MEROMORPHIC SZEGŐ FUNCTIONS AND ASYMPTOTIC SERIES FOR VERBLUNSKY

COEFFICIENTS

BARRY SIMON*

Abstract. We prove that the Szegő function, D(z), of a measure on the unit circle is entire meromorphic if and only if the Verblunsky coefficients have an asymptotic expansion in exponentials. We relate the positions of the poles of D(z)^{−1} to the exponential rates in the asymptotic expansion. Basically, either set is contained in the sets generated from the other by considering products of the form, z_{1} … z_{ℓ}z̄_{ℓ−1} … z̄_{2ℓ−1} with z_{j} in the set. The proofs use nothing more than iterated Szegő recursion at z and 1/z̄.

1. Introduction

This paper is concerned with the spectral theory of orthogonal polynomials on the unit circle (OPUC) [14, 15, 21, 7, 8] in the case of particularly regular measures. Throughout, we will consider probability measures on ∂𝔻 = {z | |z| = 1} of the form

dμ = w(θ) \frac{dθ}{2π} + dμ_{s}  (1.1)

where w obeys the Szegő condition, that is,

∫ log(w(θ)) \frac{dθ}{2π} > −∞  (1.2)

In that case, the Szegő function is defined by

D(z) = exp(∫ \frac{e^{iθ} + z}{e^{iθ} − z} log(w(θ)) \frac{dθ}{4π} )  (1.3)

Not only does w determine D but D determines w, since lim_{r↑1} D(re^{iθ}) ≡ D(e^{iθ}) exists for a.e. θ and

w(θ) = |D(e^{iθ})|^{2}  (1.4)

Date: February 23, 2005.

∗ Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125. E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0140592 and in part by Grant No. 2002068 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.

Indeed, D is the unique function analytic on 𝔻 = {z | |z| < 1} with D nonvanishing on 𝔻 so that (1.4) holds.

Given dμ, we let Φ_{n} be the monic orthogonal polynomial and φ_{n} = Φ_{n}/∥Φ_{n}∥_{L^{2}(dμ)}. The Φ_{n}’s obey the Szegő recursion:

Φ_{n+1}(z) = zΦ_{n}(z) − ᾱ_{n}Φ_{n}^{∗}(z)  (1.5)

where for P_{n} a polynomial of degree n,

P_{n}^{∗}(z) = z^{n} \overline{P_{n}(1/z̄)}  (1.6)

The α_{n} are called Verblunsky coefficients. They lie in 𝔻 and μ ↦ {α_{n}}_{n=0}^{∞} is a bijection of nontrivial measures on ∂𝔻 and 𝔻^{∞}. Our goal here is to focus on the map and its inverse. Here is the background on our first main result: A. Nevai-Totik [11] proved that lim sup_{n→∞}|α_{n}|^{1/n} ≤ R^{−1} < 1 if and

only if

(a) dμ obeys the Szegő condition and dμ_{s} = 0.

(b) D(z)^{−1} is analytic in {z | |z| < R}. B. Barrios-López-Saff [1] proved that for R > 1,

α_{n} = cR^{−n} + O(((1 − ε)R^{−1})^{n})  (1.7)

if and only if D(z)^{−1} is meromorphic in a circle of radius R(1 + δ) with a single, simple pole at z = R.

C. Simon [14] considered the functions

S(z) = − ∑_{j=0}^{∞} α_{j−1}z^{j}  (1.8)

(with α_{−1} = −1) and

r(z) = \overline{D(1/z̄)} D(z)^{−1}  (1.9)

and proved that if lim sup|α_{n}|^{1/n} ≤ R^{−1} < 1, then for some δ > 0, r(z) − S(z) is analytic in {z | 1 − δ < |z| < R^{2}} so that S(z) and r(z), which will have singularities on |z| = R if lim sup|α_{n}|^{1/n} = R^{−1}, must have the same singularities in {z | R ≤ |z| < R^{2}}. In [14], instead of S(z) as defined by (1.8), one has S(z) defined by S_{book}(z) = ∑_{j=0}^{∞} α_{j}z^{j}, and the theorem is stated as analyticity of z^{−1}r(z) + S_{book}(z), equivalent to analyticity of r − S. But, as we will explain in Section 4, (1.8) is the more natural object. Rather than 1 − δ < |z| < R^{2}, [14] has R^{−1} < |z| < R^{2}, but that is wrong since \overline{D(1/z̄)} can have poles at the Nevai-Totik zeros.

D. Using Riemann-Hilbert methods, Deift-Ostensson [4] have extended the result on analyticity of r(z) − S(z) to {z | 1 − δ < |z| < R^{3}}.

E. Barrios-López-Saff [2] have proven that if

α_{n} = cR^{−n} + O(((1 + ε)R)^{−n−nm^{2}})  (1.10)

then D(z)^{−1} is meromorphic in {z | |z| < R^{2m−1} + δ} with poles precisely at z_{k} = R^{2k−1}, k = 1, 2, …, m. In particular, if (1.10) holds for all n, then D(z)^{−1} is entire meromorphic except for poles at R^{2k−1}, k = 1, 2, . . ..

Our main goal in this paper is to give a complete analysis of what can be said about α_{n} if D(z)^{−1} is meromorphic in some disk and, contrariwise, about D(z)^{−1} if α_{n} has an asymptotic expansion as a sum of exponentials. We describe our precise results in Section 4.

Along the way, we found a direct, simple proof of the Deift-Ostensson result that is also simpler than the argument Simon used for his weaker result in [14]. So we will give this proof next, then analyze two simple examples, and return in Section 4 to a general overview and sketch of the rest of the paper.

