FINE STRUCTURE OF THE ZEROS OF ORTHOGONAL POLYNOMIALS,

II. OPUC WITH COMPETING EXPONENTIAL DECAY

BARRY SIMON*

Abstract. We present a complete theory of the asymptotics of the zeros of OPUC with Verblunsky coefficients α_{n} = ∑_{ℓ=1}^{L} C_{ℓ}b_{ℓ}^{n} + O((bΔ)^{n}) where Δ < 1 and |b_{ℓ}| = b < 1.

1. Introduction

This paper is the second in a series [7, 8] that discusses detailed asymptotics of the zeros of orthogonal polynomials with special emphasis on distances between nearby zeros. We will focus here on OPUC, orthogonal polynomials on the unit circle; see [9, 2, 5, 6] for background. The polynomials are described by the recursion coefficients, {α_{n}}_{n=0}^{∞}, called Verblunsky coefficients, that give the monic OPUC, Φ_{n}(z), by

Φ_{n}(z) = zΦ_{n}(z) − ᾱ_{n}Φ_{n}^{∗}(z)  (1.1)

where

Φ_{n}^{∗}(z) = z^{n} \overline{Φ_{n}(1/z̄)}  (1.2)

One of the examples discussed in the first paper [7] is where 0 < b < 1 and

α_{n} = Cb^{n} + O((bΔ)^{n})  (1.3)

for 0 < Δ < 1. A typical example is shown in Figure 1 where

α_{n} = (\frac{1}{2})^{n+1}  (1.4)

and zeros of Φ_{22} are shown. Figures and numeric zeros, which appear in [5, 7] and here, are computed using Mathematica and code written by M. Stoiciu.

Critical aspects of the zeros in this case are:

(a) Finitely many zeros outside |z| = b+O(log n/n) at the Nevai-Totik points, that is, solutions of D(1/z̄)^{−1} = 0 where D is the Szegő function; this is due to Nevai-Totik [4].

Date: November 8, 2004.

∗ Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, USA. E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0140592.

(b) No zeros in |z| < b−O(log n/n). This is due to Barrios-López-Saff [1].

(c) Zeros near |z| = b are asymptotically a distance 2πb/n + o(1/n) from each other except for a single gap at z = b, where the nearby zeros are be^{±2πi/n}+O(1/n^{2}), and the distance between these neighboring zeros is 2(2πb/n) + O(1/n^{2}). This is proven in [7].

Figure 1

In this paper, I will analyze the case of Verblunsky coefficients of the form

α_{n} = ∑_{ℓ=1}^{L} C_{ℓ}b_{ℓ}^{n} + O((bΔ)^{n})  (1.5)

where the b_{ℓ}’s are distinct, C_{ℓ} ≠ 0 for all ℓ, and

|b_{ℓ}| = b ℓ = 1, …, L  (1.6)

If the b_{ℓ}’s obey b_{ℓ}^{p} = b^{p} for some p, then α_{n+1}/α_{n} is periodic of period p, and this overlaps examples of BLS [1] discussed later.

A typical example is shown in Figure 2 where

α_{n} = (\frac{1}{2})^{n+1}(1 + 2 cos(\frac{π}{2}(n + 1)))  (1.7)

and again, zeros of Φ_{22} are shown. This has b_{1} = 1/2, b_{2} = i/2, b_{3} = −i/2.

Figure 2

Here is what happens to (a)–(c) above: (a′) The Nevai-Totik theory still applies. There are three NT zeros in

the example in Figure 2. (b′) There are at most L − 1 zeros in |z| < b − O(log n/n) and they

can be described explicitly. In the case of periodic α_{n+1}/α_{n}, the interior zeros for Φ_{n} with n = mp + q and m → ∞ have explicitly computable limits (limits, not merely accumulation points). For the example in Figure 2, Φ_{n}, n ≡ 2 (mod 4), have a zero approaching \frac{1}{2}(\sqrt{2} − 1) = 0.20710678118 . . .. The actual zero in Figure 2 is at 0.20710678374 . . ..

(c′) There are gaps at each b̄_{ℓ}.

