsky theorem, {λγn} corresponds to those given
by the complementary chain sequence {a(b)n+1} given by (4.6). Here {δn} and {γn} are the ones
chosen respectively by (3.10) and (3.13).
Further, when
{
a
(b)
n+1
}∞
n=1
is the constant chain sequence {1/4}, R̃(b)
n (z) are the palindromic
polynomials given by
R̃(b)
n (z) = zn + ν(λ)
(
zn−1 + · · ·+ z
)
+ 1, n ≥ 1,
where ν(λ) is a constant depending on λ. Here we study the cases λ = 0 and λ = 1 for which
the complementary chain sequence a
(b)
n+1 = 1/4.
Case 1, λ = 0. Let
R̃(b)
n (z) = zn + ν(0)
(
zn−1 + · · ·+ z
)
+ 1, n ≥ 1.
The complementary chain sequence is {1/2, 1/4, 1/4, . . . } which is known to be a SPPCS. Hence{
k
(t)
n
}∞
n=0
where k
(t)
0 = 0, k
(t)
n = 1/2, n ≥ 1 is also the maximal parameter sequence implying
that t = 0 and so
Φ̃(0)
n (z) = zn +
(
ν(0) − 1
)
zn−1.
For ν(0) = 1, Φ̃
(0)
n (z) = zn and from Remark 3.4, λ = 0 can be viewed as the limiting case for
the Verblunsky coefficients obtained in Section 3. Note that the Verblunsky coefficients are 0,
as can be verified from (4.4).
Case 2, λ = 1. Let
R̃(b)
n (z) = zn + ν(1)
(
zn−1 + · · ·+ z
)
+ 1, n ≥ 1.
The complementary chain sequence is {1/4, 1/4, 1/4, . . . } and k
(t)
0 = 0, k
(t)
n = n/2(n+ 1), n ≥ 1.
In this case, t = 1/2 and
Φ̃(1/2)
n (z) = zn +
(
ν(1) − n+ 2
n+ 1
)
zn−1 − ν(1)
n+ 1
(
zn−2 + · · ·+ z
)
− 1
n+ 1
, n ≥ 1,
so that the Verblunsky coefficients are given by 1/(n + 1). Again it can be verified from (4.4)
that the Verblunsky coefficients corresponding to λ = 1 are (1)n/(2)n = 1/(n+ 1). Finally, for
ν(1) = 0, R̃
(b)
n = zn + 1, which has been considered as Example 1 in [2].
Acknowledgements
The authors wish to thank the anonymous referees for their constructive criticism that resulted
in significant improvement of the content leading to the final version. The work of the second
author was supported by funds from CNPq, Brazil (grants 475502/2013-2 and 305073/2014-1)
and FAPESP, Brazil (grant 2009/13832-9).
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1 Preliminaries on Szego polynomials
2 Complementary chain sequences
3 An illustration involving Carathéodory functions
4 An illustration using Gaussian hypergeometric functions
References
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