Of course, included among the entire meromorphic functions are the rational functions, and there is prior literature on this case. Szabados [18] considered the case D(z)^{−1} = 1/q(z) for a polynomial q and Ismail-Ruedemann [9] and Pakula [13] discussed D(z)^{−1} = p(z)/q(z) for polynomials p and q. They have some results on asymptotics of Φ_{n} but no discussion of links to the {α_{n}}_{n=1}^{∞}. As I was completing this manuscript, I received the latest draft of a paper of Martínez-Finkelshtein, McLaughlin, and Saff [10] that has some overlap with this paper.

I would like to thank P. Deift and J. Ostensson for illuminating discussions. This research was completed during my stay as a Lady Davis Visiting Professor at Hebrew University, Jerusalem. I’d like to thank H. Farkas and Y. Last for the hospitality of the Mathematics Institute at Hebrew University.

2. The R^{3} Result

Our goal in this section is to prove

Theorem 2.1. Let

lim sup|α_{n}|^{1/n} = R^{−1} < 1  (2.1)

so that D(z)^{−1} and S(z) are analytic in {z | |z| < R}. Then for some δ > 0, r(z) − S(z) is analytic in {z | 1 − δ < |z| < R^{3}}.

Remarks. 1. Here D is given by (1.3), S(z) by (1.8), and r(z) by (1.9). 2. The proof will make repeated use of Szegő recursion (1.5).

We introduce the symbol Õ by f = Õ(g) if and only if for all ε, |f|/|g|^{1−ε} → 0.

Lemma 2.2. Let (2.1) hold. Then

(a) For all ε > 0,

sup_{n, |z|<R−ε} |Φ_{n}^{∗}(z)| < ∞  (2.2)

(b) For |z| ≤ 1,

|Φ_{n}(z)| = Õ(max(R^{−1}, |z|)^{n})  (2.3)

(c) For |z| ≤ 1,

|Φ_{n}^{∗}(z) − D(0)D(z)^{−1}| = Õ(R^{−n} max(R^{−1}, |z|)^{n})  (2.4)

Remarks. 1. There is an implicit uniformity in z in the Õ statements (2.3), (2.4).

2. (a) is due to Nevai-Totik; (b) appears in Simon [16, 17].

Proof. (a) From (1.5) and |Φ_{n}(e^{iθ})| = |Φ_{n}^{∗}(e^{iθ})|, we see

sup_{|z|=1} |Φ_{n}(z)| ≤ ∏_{j=0}^{n−1}(1 + |α_{j}|)  (2.5)

so, by ∏_{j=0}^{∞}(1 + |α_{j}|) < ∞ and the maximum principle,

sup_{n, |z|≤1} |Φ_{n}^{∗}(z)| < ∞  (2.6)

from which we get, by (1.6), that

C_{1} ≡ sup_{n, |z|≥1} |z|^{−n}|Φ_{n}(z)| < ∞  (2.7)

The ^{∗} of (1.5) is

Φ_{n+1}^{∗}(z) − Φ_{n}^{∗}(z) = −α_{n}zΦ_{n}(z)  (2.8)

so that

Φ_{n}^{∗}(z) = 1 − ∑_{j=0}^{n−1} α_{j}zΦ_{j}(z)  (2.9)

Thus, by (2.7),

|Φ_{n}^{∗}(z)| ≤ 1 + C_{1} ∑_{j=0}^{n−1} |α_{j}| |z|^{j+1}  (2.10)

Given (2.1), we see that (2.2) holds.

(b) (2.2) and (1.6) imply that for |z| > R^{−1} + ε,

|Φ_{n}(z)| ≤ C_{ε}|z|^{n} (|z| > R^{−1} + ε)  (2.11)

This plus the maximum principle implies (2.3).

(c) It is a theorem of Szegő [19, 20] (see Theorem 2.4.1 of [14]) that in |z| < 1,

lim_{n→∞} Φ_{n}^{∗}(z) = D(0)D(z)^{−1} ≡ d(z)^{−1}  (2.12)

Thus, summing (2.8) to infinity,

|d(z)^{−1} − Φ_{n}^{∗}(z)| ≤ ∑_{j=n}^{∞} |α_{j}| |z| |Φ_{j}(z)|  (2.13)

Since α_{j} = Õ(R^{−j}) and (2.3) holds, we obtain (2.4). □

Proof of Theorem 2.1. We use the function d(z) of (2.12). Since Φ_{n}^{∗}(z) → d(z)^{−1} for |z| < 1 and (1.2) holds, the Vitali theorem implies d(z)^{−1} is analytic in {z | |z| < R} and Φ_{n}^{∗}(z) → d(z)^{−1} in that region. By summing (2.8) to infinity,

d(z)^{−1} = 1 − ∑_{j=1}^{∞} α_{j−1}zΦ_{j−1}(z)  (2.14)

which we write

d(z)^{−1} = \overline{d(1/z̄)}^{−1}S(z)+[1−\overline{d(1/z̄)}^{−1}]−∑_{j=1}^{∞} α_{j−1}z[Φ_{j−1}(z)−\overline{d(1/z̄)}^{−1} z^{j−1}]

(2.15)

where this formula is valid in {z | R^{−1} < |z| < R}.