The method used in [7] to prove (a)–(c) exploits a second-order difference equation that Φ_{n} obeys, relating Φ_{n+1} to Φ_{n} and Φ_{n−1}. That method may extend to the periodic case, b_{ℓ}^{p} = b, but will have small divisor problems if α_{n}/b^{n} is only almost periodic. Instead, this paper will use a different and potentially more powerful and illuminating method that views (1.1) with Φ_{n}^{∗}(z) fixed as an inhomogeneous first-order difference equation for Φ_{n}(z).

In Section 2, we discuss asymptotics of Φ_{n} for |z| < b. In Section 3, we discuss asymptotics of Φ_{n} in the critical region bΔ_{1} < |z| < bΔ_{1}^{−1}. In Section 4, we study zeros in |z| < b, and in Section 5, the zeros near |z| = b. Given Section 2, Section 4 is straightforward. Section 5 will use the ideas in [7]. Finally, Section 6 makes various remarks about the connection to [1].

Figures 1 and 2 suggest that there might be a connection between the gaps in the clock and the Nevai-Totik zeros since in these two cases the number of NT zeros equals the number of gaps, and the zeros are near the gaps. There is certainly something to this notion. If a coefficient C_{ℓ} in (1.5) is changed from zero to nonzero, for large n, the zero that was in the gap at zero value of C_{ℓ} must stop being on the critical circle, and so presumably turns into an NT zero (this is an expectation, not a proof, since a proof involves controlling an interchange of limits). But the connection is not always there. Figure 8.3 in [5] shows an example where there is a gap but no associated NT zero. Moreover, the discussion in Section 13 of [7] makes it clear that the number of NT zeros can be arbitrary and is not, in general, the same as the number of gaps in the clock.

I thank Percy Deift and Chuck Newman for the hospitality of the Courant Institute where some some of this work was done.

2. Asymptotics in |z| < b − ε

Our main goal here is to give asymptotics of φ_{n}(z) in the region |z| < b in case (1.5) holds. Our methods will also allow us to say something when weaker asymptotics (ratio asymptotics) holds and, in particular, to improve a result of BLS [1].

We begin with an analysis of some bounds and rate of convergence of Φ_{n}^{∗}. Estimates similar to these appear in [4, 3, 1, 7]. We use q rather than b since sometimes q = b + ε.

Proposition 2.1. Suppose that

|α_{n}| ≤ Cq^{n}  (2.1)

for some q ∈ (0, 1). Then

(i) With C_{1} = ∏_{j=0}^{∞}(1 + Cq^{j}) < ∞, we have

|z| ≤ 1 ⇒ |Φ_{n}^{∗}(z)| ≤ C_{1}  (2.2)
|z| ≥ 1 ⇒ |Φ_{n}(z)| ≤ C_{1}|z|^{n}  (2.3)

(ii) For 1 ≤ |z| < q^{−1},

|Φ_{n}^{∗}(z)| ≤ 1 + C_{1}C|z|(1 − q|z|)^{−1}  (2.4)

(iii) For any q′ > q, there is C_{q′} with

|Φ_{n}(z)| ≤ C_{q′}(max(|z|, q′)^{n})  (2.5)

for |z| < 1.

(iv) For |z| ≤ q′,

|Φ_{n}^{∗}(z) − D(z)^{−1}D(0)| ≤ C̃_{q′}(qq′)^{n}  (2.6)

(v) D(z)^{−1} has an analytic continuation to {z | |z| < q^{−1}}, and in

that region,

Φ_{n}^{∗}(z) → D(0)D(z)^{−1}  (2.7)

(vi) For |z| > q,

lim_{n→∞} z^{−n}Φ_{n}(z) = D(0) \overline{D(1/z̄)}^{−1}  (2.8)

Proof. (i) For |z| = 1, |Φ_{n}(z)| = |Φ_{n}^{∗}(z)|, so (2.2) holds by induction from (1.1). (2.2) for |z| < 1 follows from the maximum principle. (2.3) then follows from (1.2).

(ii) By (2.3) and the ^{∗} of (1.1),

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

≤ 1 + CC_{1}|z| ∑_{j=0}^{∞} q^{n}|z|^{n}

proving (2.4).

(iii) By (1.2) and (2.4), we have (2.5) for q′ ≤ |z| ≤ 1. The result for |z| < q′ then follows from the maximum principle.

