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-orthogonal polynomials and the associated Szeg\"o polynomials are analyzed. Two illustrations, one involving Gaussian hypergeometric functions and the other involving Carath\'eodory functions are also provided. A connection between these two illustrations by means of complementary chain sequences is also observed.

where Φ * n (z) = z n Φ n (1/z). The complex numbers α n−1 = −Φ n (0) are called the Verblunsky coefficients [22]. The Verblunsky coefficients completely characterize the Szegő polynomials in the sense that any sequence {α n−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].
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, ω n ∈ C with |ω n | = 1, have the representation X n (z, ω n ) = Φ n (z) + ω n Φ * n (z), n ≥ 1.
The polynomials P n (ω; z) are τ n (w)-invariant sequences of polynomials which can be easily verified from (1.3). Note that a sequence of polynomials {Y n } is called τ n -invariant if [15] Y * n (z) = τ n Y n (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.
is called a positive chain sequence [5] (see also [12,Section 7.2]). Here {g n } ∞ n=0 , called the parameter sequence is such that 0 ≤ g 0 < 1, 0 < g n < 1 for n ≥ 1. This is a stronger condition than the one used in [26], in which d n is also allowed to be zero. The parameter sequence {g n } ∞ n=0 is called a minimal parameter sequence and denoted by {m n } ∞ n=0 if m 0 = 0. Every chain sequence has a minimal parameter sequence [5, pp. 91-92]. Further, for a fixed chain sequence {d n } n≥1 , let G be the set of all parameter sequences {g k } of {d n } n≥1 . Let the sequence {M n } ∞ n=0 be defined by where inf is infimum of the set. Then, {M n } is called the maximal parameter sequence of {d n }.
The role of chain sequences in the study of orthogonal polynomials on the real line is well known. Similarly, a positive chain sequence {d n } appears in the three term recurrence relation for the polynomials R n (z), that turn out to be scaled versions of the (kernel) polynomials P n (1, z), namely, with R 0 (z) = 1 and R 1 (z) = (1 + ic 1 )z + (1 − ic 1 ). It is indeed shown in [6], that on condition that is a chain sequence with parameter sequence It is also not difficult to verify that in this case R n (z) has r n,n = n k=1 (1 + ic k ) as the leading coefficient and r n,0 =r n,n = n k=1 (1 − ic k ) as the constant term.
It is known that {R n (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 {α n−1 } ∞ n=1 as the Verblunsky coefficients. A further interesting fact is that the above parameter sequence {g n+1 } ∞ n=0 is such that n=0 is the maximal parameter sequence of {d n+1 } ∞ n=1 and that , is the size of the pure point at z = 1 in the probability measure µ(z) associated with the Verblunsky coefficients {α n−1 } ∞ n=1 . This means, if the measure does not have a pure point at z = 1 then {g n+1 } ∞ n=0 is the maximal parameter sequence of {d n+1 } ∞ n=1 . Consider now the Uvarov transformation of the measure µ(z), [6, p. 11], 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 {d n } ∞ n=1 obtained from {d n+1 } ∞ n=1 by including the additional term We also denote the generalized sequence of Verblunsky coefficients associated with Note 1.1. Since the measure µ (t) (z) has a parameter 't', the notations for the polynomials R n (z) and the sequences {c n }, {d n } should have involved a t . However, it has been proved [6, p. 7], that the kernel polynomials P n (1; z) and hence R n (z) as well as the sequences {c n } and {d n } are independent of 't' and so their notations are devoid of 't'. But the minimal parameters depend on d 1 and this has been reflected in the notation m As shown in [6, Theorem 1.1], µ (t) (z) can also be given by As is obvious from the notation, µ (0) (z) are the measures arising when t = 0. This is the case when d 1 = M 1 , 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 where m It can be verified from (1.4) that if c k = 0, k ≥ 0, α n−1 , n ≥ 1, are all real. The R n (z) are then the singular predictor polynomials of the second kind given in [7]. Indeed, if c n = 0, n ≥ 1, it can be easily shown from (1.7) that We would like to mention here that the Szegő polynomials, Verblunsky coefficients and the related 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 {d n }, 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 {d n }. 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, perturbations 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 characterizing 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.