Apply ^{∗} to (2.4) and see that in |z| ≥ 1,

|Φ_{n}(z) − \overline{d(1/z̄)}^{−1}z^{n}| ≤ Õ(R^{−n} max(|z|R^{−1}, 1)^{n})  (2.16)

Thus the summand in (2.15) is bounded in {z | |z| ≥ 1} by |z|Õ(R^{−2n} max(|z|R^{−1}, 1)^{n}). In 1 ≤ |z| ≤ R, this is bounded by Õ(RR^{−2n}) and in R ≤ |z| by Õ(|z|^{n+1}R^{−3n}). Thus, the sum in (2.15), which is a sum of functions each analytic in {z | |z| > R^{−1}}, converges uniformly in \{z | 1 ≤ |z| < R^{3}\}. Multiplying by \overline{d(1/z̄)}, which is analytic in {z | |z| > 1 − δ}, implies the result. □

3. Two Examples

We want to analyze two examples from [14] from the point of view of singularities of D(z)^{−1} and asymptotics of α_{n}. The first is already mentioned in this context in [2].

Example 3.1 (Rogers-Szegő polynomials; Example 1.6.5 of [14]). Here 0 < q < 1,

α_{n} = (−1)^{n}q^{(n+1)/2}  (3.1)

6 B. SIMON

and

D(z) = ∏_{j=0}^{∞}(1 − q^{j})^{1/2}(1 + q^{j+1/2}z)  (3.2)

Let R = q^{−1/2}. Then

S(z) = − ∑_{j=0}^{∞}(−1)^{j−1}q^{j/2}z^{j}
= (1 + zR^{−1})^{−1}  (3.3)

has a single pole at

z_{1} = −R  (3.4)

On the other hand, by (3.2), D(z) has a zero and so D(z)^{−1} a pole at

z_{ℓ} = −R^{2ℓ−1} ℓ = 1, 2, …  (3.5)

Example 3.2 (Single nontrivial moment; Example 1.6.4 of [14]). Fix 0 < a < 1 and let

dμ_{a}(θ) = (1 − a cos θ) \frac{dθ}{2π}  (3.6)

Let

μ_{±} = \frac{1}{a} ± \sqrt{(\frac{1}{a})^{2} − 1}  (3.7)

so μ_{−}μ_{+} = 1 and μ_{−} < 1. Then

D(z) = \sqrt{\frac{a}{2μ_{−}}} (1 − μ_{−}z)  (3.8)
= \sqrt{\frac{a}{2μ_{−}}} (1 − \frac{z}{μ_{+}} )  (3.9)

so D(z)^{−1} has a single pole at

z_{1} = μ_{+}  (3.10)

On the other hand,

α_{n} = \frac{−(μ_{+} − μ_{−})}{(μ_{+}^{n+2} − μ_{−}^{(n+2)})}
= −(μ_{+} − μ_{−})μ_{+}^{−n−2}(1 − μ_{+}^{−(2n+4)})^{−1}
= −(μ_{+} − μ_{−}) ∑_{j=1}^{∞}(μ_{+}^{−n−2})^{2j−1}  (3.11)

and so S(z) has poles at

z_{j} = μ_{+}^{2j−1} j = 1, 2, …  (3.12)

In these examples, the set of singularities of S and of D(z)^{−1} are distinct and one or the other might be larger. If {z_{j}} are the singularities, then {z_{j}^{2ℓ−1}}_{j,ℓ=1,2,...} are identical for S and D^{−1}, which motivates the 𝔾 construction of the next section.

4. Overview and Discussion of Further Results

Definition. A sequence {A_{n}}_{n=−1}^{∞} of complex numbers is said to have an asymptotic series with error R^{−n} for some R > 1 if and only if there exists a finite number of points {μ_{j}}_{j=1}^{J} in {w | 1 < |w| < R} and polynomials {P_{j}}_{j=1}^{J} so that

lim sup_{n→∞} |A_{n} − ∑_{j=1}^{J} P_{j}(n)μ_{j}^{−(n+1)}|^{1/n} ≤ R^{−1}  (4.1)

Equivalently,

A_{n} = ∑_{j=1}^{J} P_{j}(n)μ_{j}^{−n−1} + Õ(R^{−n})

We say A_{n} has a complete asymptotic series if it has an asymptotic series with error R^{−n} for all R > 1.

In many ways, our main result in this paper is:

Theorem 4.1. Let dμ be a nontrivial probability measure on ∂𝔻 with Verblunsky coefficients, α_{n}. Then α_{n} has a complete asymptotic series if and only if

(1) dμ_{s} = 0 and dμ obeys the Szegő condition.

(2) D(z)^{−1} is an entire meromorphic function.

Of course,

∑_{n=0}^{∞} z^{n}μ_{j}^{−n} = (1 − \frac{z}{μ_{j}} )^{−1}  (4.2)

and so, taking derivatives, for ℓ = 1, 2 …,

∑_{n=0}^{∞}(n + ℓ)(n + ℓ − 1) … (n + 1)z^{n}μ_{j}^{−n} = ℓ!(1 − \frac{z}{μ_{j}} )^{−ℓ−1} (4.3)

So (4.1) is equivalent to a sum of explicit pole terms:

Proposition 4.2. {A_{n}}_{n=−1}^{∞} has an asymptotic series with error R^{−n} if and only if

F (z) = ∑_{n=0}^{∞} A_{n−1}z^{n}  (4.4)

is meromorphic in {z | |z| < R} with a finite number of poles, all in {z | 1 < |z| < R}. In particular, {A_{n}}_{n=−1}^{∞} has a complete asymptotic series if and only if F (z) is an entire meromorphic function.

Thus, Theorem 4.1 is equivalent to

Theorem 4.3. (1) and (2) of Theorem 4.1 are equivalent to the function S of (1.8) being an entire meromorphic function.