(iv) By (2.9) and (2.5),

∑_{m=n}^{∞} |Φ_{m+1}^{∗}(z) − Φ_{m}^{∗}(z)| ≤ ∑_{m=n}^{∞} C_{q}Cq^{m}|z|(q′)^{m}
≤ C̃_{q′}(qq′)^{n}

Since Φ_{m}^{∗}(z) → D(0)D(z)^{−1}, (2.6) holds.

(v) Following [4], we note that (2.9) implies that if |z| < q^{−1}, then ∑_{n}|Φ_{n+1}^{∗}(z)−Φ_{n}^{∗}(z)| < ∞, so Φ_{n}^{∗}(z) has a limit. Since Szegő’s theorem holds if |z| < 1, we conclude that D(z)^{−1} has a continuation and (2.7) holds.

(vi) is immediate from (v) and (1.2). □

By iterating (1.1), we obtain

Φ_{n}(z) = z^{n} − ∑_{j=1}^{n} ᾱ_{n−j}z^{j−1}Φ_{n−j}^{∗}(z)  (2.10)

We conclude that Theorem 2.2. Suppose (1.5) holds. Then, uniformly in each disk |z| < b − ε, we have that

\begin{aligned}φ_{n}(z) − [ & ∑_{ℓ=1}^{L} C̄_{ℓ}b̄_{ℓ}^{n}(z − b̄_{ℓ})^{−1}]D(z)^{−1} \\ & = O(n(bΔ)^{n}) + O(nb^{2n}) + O(nb^{n}(1 − \frac{ε}{b})^{n}) \end{aligned} (2.11)

In particular, uniformly on the disk,

lim_{n→∞} |b^{−n}φ_{n} − Q̃_{n}(z)| = 0

where

Q̃_{n}(z) = [ ∑_{ℓ=1}^{L} C̄_{ℓ}ω_{ℓ}^{n}(b̄_{ℓ} − z)^{−1}]D(z)^{−1}  (2.12)

and

ω_{ℓ} = \frac{b̄_{ℓ}}{b}  (2.13)

Remarks. 1. In (2.11), the b^{2n} is actually b^{(3−δ)n} for any δ > 0.

2. D(z)Q̃_{n}(z) ∏_{ℓ=1}^{L}(b̄_{ℓ} − z) is a polynomial of degree L − 1 with almost periodic coefficients (periodic if b_{ℓ}^{p} = 1 for some p and all ℓ).

Proof. We begin by noting that

∑_{j=1}^{∞} b̄_{ℓ}^{n−j}z^{j−1} = b̄_{ℓ}^{n−1}(1 − zb̄_{ℓ}^{−1})^{−1}
= b̄_{ℓ}^{n}(b̄_{ℓ} − z)^{−1}  (2.14)

and by an identical calculation,

∑_{j=n+1}^{∞} b̄_{ℓ}^{n−j}z^{j−1} = z^{n}(b̄_{ℓ} − z)^{−1}  (2.15)

It follows by (2.10) that if |z| < b − ε, then

\begin{aligned}|Φ_{n} & (z) − D(z)^{−1}D(0) ∑_{ℓ=1}^{L} c̄_{ℓ}b̄_{ℓ}^{n}(z − b̄_{ℓ})^{−1}| \\ & ≤ C_{ε}|z|^{n} + ∑_{j=1}^{n} |z|^{j−1}|ᾱ_{n−j}Φ_{n−j}^{∗}(z) − ∑_{ℓ=1}^{L} C̄_{ℓ}b_{ℓ}^{n}D(0)D(z)^{−1}| \end{aligned} (2.16)

The |z|^{n} term is O(b^{n}(1 − \frac{ε}{b})^{n}). By (1.5) and (2.5), the term in | · | in (2.16) is bounded by

C[b^{2(n−j)} + (bΔ)^{n−j}]  (2.17)

where the first term comes from |[Φ_{n}^{∗} − D(0)D(z)^{−1}]α_{n}| and the second from |D(z)^{−1}D(0)(α_{n} − ∑_{ℓ=1}^{L} C_{ℓ}b_{ℓ}^{n})|. Since |z| < b, the sum in (2.16) is thus bounded by nC′[b^{2n} + max(bΔ, z)^{n}]. It follows that

D(0)^{−1}Φ_{n}(z) − [ ∑_{ℓ=1}^{L} C̄_{ℓ}b̄_{ℓ}^{n}(z − b̄_{ℓ})^{−1}]D(z) = RHS of (2.11)