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, M 0 = 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 Thus, introducing a perturbation in the minimal parameters m n will lead to a uniquely defined change in the chain sequence.
is a chain sequence with {m n } ∞ n=0 as its minimal parameter sequence. Let {k n } ∞ n=0 be another sequence given by k 0 = 0 and k n = 1 − m n for n ≥ 1. Then the chain sequence {a n } ∞ n=1 having {k n } ∞ n=0 as its minimal parameter sequence is called the complementary chain sequence of {d n }.
Such chain sequences enjoy interesting relations like [26, equation ( They also satisfy where and ∇ are the forward and backward difference operators respectively. Further of particular interest is the ratio of these two chain sequences given by This implies Remark 2.3. The above lemma is useful while considering a chain sequence and its complementary chain sequence without using the information on the corresponding minimal parameters.
Hence, lim Thus, concluding the proof of the lemma.
Lemma 2.5. Let {d n } ∞ n=1 be a chain sequence and {a n } ∞ n=1 be its complementary chain sequence with minimal parameter sequences {m n } ∞ n=0 and {k n } ∞ n=0 respectively.
The results now follow from (2.1).
It is known that [26, p. 79] if d n ≥ 1/4, n ≥ 1, every parameter sequence {g n }, in particular the minimal parameter sequence {m n } of {d n } is non-decreasing. For the special case when d n = 1/4, n ≥ 1, m n → 1/2 as n → ∞. This implies 0 < m n < 1/2, n ≥ 1. By Lemma 2.5, {a n } is a SPPCS. In other words, the chain sequence complementary to the constant chain sequence {1/4} determines its parameters g n uniquely, which are further given by Moreover, if d n ≥ 1/4, there exist some n ∈ N such that a n < 1/4 ≤ d n . Indeed, with the sign of the difference of d 1 and a 1 depending on whether m 1 ∈ (0, 1/2) or (1/2, 1). If a n ∈ (1/4, 1) for n ≥ 1, k n has to be non-decreasing. This is a contradiction as k n = 1 − m n for n ≥ 1.
The effect of complementary chain sequences in studying perturbation of Verblunsky coefficients given by (1.6) has interesting consequences. In this context, we give the following result.
for n ≥ 1, with τ 0 = 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: 4. If c n = 0, n ≥ 1 then the Verblunsky coefficients, which are real, are such that β Proof . First we observe that α (t) n−1 are the generalized Verblunsky coefficients of the measure 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 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 By conjugation of the expression for α (t) which leads to the second part of the theorem. Clearly with c n = (−1) n c, n ≥ 1 we have τ 2n = 1 and τ 2n+1 = 1−ic 1+ic . Thus, the third part of the theorem is established.
The last part follows by taking τ n τ n−1 = 1, n ≥ 1. This is only possible if c n = 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 c n = 0, n ≥ 1, we construct the para-orthogonal polynomials and the related Szegő polynomials to illustrate our results.

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 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 δ n ∈ C, where δ 0 = 0 and |δ n | = 1 for n ≥ 1. Under the stronger conditions δ 0 > 0 and |δ n | < 1, for n ≥ 1, (3.1) is called a positive PC fraction (PPC-fractions). Let P n (z) and Q n (z) be respectively the numerator and denominator of the n th approximant of a PPC-fraction where Q n (z) is a polynomial of degree n and P n (z) of degree at most n. Then [15, Theorems 3.1 and 3.2] Φ n (z) are precisely the odd ordered denominators Q 2n+1 (z) and Φ * n (z) the even ordered denominators Q 2n (z). The δ n s are then given by δ n = Φ n (0) and are called the Schur parameters or the reflection coefficients. This gives the following equivalent set of recurrence relations for the Szegő polynomials: Further, if (3.1) is a positive PC-fraction, there exists a pair of formal power series where µ k are the moments as defined earlier and such that Here, Λ 0 (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 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−z dµ(ζ) with Re C(z) > 0 such that C(z) is called the Carathéodory function associated with the PPC-fraction (3.1) or with the Szegő polynomials Φ n (z) obtained from this PPC-fraction. The ratio Ψ n (z)/Φ n (z) also converges to a functionĈ(z) called the Carathéodory reciprocal of C(z) [14] and is defined by The convergence is uniform on compact subsets of |z| < 1 and |z| > 1 respectively. Also, L 0 is the Taylor series expansion of C(z) about 0 and L ∞ is that ofĈ(z) about ∞.