Note that the μ_{j}’s and P_{j}’s are determined uniquely by the A_{n}’s. Both to prove the results and for its intrinsic interest, we are inter-

ested in the relation between the poles of S(z) and of D(z)^{−1} and in results in fixed circles. By a discrete exterior set, we mean a subset, T, of {w | 1 < |w| < ∞} so that #[{w | 1 < |w| < R} ∩ T] is finite for each R > 1. Given a discrete exterior set T, define for k = 1, 2, …,

𝔾^{(2k−1)}(T) = {λ_{i_{1}} … λ_{i_{k}} λ̄_{i_{k+1}} … λ̄_{i_{2k−1}} | λ_{j} ∈ T }  (4.5)
𝔾_{2k−1}(T) = ⋃_{j=k}^{∞} 𝔾^{(2j−1)}(T)  (4.6)
𝔾(T) = 𝔾_{1}(T)  (4.7)

𝔾(T) will be called the generated set. Note that

𝔾(𝔾(T)) = 𝔾(T)  (4.8)

We will prove the following:

Theorem 4.4. Let α_{n} be a set of Verblunsky coefficients with complete asymptotic series and let T be the set of λ_{j}’s that enter in the series. Let P be a set of poles of D(z)^{−1}. Then

T ⊂ 𝔾(P) and P ⊂ 𝔾(T)  (4.9)

This implies the following refined form of Theorem 4.1:

Theorem 4.5. Let Q be an exterior discrete set with 𝔾(Q) = Q. Then α_{n} is a set of Verblunsky coefficients with λ_{j}’s in Q if and only if condition (1) of Theorem 4.1 holds and the poles of D(z)^{−1} lie in Q.

Theorems 4.1, 4.4, and 4.5 are equivalence results and thus both a direct (going from α to D) and inverse (going from D to α) aspect. Generally, direct arguments are simpler than inverse. We will actually deduce everything from direct arguments and a bootstrap. An inverse argument is only used to start the analysis, and that was already done by Nevai-Totik. Here is the master stepping stone we will need. Throughout, we suppose there is R > 1 so

lim sup_{n→∞} |α_{n}|^{1/n} = R^{−1}  (4.10)

Theorem 4.6. Fix ℓ = 1, 2, . . .. Suppose S(z) is meromorphic in

R_{ℓ} = {z | 0 < |z| < R^{2ℓ−1}}  (4.11)

Then D(z)^{−1} is meromorphic there and the poles of D(z)^{−1} there lie in 𝔾(T_{ℓ}) where T_{ℓ} is the set of poles of S(z) in R_{ℓ}. Moreover, r(z) − S(z) has a meromorphic continuation to R_{ℓ+1} ∩ {z | 1 > |z| − δ} and the poles of this difference lie in 𝔾_{3}(T_{ℓ}).

We are heading towards a proof that Theorem 4.6 implies the earlier Theorems 4.1, 4.4, and 4.5. We need a preliminary notion and fact.

Definition. Let Q be an exterior discrete set with 𝔾(Q) = Q. We say that W ⊂ Q is a set of minimal generators if and only if 𝔾(W) = Q and 𝔾_{3}(W) ∩ W = ∅.

Proposition 4.7. Any exterior discrete set Q with 𝔾(Q) = Q has a minimal set of generators.

Proof. Order the points in Q, w_{1}, w_{2}, … so |w_{n}| ≤ |w_{n+1}|. Define W inductively by putting w_{n} in W if and only if w_{n} ∉ 𝔾_{3}({w_{1}, …, w_{n−1}}). It is easy to see that W is a set of minimal generators. □

Proof that Theorem 4.6 implies Theorems 4.4, 4.5, and 4.1. It suffices to prove that D^{−1} is entire meromorphic if and only if S is, and to prove Theorem 4.4 since it in turn implies Theorem 4.5, which implies Theorem 4.1. If S(z) is entire meromorphic, it is meromorphic in each R_{ℓ}, so D(z)^{−1} is meromorphic in each R_{ℓ} and, clearly, P ⊂ 𝔾(T).

Conversely, if D(z)^{−1} is entire meromorphic, we prove S(z) is entire meromorphic by proving inductively that it is meromorphic in each R_{ℓ}. S(z) is meromorphic in R_{1} by the Nevai-Totik theorem. If we know S(z) is meromorphic in R_{ℓ}, then by Theorem 4.6, r(z) − S(z) is meromorphic in R_{ℓ+1}\R_{ℓ} so, since r(z) is meromorphic on R_{ℓ+1}, we conclude that S(z) is meromorphic there also.

Finally, to identify the points of T, as lying in 𝔾(P) with P the poles of D^{−1}(z), suppose W is a set of minimal generators of T. If w_{j} ∈ W, then w_{j} ∉ 𝔾_{3}(T), so S − r is regular at w_{j} by Theorem 4.6. Since w_{j} is a singularity of S, it must be a singularity of r, that is, w_{j} ∈ P. Thus, T = 𝔾(W) ⊂ 𝔾(P). □

Our proof of Theorem 4.6 will also show

Theorem 4.8. Suppose z_{0} ∈ 𝔾^{3}(T) has a unique expression as z_{0} = μ_{1}^{2}μ̄_{2} with μ_{1}, μ_{2} ∈ T. Suppose also that z_{0} ∉ T ∪ 𝔾_{5}(T). Then r(z) − S(z) has a singularity at z_{0}.

Remarks. 1. For example, if S(z) has a single pole, z_{0}, with |z_{0}| = R, then either S or D^{−1} or both have a pole at z_{0}R^{2}.

2. Our proof shows that if the poles of S(z) have {log|z_{j}|} independent over the rationals, D^{−1} has a pole at every point in 𝔾(T).

3. Our proof also allows the precise calculation of the singularity in S − r at any point in 𝔾(T). There can be cancellations if z_{0} can be written as a product in 𝔾(T) in more than one way. So one cannot guarantee a singularity of r(z)−S(z) at every point of 𝔾_{3}(T), but that will happen in some generic sense.