(2.11) then follows from

D(0)^{−1}Φ_{n}(z) = φ_{n}(z) + O(b^{n}| ∏_{j=1}^{n−1}(1 − |α_{j}|^{2})^{1/2} − D(0)|)
= φ_{n}(z) + O(b^{2n})  □

From (2.10), we also get a result that only depends on ratio asymptotics:

Theorem 2.3. Suppose that α_{n} is a sequence of Verblunsky coefficients and n_{j} a subsequence so that (i)

lim sup_{n→∞} |α_{n}|^{1/n} = b ∈ (0, 1)  (2.18)

(ii)

lim inf_{j→∞} |α_{n_{j}}|^{1/n_{j}} = b  (2.19)

(iii) For all k = 0, 1, 2, … and suitable β_{k} ∈ ℂ,

lim_{j→∞} \frac{ᾱ_{n_{j}−k−1}}{ᾱ_{n_{j}−1}} = β_{k}  (2.20)

exists. (iv) For every ε, there is C_{ε} so that for all j and k = 1, 2, …, we have

|α_{n_{j}−k−1}|(b − ε)^{k} ≤ C_{ε}|α_{n_{j}−1}|  (2.21)

Then for |z| < b, ∑_{j=0}^{∞} β_{j}z^{j} converges absolutely and uniformly on each disk |z| < b − ε and

lim_{j→∞} \frac{Φ_{n_{j}}(z)}{ᾱ_{n_{j}−1}} = −D(z)^{−1} ∑_{k=0}^{∞} β_{k}z^{k}  (2.22)

Proof. By (2.21), we have

|β_{k}| ≤ C_{ε}(b − ε)^{−k}  (2.23)

proving that ∑_{j=0}^{∞} β_{j}z^{j} has radius of convergence at least b. In (2.10) divided by α_{n_{j}−1}, the summand is bounded by

|z|^{j−1}(b − ε)^{−j} sup_{m, |z|≤b} |Φ_{m}^{∗}(z)|

which is summable for |z| < b − ε, so by the dominated convergence theorem, (2.22) holds. □

Corollary 2.4. Let α_{n} be a sequence of Verblunsky coefficients so that

lim_{n→∞} \frac{α_{n}}{α_{n−1}} = b ∈ (0, 1)  (2.24)

Then uniformly for |z| < b − ε,

lim_{n→∞} \frac{Φ_{n}(z)}{ᾱ_{n−1}} = −D(z)^{−1}(1 − \frac{z}{b} )^{−1}  (2.25)

Remarks. 1. By rotational covariance, if (2.24) holds for some b ∈ 𝔻, we can find a rotated problem with the ratio in (0, 1), so this implies a result whenever ratio asymptotics holds.

2. This is related to results of BLS [1]; see the discussion in Section 6.

Proof. Clearly, (2.20) holds with β_{k} = b^{−k} so we need only prove (2.21). For any δ, we have

|\frac{α_{m−1}b}{α_{m}} | ≤ C_{m}^{(δ)}(1 + δ)

where C_{m}^{(δ)} = 1 for m ≥ M_{δ} for some M_{δ}. It follows that

|\frac{α_{m−k}b^{k}}{α_{m}} | ≤ [ ∏_{m=1}^{M_{δ}} C_{m}^{(δ)}](1 + δ)^{k}

which implies (2.21). □

Similarly, we obtain

Corollary 2.5. Let α_{n} be a sequence of Verblunsky coefficients, b ∈ (0, 1), and c_{1}, c_{2}, …, c_{p} a sequence so that

∏_{j=1}^{p} c_{j} = 1  (2.26)

and

lim_{m→∞} \frac{α_{mp+ℓ}}{α_{mp+ℓ−1}} = bc_{ℓ}  ℓ = 1, 2, …, p  (2.27)

Then uniformly for |z| < b − ε,

lim_{m→∞} \frac{Φ_{mp+ℓ}(z)}{ᾱ_{mp+ℓ−1}} = −D(z)^{−1}G_{ℓ}(z)

where

\begin{aligned}G_{ℓ}(z)(1 − \frac{z^{ℓ}}{b^{ℓ}} ) = 1 + (bc_{ℓ−1})^{−1}z & + (bc_{ℓ−1})^{−1}(bc_{ℓ−2})^{−1}z^{2} \\ & + ⋯ + ∏_{j=1}^{p−1}(bc_{ℓ−j})^{−1}z^{p−1} \end{aligned} (2.28)

One can also say something when b = 1 if we also have

lim_{n→∞} α_{n} = 0  (2.29)

A key issue is that it may not be true that ∑_{n=1}^{∞}|α_{n}|^{2} < ∞, so D(z) may not exist.