(3.2)
Our aim in this section is to use a chain sequence to construct the Szegő polynomials Φ (t) n (z), having δ n ∈ R and satisfying (3.2) as the Verblunsky coefficients. We will also use the complementary chain sequence to get another sequence of Szegő polynomialsΦ (t) n (z) which has −δ n 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 n = (1 − δ n )/2, n ≥ 1. These minimal parameters are obtained by first substituting c k = 0, k ≥ 1 in the Verblunsky coefficients (1.6) and then equating them to δ n . The corresponding chain sequence is The following are two algebraic relations of δ n which will be needed later and can be proved by simple induction using (3.2).
Proof . First, note that R 1 (z) given by (3.4) satisfies the initial condition. Suppose R n (z) has this form and satisfies the recurrence relation for n = 1, 2, . . . , j. We shall now show R j+1 (z) + (1 − 2δ j δ j+1 )zR j−1 (z) = (z + 1)R j (z). (3.5) Using (3.3), the coefficient of z k in the left-hand side of (3.5) is It is easy to verify that the coefficients of δ j+1 and δ j−1 vanish in (3.6). The coefficient of Similarly, the coefficient of Using (3.7) and (3.8) in (3.6), the coefficient of z k in the left-hand side of (3.5) is given by which is nothing but the coefficient of z k 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 R n (z) given by (3.4). Using (1.7) and (3.4), it can be seen that the coefficient of 2kδ 1 ). Hence, the Szegő polynomials are given by with α (t) n−1 = −δ n . We now give the Carathéodory function associated with the parameters δ n 's given by (3.2). Consider where 0 < σ < 1. That C(z) corresponds to a PPC-fraction with the parameter γ n , where can be shown by applying the algorithm [14] which is similar to the Schur algorithm. With the initial values C 0 (z) = (1 − z)/(1 + (1 − 2σ)z), γ 0 = C 0 (0) = 1, define Assume for k ≥ 1 the following This is true for k = 1. Now define It can be shown that which is also true for k = 1. Simplifying (3.11), we obtain 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 δ n = −γ n satisfies (3.2) and so Φ (t) From the power series expansion of C(z), we also obtain the moments as Using the fact that the Verblunsky coefficients are all real, from (1.2), we have Further and we obtain which yields the fact that Rewriting the right-hand expression as σ 1 + 1−σ n(1−σ)+σ gives which tends to σ > 0 as n → ∞. Consider now the parameter sequence k  From (3.2), it is easy to check that 1 + δ n+1 = 1/(1 − δ n ), n ≥ 1. In this case, the constant sequence {1/4} becomes the complementary chain sequence so that equation (1.4) assumes the form n ≥ 1.
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 {d n } ∞ n=1 . Since its complementary chain sequence {1/4} is not a SPPCS, by Lemma (2.4) {d n } ∞ 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, is a point measure with massμ 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 R n (z) satisfying (1.4) with c n = 0 for n ≥ 1 will be given in terms of ε. Since, it is clear that R n (z) is palindromic for the chain sequence {d n } = {1/4}, R n (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 d n = 1/4.
In case the series is terminating, we have the Chu-Vandermonde identity [1] F Two hypergeometric functions F (a 1 , b 1 ; c 1 ; z) and F (a 2 , b 2 ; c 2 , 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] ( which as shown in [24], can be transformed to the three term recurrence relation satisfied by the monic polynomial 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) = R n (z) = (2λ + 2) n (λ + 1) n F (−n, λ + 1; 2λ + 2; 1 − z), n ≥ 1. n(2λ + n + 1) (λ + n)(λ + n + 1) , n ≥ 1.
Using the identity (4.1), the Verblunsky coefficients are given by The Verblunsky coefficients α (0) n−1 are associated with the non-trivial probability measure given by [24] 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 polynomials. Since {R n (z)}, depends on the parameter b (= λ+1), in what follows, we denote R n (z) by R with Q n (z) . Further, observe that the three term recurrence for Q can also be given in the shifted form 2 . Consider now the parameter sequence given by k (n + 1)(2λ + n) (λ + n)(λ + n + 1) , is a SPPCS or not. Let ν (t) (z) be the measure associated with the Verblunsky coefficients β Following Theorem 2.6, the corresponding OPUC arẽ Φ (t) n (z) =R n−1 (z), n ≥ 1, (4.7) withR (b) 0 (z) = 1 andR n+2 , n ≥ 1, we have from (4.5) and (4.7) and thus That is, if R  for n ≥ 1. Hence dν (t) (z) are the Aleksandrov measures associated with dµ (0) (z) [22].