Note that our results generalize those of Barrios-López-Saff [2] in three ways:

(a) They only have results on the the direct problem, that is, going from α to D^{−1}, while we have results in both directions.

(b) They allow only a single term in the α asymptotics.

(c) Their error assumptions in case of disks are much stronger (R^{−nm^{2}} vs. R^{−n(2m+1)}) than ours.

This concludes the description of our main results — and reduces everything to proving Theorems 4.6 and 4.8. We will do this for 2ℓ−1 = 3 in Section 5 and general 2ℓ − 1 in Section 6.

One could analyze other situations such as where S(z) has a branch cut associated with specific asymptotics for α_{n} such as n^{β}R^{−n} with β nonintegral.

We close this section, which is the continuation of the introduction, with two remarks. First, there is a scattering theoretic interpretation of S and r. Since ∑_{n=0}^{∞}|α_{n}|^{2} < ∞, one can define wave operators (see Geronimo-Case [6] and Section 10.7 of [15]), Ω^{±} : L^{2}(∂𝔻, \frac{dθ}{2π}) → L^{2}(∂𝔻, |D|^{2} \frac{dθ}{2π}), which obey

(Ω^{+}f)(θ) = D(e^{iθ})^{−1}f(θ) (Ω^{−}f)(θ) = \overline{D(e^{iθ})}^{−1}f(θ)  (4.12)

Thus, the reflection coefficient is given by

((Ω^{−})^{−1}Ω^{+}f)(θ) = \overline{D(e^{iθ})} D(e^{iθ})^{−1}f(θ)  (4.13)

so r(z) is the analytic continuation of the reflection coefficient. S(z) is the leading Born approximation to r (see Newton [12] and Chadan-Sabatier [3] for background on scattering theory). While we will not study it from this point of view, it is presumably true that the arguments in the next two sections can be interpreted as use of some kind of Born series.

The second issue concerns a comparison between the basic formula used by Nevai-Totik [11] to do the inverse problem and a different, but similar-looking, formula used in our discussion, namely (2.14). The formula they use, where they quote Freud [5], is also in Geronimus [8]:

α_{n} = −κ_{∞} ∫ \overline{Φ_{n+1}(e^{iθ})} D(e^{iθ})^{−1} dμ(θ)  (4.14)

where κ_{∞} = lim_{n→∞} κ_{n} with κ_{n} = ∥Φ_{n}∥^{−1}, so κ_{n} = φ_{n}^{∗}(0) and

κ_{∞} = D(0)^{−1}  (4.15)

(4.14) only holds if dμ_{sing} = 0.

Since φ_{n} = κ_{n}Φ_{n}, (4.14) can be rewritten as

α_{n} = −κ_{∞}κ_{n}^{−1}⟨φ_{n+1}, D^{−1}⟩  (4.16)

Since

⟨1, D^{−1}⟩ = ∫ \overline{D}(e^{iθ}) \frac{dθ}{2π} = \overline{D(0)} = κ_{∞}^{−1}

(4.16) also holds if we interpret α_{−1} = −1 and κ_{−1} = 1. Thus, with α_{−1} = −1, (4.16) is equivalent to

d(z)^{−1} ≡ D(0)D(z)^{−1} = −κ_{∞}^{−2} ∑_{n=−1}^{∞} κ_{n}α_{n}φ_{n}(z)  (4.17)

On the other hand, (2.14) says

(d(z)^{−1} − 1)z^{−1} = − ∑_{n=0}^{∞} α_{n}κ_{n}^{−1}φ_{n}(z)  (4.18)

or equivalently,

α_{n} = −κ_{∞}^{−1}κ_{n}^{2} ∫ \overline{Φ_{n}(e^{iθ})} [D(e^{iθ})^{−1} − D(0)^{−1}]e^{−iθ} dμ(θ)  (4.19)

These formulae are distinct, and it is striking that both are true and their proofs (see (2.4.35) of [14]) are so different. Where Nevai-Totik [11] use (4.14), one could just as well use (4.19).

5. The R^{5} Result

In this section, as a warmup and also as the start of induction for the general case, we consider the case 2ℓ − 1 = 3, that is, ℓ = 2 where we deal with induced singularities in {z | R^{3} ≤ |z| < R^{5}}. Thus, we should suppose

α_{n} = ∑_{k=1}^{K} P_{k}(n)μ_{k}^{−n−1} + Õ(R^{−3n})  (5.1)

with R ≤ |μ_{k}| < R^{3}. Here, P_{k}(n) are polynomials. We will instead suppose that

α_{n} = ∑_{k=1}^{K} c_{k}μ_{k}^{−n−1} + Õ(R^{−3n})  (5.2)

The consideration of general P_{k}’s rather than constants presents no difficulties other than notational ones, so we spare the reader. Our goal is to prove

Theorem 5.1. If (5.2) holds, then D(z)^{−1} is meromorphic in {z | |z| < R^{3}} with poles precisely at {μ_{k}}_{k=1}^{K}. In addition, S(z) − r(z) is meromorphic in {z | 1 − δ < |z| < R^{5}} with poles contained in 𝔾^{(3)}({μ_{k}}_{k=1}^{K}). Moreover, if z_{0} = μ_{i_{1}}^{2}μ̄_{i_{2}} in precisely one way and |z_{0}| ≤ R^{5}, then S(z) − r(z) has a pole at z_{0}.