Theorem 2.6. Let α_{n} be a sequence of Verblunsky coefficients and c_{1}, c_{2}, …, c_{p} is a sequence so that (2.26) holds and

lim_{m→∞} \frac{α_{mp+ℓ}}{α_{mp+ℓ−1}} = c_{ℓ}  ℓ = 1, 2, …, p  (2.30)

Then, uniformly in |z| < 1 − ε,

lim_{m→∞} \frac{φ_{mp+ℓ}(z)}{ᾱ_{mp+ℓ−1}φ_{mp+ℓ}^{∗}(z)} = −G_{ℓ}(z)  (2.31)

where G is given by (2.28) with b = 1.

Proof. (2.29) and (2.9), together with |Φ_{n}(z)| ≤ |Φ_{n}^{∗}(z)| on 𝔻̄, implies that on 𝔻̄,

lim_{n→∞} \frac{Φ_{n}^{∗}(z)}{Φ_{n+1}^{∗}(z)} = 1

Dividing (2.10) by ᾱ_{mp+ℓ−1}Φ_{mp+ℓ}^{∗}, we obtain the result by the same argument that led to Corollary 2.5. □

3. Asymptotics in the Critical Region

In this section, we will determine asymptotics of Φ_{n}(z) in an annulus about |z| = b when (1.5) holds. The idea will be to view (1.1) as an inhomogeneous equation, so we first look at some solutions with particular inhomogeneities. Define for z ≠ b̄_{ℓ} and n = 0, 1, 2, …,

u_{n}^{(ℓ)} = b̄_{ℓ}^{n}(z − b̄_{ℓ})^{−1}  (3.1)

Proposition 3.1. u_{n}^{(ℓ)} obeys

u_{n+1}^{(ℓ)} = zu_{n}^{(ℓ)} − b̄_{ℓ}^{n}  (3.2)

for all z ∈ ℂ, z ≠ b̄_{ℓ}, and all n = 0, 1, 2, . . ..

Proof. u_{n+1}^{(ℓ)} − zu_{n}^{(ℓ)} = (b̄_{ℓ}^{n} − z)u_{n}^{(ℓ)} = −b̄_{ℓ}^{n}. □

Next, define

R_{n}(z) = ᾱ_{n}Φ_{n}^{∗}(z) − ∑_{ℓ=1}^{L} C̄_{ℓ}b_{ℓ}^{n}D(z)^{−1}D(0)  (3.3)

and also define

s_{n}(z) = ∑_{j=0}^{∞} z^{−j−1}R_{n+j}(z)  (3.4)

We have

Proposition 3.2. Let α_{n} obey (1.5). Then there is Δ_{1} < 1, (i)

sup_{|z|≤1} |R_{n}(z)| ≤ C(bΔ_{1})^{n}  (3.5)

(ii) The sum in (3.4) converges uniformly in

𝔸 = {z | 1 > |z| > bΔ_{1}}  (3.6)

and s_{n}(z) is analytic there. (iii) We have in 𝔸 that

|s_{n}(z)| ≤ C(bΔ_{1})^{n}(|z| − bΔ_{1})^{−1}  (3.7)

(iv) s_{n} obeys

s_{n+1}(z) = zs_{n}(z) − R_{n}(z)  (3.8)

Proof. (i) follows from (1.5) and (2.6).

(ii), (iii) Since

|z^{j−1}R_{n+j}(z)| ≤ |z|^{−1}(bΔ_{1})^{n}(|z|^{−1}bΔ_{1})^{j}

we have a geometric series which yields (ii) and (iii).