We note that the first statement is immediate from Theorem 2.1, so we will focus on R^{3} ≤ |z| < R^{5}. We will follow the same three-step strategy used in Section 2:

(i) Estimate Φ_{n} in {z | |z| < R^{−3}(1 + δ)}. (ii) Estimate Φ_{n}^{∗} − d(z)^{−1} in {z | |z| < R^{−3}(1 + δ)} using (2.8) and

Step (i). (iii) Estimate S(z) − r(z) in |z| > R^{3}/(1 + δ) using (2.8), the formula

Φ_{n}(z)z^{−n} = \overline{Φ_{n}^{∗}(1/z̄)}, and the estimate in Step (ii). What will be different from Section 2 is that we will find the leading

asymptotics of Φ_{n} rather than just use |Φ_{n}(z)| ≤ Õ(R^{−n}) in |z| < R. In essence, this leading asymptotics was discussed in [17] and we will use the techniques from there, although in a slightly more general context.

Theorem 5.2. Suppose (5.2) holds. Choose δ, so for all k, |μ_{k}| < R^{3}/(1 + δ). Define E_{n}(z) in {z | |z| < R^{−3}(1 + δ)} by

Φ_{n}(z) = −d(z)^{−1} ∑_{k=1}^{K} c̄_{k}μ̄_{k}^{−n}(1 − zμ̄_{k})^{−1} + E_{n}(z)  (5.3)

Then, for |z| < R^{−3}(1 + δ),

|E_{n}(z)| = Õ(max(R^{−3}, |z|)^{n})  (5.4)

Remark. Since |μ_{k}| < R^{3}(1 + δ)^{−1} and |z| < R^{−3}(1 + δ), ||z|μ_{k}| < 1.

Proof. Iterating (1.5) from j = n − 1 down to j = 0 yields

Φ_{n}(z) = z^{n} − ∑_{j=1}^{n} ᾱ_{n−j}z^{j−1}Φ_{n−j}^{∗}(z)  (5.5)

Write (5.2) as

α_{n} = ∑_{k=1}^{K} c_{k}μ_{k}^{−n−1} + (δα)_{n}  (5.6)
(δα)_{n} = Õ(R^{−3n})  (5.7)

In (5.5), do the following:

E_{0}^{(n)} = z^{n}  (5.8)
E_{1}^{(n)} = − ∑_{j=1}^{n} ᾱ_{n−j}z^{j−1}[Φ_{n−j}^{∗}(z) − d(z)^{−1}]  (5.9)

E_{2}^{(n)} = − ∑_{j=1}^{n} \overline{(δα)}_{n−j} z^{j−1} d(z)^{−1}  (5.10)

E_{3}^{(n)} = ∑_{k=1}^{K} c̄_{k} ∑_{j=n+1}^{∞} μ̄_{k}^{−(n+1−j)}z^{j−1} d(z)^{−1}  (5.11)

Since ∑_{j=1}^{∞} μ̄_{k}^{−(n+1−j)}z^{j−1} = μ̄_{k}^{−1}(1 − zμ̄_{k})^{−1}, (5.3) holds where

E_{n}(z) = E_{0}^{(n)} + E_{1}^{(n)} + E_{2}^{(n)} + E_{3}^{(n)}

We need to show that for j = 0, 1, 2, 3,

|E_{j}^{(n)}(z)| = Õ(max(R^{−3}, |z|)^{n})  (5.12)

This is trivial for j = 0. For j = 1, we use (2.4) and |α_{n−j}| = Õ(R^{−(n−j)}) to see if |z| < R^{−1}:

|E_{1}^{(n)}| ≤ ∑_{j=1}^{n} |z|^{j−1}Õ(R^{−(n−j)})Õ(R^{−2(n−j)})
= nÕ(max(|z|, R^{−3})^{n})
= Õ(max(|z|, R^{−3})^{n})

By (5.7), d(z)^{−1} is bounded in {z | |z| < R^{−1}} and nÕ(max(|z|, R^{−3})^{n}) = Õ(max(|z|, R^{−3})^{n}), we have that (5.12) holds for j = 2. For j = 3, we note that the sum of the geometric series is z^{n}(1 − μ̄_{k}z)^{−1}, so in |z| < R^{−3}(1 + δ), we get a |z|^{n} bound. This proves (5.12). □

Theorem 5.3. Let (5.2) hold and let δ be as in Theorem 5.2. Define Ẽ_{n} for |z| < R^{−3}(1 + δ),

Φ_{n}^{∗}(z)−d(z)^{−1} = −[d(z)^{−1} ∑_{k,ℓ=1}^{K} c̄_{k}c_{ℓ}μ_{ℓ}^{−1}(1−zμ̄_{k})^{−1}(1−μ̄_{k}^{−1}μ_{ℓ}^{−1})(μ̄_{k}μ_{ℓ})^{−n}]+Ẽ_{n}(z)

(5.13)

Then for |z| < R^{−3}(1 + δ),

Ẽ_{n}(z) = Õ(R^{−n} max(|z|, R^{−3})^{n})  (5.14)

Proof. We iterate (2.8) to get

Φ_{n}^{∗}(z) − d(z)^{−1} = ∑_{j=n}^{∞} α_{j}zΦ_{j}(z)  (5.15)

In (5.15), first replace α_{j} by (5.6) and then, in the main term, replace Φ_{j} by (5.3). Noting that ∑_{j=n}^{∞} μ̄_{k}^{−j}μ_{ℓ}^{−j−1} = μ_{ℓ}^{−1}(1−μ̄_{k}^{−1}μ_{ℓ}^{−1})^{−1}(μ̄_{k}μ_{ℓ})^{−n}, we see that (5.13) holds, where

Ẽ_{n} = Ẽ_{1}^{(n)} + Ẽ_{2}^{(n)}

with

Ẽ_{1}^{(n)} = ∑_{j=n}^{∞} (δα_{j})zΦ_{j}(z)  (5.16)