(iv) Since the sum converges absolutely,

s_{n+1}(z) − zs_{n}(z) = ∑_{j=1}^{∞} z^{−j}R_{n+j}(z) − ∑_{j=0}^{∞} z^{−j}R_{n+j}
= −R_{n}(z)  □

The main result of this section is

Theorem 3.3. Let α_{n} obey (1.5). Then for some Δ_{1} < 1 and z ∈ 𝔸 given by (3.7), we have that

Φ_{n}(z) = s_{n}(z) + [ ∑_{ℓ=1}^{L} C̄_{ℓ}b̄_{ℓ}^{n}(z − b̄_{ℓ})^{−1}]D(0)D(z)^{−1} + z^{n}\overline{D(1/z̄)}^{−1} (3.9)

Remarks. 1. Since φ_{n} = κ_{n}Φ_{n}(z) and κ_{n} = D(0)^{−1}(1 + O(b^{n})), this also gives us asymptotics for φ_{n}.

2. Since Φ_{n} is analytic in 𝔸, the poles at b̄_{ℓ} in the second and third terms of (3.9) must cancel.

3. In 𝔸, (3.7) implies s_{n} is small compared to both z^{n} and b^{n}, so the asymptotics of Φ_{n} comes from the competition between the second and third terms in (3.9).

Proof. Let

Q_{n}(z) = Φ_{n}(z) − s_{n}(z) − [ ∑_{ℓ=1}^{L} C̄_{ℓ}b̄_{ℓ}^{n}(z − b̄_{ℓ})^{−1}]D(0)D(z)^{−1}

By (1.1), (2.2), and (3.8), we have

Q_{n+1}(z) = zQ_{n}(z)

so

Q_{n}(z) = f(z)z^{n}

Since Q_{n} is analytic in 𝔸\{b̄_{ℓ}}_{ℓ=0}^{L}, f(z) is analytic there.

By (3.7),

lim_{n→∞} |z|^{−n}|s_{n}(z)| = 0

in 𝔸, and if |z| > b, |z|^{−n} ∑_{ℓ=1}^{L} C̄_{ℓ}b̄_{ℓ}^{n}(z − b̄_{ℓ})^{−1} → 0, so for |z| > b,

f(z) = lim_{n→∞} z^{−n}Q_{n}(z) = lim_{n→∞} z^{−n}Φ_{n}(z)
= D(0) \overline{D(1/z̄)}^{−1}

by (2.8). □

4. Zeros in |z| < b − ε

In this section, we use the asymptotic result from Section 2 to analyze zeros of φ_{n} in the region where |z| < b. We initially focus on the case where (1.5) holds. A key role is played by the polynomials

P_{n}(z) = ∑_{ℓ=1}^{L} C̄_{ℓ}ω_{ℓ}^{n} ∏_{k≠ℓ}(z − b̄_{k})  (4.1)

of degree at most L − 1. Here ω_{ℓ} = b̄_{ℓ}/b.

The P_{n} are almost periodic in n and, in particular, for any sequence n_{j}, there is a subsequence n_{j(k)} so P_{∞} ≡ lim P_{n_{j(k)}} exists and is a nonzero polynomial (since P_{∞}/ ∏_{ℓ}(b̄_{ℓ} − z) has poles at each b̄_{ℓ}).

Theorem 4.1. Let (1.5) hold. Then for any ε > 0, there is an N so that for n ≥ N, φ_{n}(z) has at most L − 1 zeros in {z | |z| < b − ε} ≡ 𝕊.

Proof. If not, we can find a sequence n(j) → ∞ so that P_{n(j)}(z) has at least L zeros in 𝕊̄. By passing to a further subsequence, we can suppose P_{n(j)} → P_{∞} and that the L zeros have limits z_{1}, …, z_{L} in 𝕊̄ (maybe not distinct). By Theorem 2.2,

lim_{j→∞} φ_{n(j)}D(z) ∏_{ℓ=1}^{L}(z − b̄_{ℓ}) = P_{∞}(z)  (4.2)

in a neighborhood of 𝕊̄, so by Hurwitz’s theorem, P_{∞} has L zeros (counting multiplicity). Since P_{∞} has degree L−1 and is not identically zero, we have a contradiction. □

Using Hurwitz’s theorem and (4.2), we also have an existence result for zeros:

Theorem 4.2. Let (1.5) hold and let ω_{ℓ} = b̄_{ℓ}/b. Suppose n(j) is a subsequence so that lim ω_{ℓ}^{n(j)} exists, call it ω_{ℓ}^{(∞)}. Let

P_{∞}(z) = ∑_{ℓ=1}^{L} C̄_{ℓ}ω_{ℓ}^{(∞)} ∏_{k≠ℓ}(z − b̄_{k})  (4.3)

and let {w_{j}}_{j=1}^{J} be its zeros in {z | |z| < b}. Then for all sufficiently small δ and j ≥ N_{δ}, φ_{n(j)}(z) has one zero within δ of each w_{j} and no other zero in {z | |z| < b − δ}.