Ẽ_{2}^{(n)} = ∑_{j=n}^{∞}( ∑_{k=1}^{K} c_{k}μ_{k}^{−j−1})zE_{j}(z)  (5.17)

By (5.7) and (2.3),

|E_{1}^{(n)}| ≤ c ∑_{j=n}^{∞} Õ(R^{−3n})Õ(R^{−n}) = Õ(R^{−4n})  (5.18)

By (5.4),

|E_{2}^{(n)}| ≤ c ∑_{j=n}^{∞} R^{−j−1} Õ(max(R^{−3}, |z|)^{j})  (5.19)
= Õ(R^{−n} max(|z|, R^{−3})^{n})

proving (5.14). □

Theorem 5.4. Suppose (5.2) holds and δ is chosen as in Theorem 5.2. Then in {z | R^{3}(1 + δ)^{−1} < |z| < R^{5}}, we have that r(z) − S(z) − q_{3}(z)

MEROMORPHIC SZEGŐ FUNCTIONS 15

is analytic where

q_{3}(z) = − ∑_{k,ℓ,r=1}^{K} c_{k}c̄_{ℓ}c_{r} z(z − μ_{k})^{−1}(1 − μ_{k}^{−1}μ̄_{ℓ}^{−1})^{−1}μ_{k}(1 − zμ_{k}^{−1}μ̄_{ℓ}^{−1}μ_{r}^{−1})^{−1}

(5.20)

Proof. By (2.15),

r(z) − S(z) = [ \overline{d(1/z̄)} − 1] − \overline{d(1/z̄)} ∑_{j=1}^{∞} α_{j−1}z[Φ_{j−1} − \overline{d(1/z̄)}^{−1}z^{−j−1}]

(5.21)

Because q_{3}(z) is obtained by summing

∑_{j=1}^{∞} (α_{j−1} − δα_{j−1})[Φ_{j−1} − \overline{d(1/z̄)} z^{j−1} − z^{j−1} \overline{Ẽ_{j−1}(1/z̄)} ] (5.22)

we see that

r(z) − S(z) − q_{3}(z) = −\overline{d(1/z̄)} ∑_{j=1}^{∞} F_{1,j}(z) + F_{2,j}(z)  (5.23)

where

F_{1,j}(z) = δα_{j−1}z[Φ_{j−1} − d(1/z̄)z^{j−1}]  (5.24)
F_{2,j}(z) = [α_{j−1} − δα_{j−1}]z^{j−1} \overline{Ẽ_{j−1}(1/z̄)}  (5.25)

By (2.16) and (5.7),

|F_{1,j}(z)| = Õ(R^{−3j}R^{−j} max(|z|R^{−1}, 1)^{j})  (5.26)

so if R^{3}/(1 + δ) < |z| < R^{5},

∑_{j=1}^{∞} |F_{1,j}(z)| ≤ ∑_{j=1}^{∞} R^{−5j}|z|^{j} < ∞  (5.27)

By (5.14), if |z| > R^{3}/(1 + δ),

z^{k}Ẽ_{k}(1/z̄) = Õ(R^{−n} max(1, |z|R^{−3})^{n})  (5.28)

and thus,

|F_{2,j}(z)| ≤ Õ(R^{−2j} max(1, |z|R^{−3})^{j})  (5.29)
≤ \begin{cases}Õ(|z|^{j}R^{−5j}) & if |z| ≥ R^{3} \\ Õ(R^{−2j}) & if |z| ≤ R^{3} \end{cases}  (5.30)

so ∑_{j=1}^{∞}|F_{2,j}(z)| < ∞ uniformly on compacts of {z | R^{3}/(1+δ) ≤ |z| < R^{5}}. This implies r(z) − S(z) − q_{3}(z) is analytic there. □

Proof of Theorem 5.1. As already noted, Theorem 2.1 proves the results in {z | 1 < |z| < R^{3}}. In {z | R^{3}/(1+δ) < |z| < R^{5}}, Theorem 5.4 shows r−S is meromorphic with poles contained in 𝔾^{(3)}({μ_{k}}_{k=1}^{K}). The explicit formula shows that there is a pole if there is a single summand contributing to the potential pole. □

6. The R^{2ℓ−1} Result

In this section, we will prove the following (again, for simplicity of exposition, we replace general P_{j}(n) by constants), which clearly implies Theorems 4.6 and 4.8.

Theorem 6.1. Let {μ_{k}}_{k=1}^{K} obey R ≤ |μ_{k}| < R^{2ℓ−1} with min_{k}|μ_{k}| = R. Suppose that

α_{n} = ∑_{k=1}^{K} c_{k}μ_{k}^{−n−1} + Õ(R^{−(2ℓ−1)})  (6.1)

Then D(z)^{−1} is meromorphic in {z | |z| < R^{2ℓ−1}} with poles contained in 𝔾({μ_{k}}_{k=1}^{K}). In addition, S(z)−r(z) is meromorphic in {z | 1−δ_{0} < |z| < R^{2ℓ+1}} and the only poles in {z | R^{2ℓ−1} ≤ |z| < R^{2ℓ+1}} lie in 𝔾_{3}({μ_{k}}_{k=1}^{K}). If z_{0} obeys z_{0} = μ_{i_{1}}^{2}μ̄_{i_{2}} with |z_{0}| < R^{2ℓ+1} and z_{0} cannot be written as any other 𝔾({μ_{k}}_{k=1}^{K}) product, then s(z)−r(z) has a pole at z_{0}.

The strategy is the same as in the last section. Pick δ > 0 so that |μ_{k}| < R^{2ℓ−1}/(1 + δ) for all k. We will prove the following estimate on the Φ’s and Φ^{∗}’s inductively:

Theorem 6.2. Under the hypothesis of Theorem 6.1, in Q ≡ {z | |z| ≤ R^{−(2ℓ−1)}(1 + δ)}, we have

Φ_{n}(z) = ∑_{p} f_{p,ℓ}^{(ℓ)}(z)w̄_{p,ℓ}^{−n} + E_{n,ℓ}(z)  (6.2)

where the sum is over all points w in

[ ⋃_{m=1}^{2ℓ−3} 𝔾^{(2m−1)}(\{μ_{k}\}_{k=1}^{K})]⋂\{z | R ≤ |z| < \frac{R^{2ℓ−1}}{1 + δ} \}

each f_{p,ℓ}^{(ℓ)} is analytic in Q, and on Q,

|E_{n,ℓ}(z)| = Õ(max(R^{−(2ℓ−1)}, |z|)^{n})  (6.3)

In addition, in Q,

Φ_{n}^{∗}(z) − d(z)^{−1} = ∑_{p} g_{p,ℓ}(z)y_{p,ℓ}^{−n} + Ẽ_{n,ℓ}(z)  (6.4)

where the sum is over all products, p = μ_{i_{1}} … μ_{i_{m}} μ̄_{i_{m+1}} … μ̄_{i_{2n}} with i_{1}, …, i_{2m} ∈ {1, …, K} and with the product lying in {z | 2R ≤ |z| < R^{2ℓ}/(1 + δ)} and on Q, each g_{p,ℓ} is analytic and

|Ẽ_{n,ℓ}(z)| = Õ(R^{−n}(max(R^{−(2ℓ−1)}, z))^{n})  (6.5)

Proof. The proof is by induction in ℓ. Theorems 5.2 and 5.3 establish the case ℓ = 2. Suppose that we have the result for ℓ − 1 with ℓ ≥ 3 and that (6.1) holds. Write α_{n} as in (5.6) where now

(δα)_{n} = Õ(R^{−(2ℓ−1)n})  (6.6)

In (5.5), do the following:

E_{0,ℓ}^{(n)} = z^{n}  (6.7)
E_{1,ℓ}^{(n)} = − ∑_{j=1}^{n} ᾱ_{n−j}z^{j−1}[Φ_{n−j}^{∗} − d(z)^{−1} − ∑_{p} g_{p,ℓ−1}(z)y_{p,ℓ−1}^{−(n−j)}] (6.8)

E_{2,ℓ}^{(n)} = − ∑_{j=1}^{n} \overline{(δα)}_{n−j}z^{j−1}[d(z)^{−1} + ∑_{p} g_{p,ℓ−1}(z)y_{p,ℓ−1}^{(n−j)}]  (6.9)

Then

Φ_{n}−∑_{k=0}^{2} E_{k,ℓ}^{(n)} = − ∑_{j=1}^{n}( ∑_{k=1}^{K} c̄_{k}μ̄_{k}^{−(n−j)−1})[d(z)^{−1}+∑_{p} g_{p,ℓ−1}(z)y_{p,ℓ−1}^{−(n−j)}]

(6.10)

Define E_{3,ℓ}^{(n)} to be the summand on the right side of (6.10) with the sum from n + 1 to ∞.

The infinite sum yields geometric series which precisely have the form ∑_{p} g_{p,ℓ}(z)y_{p,ℓ}^{−n}, and as in the proof of Theorem 5.1,

|E_{0,ℓ}^{(n)}| + |E_{1,ℓ}^{(n)}| + |E_{2,ℓ}^{(n)}| + |E_{3,ℓ}^{(n)}| = Õ(max(|z|, R^{−(2ℓ−1)}))

since Ẽ_{n,ℓ−1} by induction has Õ(max(R^{−2ℓ}, |z|)^{n}) decay and other terms are bounded by Õ(max(R^{−1}, |z|)^{n}). This proves (6.3) for ℓ.

To bound Φ_{n}^{∗}(z) − d(z)^{−1}, we use (5.15), replace Φ_{j}(z) by (6.2), α_{n} by (δα)_{n} plus the asymptotic exponentials, and obtain (6.4) and (6.5) for ℓ by the same estimate as in Theorem 5.3. □

Basically, using Φ_{n}^{∗} to order R^{−2(ℓ−1)n} in the expansion of Φ_{n} gets us Φ_{n} to order R^{−(2ℓ−1)n}, and then plugging that into the expansion of Φ_{n}^{∗} gets us Φ_{n}^{∗} to order R^{−2ℓn}. Each full iteration improves by R^{−2n}.

Proof of Theorem 6.1. By induction, S −r is meromorphic in {z | |z| < R^{2ℓ−1}}, so knowing S meromorphic implies meromorphicity of r and so D^{−1} there, and the poles of both S − r and S lie in 𝔾({μ_{k}}_{k=1}^{K}).

Using (6.4) in |z| < R^{−(2ℓ−1)}/(1 + δ) yields an expansion of Φ_{n}(z) in |z| > R^{2ℓ−1}/(1 + δ). Plug this into (5.21) and use (5.6). The purely geometric terms sum to poles in {z | R^{2ℓ−1} < |z| < R^{2ℓ+1}}. The bounds on the errors as in the proof of Theorem 5.4 converge to an analytic function in the annulus. The poles are clearly in 𝔾_{3}({μ_{k}}_{k=1}^{K}).

Tracking the contribution of a single μ_{1}^{2}μ̄_{2} shows that it yields a nonvanishing pole which, by the unique product hypothesis, cannot be cancelled. □

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