Remark. By “one zero within δ of w_{j},” we actually mean exactly k zeros if some w_{j} occurs k times in the list of zeros counting multiplicity.

Since the right side of (2.25) is nonvanishing on {z | |z| < b}, we recover a result of BLS [1] from Corollary 2.4, Theorem 2.6, and Hurwitz’s theorem:

Theorem 4.3. Let α_{n} be a sequence of Verblunsky coefficients so that

lim_{n→∞} \frac{α_{n}}{α_{n−1}} = b ∈ (0, 1]

Then for any ε > 0, there is an N_{ε} so φ_{n}(z) has no zeros in {z | |z| < b − ε} if n ≥ N_{ε}.

Finally, Corollary 2.5 and Hurwitz’s theorem imply

Theorem 4.4. Let α_{n} be a sequence of Verblunsky coefficients, b ∈ (0, 1), and c_{1}, c_{2}, …, c_{p} a sequence so that (2.26) and (2.27) hold. Let G_{ℓ} be given by (2.25), let W_{ℓ} = G_{ℓ}(1 − z^{ℓ}/b^{ℓ}) = RHS of (2.25), and let {w_{j}^{(ℓ)}}_{j=1}^{N_{ℓ}} be the zeros of W_{ℓ} in {z | |z| < b}. Then for any sufficiently small δ, there is an N so for mp + ℓ ≥ N, we have that the only zeros of φ_{mp+ℓ} in {z | |z| < b − δ} are one each within δ of each w_{j}^{(ℓ)}.

Remark. As we will explain in Section 6, that the only possible limit points of zeros are the w_{j}^{(ℓ)} is a result of BLS [1], but they do not prove there actually are zeros there.

Example 4.5. Let α_{n} be given by (1.7). We have b = \frac{1}{2}, p = 4, and c_{1} = −1, c_{2} = −1, c_{3} = 3, c_{4} = \frac{1}{3}. Thus

W_{2}(z) = 1 − 2z − 12z^{2} − 8z^{3}

which has zeros at −\frac{1}{2} and at \frac{1}{2}(−1 ± \sqrt{2}). Only (\sqrt{2} − 1)/2 is in \{z | |z| < \frac{1}{2}\}. The comparision of the limit and the zeros of Φ_{22} appears in Section 1 just after Figure 2. It is not coincidental that W_{2} has a zero at z = −\frac{1}{2}. In this case, the second term in (3.9) is, for n ≡ 2 (mod 4), C(\frac{1}{2})^{n}W_{2}(z)/(z^{4} − \frac{1}{16}) with poles only at \frac{1}{2}, ±\frac{1}{2}i. The potential pole at z = −\frac{1}{2} has to be cancelled by a zero in W_{2}. □

As in [7], one can analyze how close the zeros of φ_{n} are to the points w_{j}^{(ℓ)}. In general, they are exponentially close. If the w_{j}^{(ℓ)} are in the annulus where (3.9) holds, one can write down the leading asymptotic exactly. For example, if w_{j}^{(ℓ)} is a k-fold zero and D(1/w̄_{j}(z)) ≠ 0, then the zeros have a clock structure as in Theorem 4.5 of [7].

By using Theorem 2.6, we see that Theorem 4.4 extends to the case b = 1 if lim α_{n} = 0. In particular, if lim α_{n} = 0 and lim_{n→∞} \frac{α_{n+1}}{α_{n}} = 1 (e.g., α_{n} = (n + 2)^{−β}), then there are no zeros of φ_{n} in {z | |z| < 1 − δ} for n large.

5. Zeros in the Critical Region

Given Theorem 3.3 and the estimate (3.7), the analysis of zeros of φ_{n} in the region {z | bΔ_{1} < |z| < bΔ_{1}^{−1}} is identical to the analysis in [7] of the zeros in case L = 1. The gap in that case if z = b comes from the analysis of \overline{D(1/z̄)} φ_{n}(z) which has zeros with no gap. The gap in zeros of φ_{n}(z) comes from the fact that \overline{D(1/z̄)} has zeros at z = b. In our case, when (1.5) holds, \overline{D(1/z̄)} has a zero at each b̄_{ℓ}, so there are gaps at all those points.

The following extends Theorem 4.3 of [7] and has the same proof:

Theorem 5.1. Let α_{n} be a sequence of Verblunsky coefficients obeying (1.5). Then for some δ, all the zeros {z_{j}^{(n)}}_{j=1}^{N_{n}} of φ_{n}(z) with ||z|−b| < δ obey (1)

sup_{j} ||z_{j}^{(n)}| − b| = O(\frac{log n}{n})  (5.1)

(2) For n large, the z_{j}^{(n)} can be ordered in increasing arguments and

\frac{|z_{j+1}^{(n)}|}{|z_{j}^{(n)}|} = 1 + O(\frac{1}{n log n})  (5.2)

(3) Let {z̃_{j}^{(n)}}_{j=1}^{N_{n}+L} be the sequence of z_{j}^{(n)}’s with L points added at {b̄_{ℓ}}_{ℓ=1}^{L} still listed in increasing order.

Then

arg z_{j+1}^{(n)} − arg z_{j}^{(n)} = \frac{2π}{n} + O(\frac{1}{n log n})  (5.3)

for j = 1, 2, …, N_{n} + L with arg z_{N_{n}+L+1}^{(n)} ≡ 2π + arg z_{1}^{(n)}. Moreover, if D(z)^{−1} is nonvanishing on {z | |z| = b^{−1}}, then O(1/n log n) in (5.2) and (5.3) can be replaced by O(1/n^{2}), and O(log n/n) in (5.1) can be replaced by O(1/n).

Remark. In particular, the zeros nearest b̄_{ℓ} are b̄_{ℓ}e^{±2πi/n} + O(1/n^{2}) with the difference in the args equal to 4π/n + O(1/n^{2}).

6. Connection to the Results of Barrios-López-Saff

In this final section, we want to relate the results of [1] to ours. In their work, determinants of the following form enter:

Δ_{m}(z) = | \begin{matrix}z + x_{1} & zx_{2} & 0 \\ 1 & z + x_{2} & zx_{3} & ... \\ 0 & 1 & z + x_{3} & ... & ... \\ & & ... & ... & xz_{m} \\ & & & 1 & z + x_{m} \end{matrix} |  (6.1)

where we also define Δ_{0}(z) ≡ 1. We need the following:

Proposition 6.1. (i) For m = 2, 3, …

Δ_{m}(z) = (z + x_{m})Δ_{m−1} − zx_{m}Δ_{m−2}(z)  (6.2)

(ii) For m = 1, 2, …

Δ_{m}(z) = zΔ_{m−1} + x_{1}x_{2} … x_{m}  (6.3)

(iii) For m = 1, 2, …

Δ_{m}(z) = z^{m} + x_{1}z^{m−1} + x_{1}x_{2}z^{m−2} + ⋯ + x_{1} … x_{m}  (6.4)

Proof. (i) (6.2) comes from expanding Δ_{m} in minors along the last row. (ii) (6.2) reads

Δ_{m}(z) = zΔ_{m−1}(z) + x_{m}[Δ_{m−1}(z) − zΔ_{m−2}]

which implies (6.3) inductively once one notes that (6.3) holds for m = 1 since Δ_{1}(z) = z + x_{1}.

(iii) This follows by induction from (6.3). □

In [1], they consider sequences of Verblunsky coefficients where (2.26) and (2.27) hold to prove that the only accumulation points of zeros of φ_{mp+ℓ} are given by zeros of a polynomial that has the form of (6.1). Using (6.4) and

(x_{1} … x_{m})^{−1}Δ_{m}(z) = 1 + x_{m}^{−1}z + x_{m}^{−1}x_{m−1}^{−1}z^{2} + ⋯ + (x_{1} … x_{m})^{−1}z^{m}

one sees their polynomials are up to a constant, our polynomial W_{ℓ}. Thus our results extend theirs (in that we prove there are, in fact, always limit points).

References

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