Fine Structure of the Zeros of Orthogonal Polynomials, III. Periodic Recursion Coefficients

Barry Simon

California Institute of Technology

Abstract We discuss asymptotics of the zeros of orthogonal polynomials on the real line and on the unit circle when the recursion coefficients are periodic. The zeros on or near the absolutely continuous spectrum have a clock structure with spacings inverse to the density of zeros. Zeros away from the a.c. spectrum have limit points mod p and only finitely many of them. \begin{matrix}c \\ ◯ \end{matrix} 2000 Wiley Periodicals, Inc.

1 Introduction

This paper is the third in a series [18, 19] that discusses detailed asymptotics of the zeros of orthogonal polynomials with special emphasis on distances between nearby zeros. We discuss both orthogonal polynomials on the real line (OPRL) where the basic recursion for the orthonormal polynomials, p_{n}(x), is

(1.1)  xp_{n}(x) = a_{n+1}p_{n+1}(x) + b_{n+1}p_{n}(x) + a_{n}p_{n−1}(x)

(a_{n} > 0 for n = 1, 2, …, b_{n} real, and p_{−1}(x) ≡ 0), and orthogonal polynomials on the unit circle (OPUC) where the basic recursion is

(1.2)  φ_{n+1}(z) = ρ_{n}^{−1}(zφ_{n}(z) − ᾱ_{n}φ_{n}^{∗}(z))

Here α_{n} are complex coefficients lying in the unit disk 𝔻 and

(1.3)  φ_{n}^{∗}(z) = z^{n} \overline{φ_{n}(1/z̄)}

and

(1.4)  ρ_{n} = (1 − |α_{n}|^{2})^{1/2}

In this paper, we focus on the case where the Jacobi coefficients {a_{n}}_{n=1}^{∞}, {b_{n}}_{n=1}^{∞} or the Verblunsky coefficients {α_{n}}_{n=0}^{∞} are periodic, that is, for some p,

(1.5)  a_{n+p} = a_{n}  b_{n+p} = b_{n}

or

(1.6)  α_{n+p} = α_{n}

It should be possible to say something about perturbations of a periodic sequence, say α_{n}^{(0)}, which obeys (1.6) and α_{n} = α_{n}^{(0)} + δα_{n} with |δα_{n}| → 0 sufficiently fast. We leave the details to be worked out elsewhere.

Communications on Pure and Applied Mathematics, Vol. 000, 0001–0021 (2000) \begin{matrix}c \\ ◯ \end{matrix} 2000 Wiley Periodicals, Inc.

To describe our results, we begin by summarizing some of the basics of the structure of the measures and recursion relations when (1.5)or (1.6)holds. We will say more about this underlying structure in the sections below. In this introduction, we will assume that all gaps are open, although we don’t need and won’t use that assumption in the detailed discussion.

When (1.5) holds, the continuous part of the underlying measure, dρ, on ℝ is supported on p closed intervals [α_{j}, β_{j}], j = 1, …, p, called bands, with gaps (β_{j}, α_{j+1}) in between. Each gap has zero or one mass point. The m-function of the measure dρ,

(1.7)  m(z) = ∫ \frac{dρ(x)}{x − z}

has a meromorphic continuation to the genus p − 1 hyperelliptic Riemann surface, S, associated to [∏_{j=1}^{p}(x − α_{j})(x − β_{j})]^{1/2}. This surface has a natural projection π : S → ℂ, a twofold cover except at the branch points {α_{j}}_{j=1}^{p} ∪ {β_{j}}_{j=1}^{p}. π^{−1}[β_{j}, α_{j+1}] is a circle and m(z) has exactly one pole γ_{1}, …, γ_{p−1} on each circle.

It has been known for many years (see Faber [2]) that the density of zeros dk is supported on ∪_{j=1}^{p}[α_{j}, β_{j}] ≡ B and is the equilibrium measure for B in potential theory. We define k(E) = ∫_{α_{1}}^{E} dk. Then k(β_{j}) = j/p. Our main results about OPRL are:

(1) We can describe the zeros of p_{np−1}(x) exactly (not just asymptotically) in terms of π(γ_{j}) and k(E).

(2) Asymptotically, as n → ∞, the number of zeros of p_{n} in each band [α_{j}, β_{j}], N^{(n,j)}, obeys sup_{n}|\frac{n}{p} − N^{(n,j)}| < ∞, and the zeros \{x_{ℓ}^{(n,j)}\}_{ℓ=1}^{N(n,j)} obey

(1.8)  sup_{\begin{matrix}j \\ ℓ=1,2,...,N^{(n,j)}−1 \end{matrix}} n|k(x_{ℓ+1}^{(n,j)}) − k(x_{ℓ}^{(n,j)}) − \frac{1}{n}| → 0

as n → ∞.

(3) z ∈ ℂ is a limit of zeros of p_{n} (all n) if and only if z lies in supp(dρ).

(4) Outside the bands, there are at most 2p + 2b − 3 points which are limits of zeros of p_{mp+b−1} for each b = 1, …, p and, except for these limits, zeros have no accumulation points in ℂ\bands. While this is our bound, an optimal bound might be 2p − 2. For OPUC, the continuous part of the measure, dμ, is supported on p disjoint

intervals {e^{iθ} | x_{j} ≤ θ ≤ y_{j}}, j = 1, …, p, in ∂𝔻 with p gaps in between {e^{iθ} | y_{j} ≤ θ ≤ x_{j+1}} with x_{p+1} ≡ 2π +x_{1}. Each gap has zero or one mass point. The Carathéodory function of the measure dμ,

(1.9)  F (z) = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ)

has a meromorphic continuation from 𝔻 to the genus p − 1 hyperelliptic Riemann surface, S, associated to [∏_{j=1}^{p}(z − e^{ix_{j}})(z − e^{iy_{j}})]^{1/2}. The surface has a natural projection π : S → ℂ, and the closure of each gap has a circle as the inverse image. F has a single pole in each such circle, so p in all at γ_{1}, …, γ_{p}.

Again, the density of zeros is the equilibrium measure for the bands and each band has mass 1/p in this measure. See [17], especially Chapter 11, for a discussion of periodic OPUC. Our main results for OPUC are: (1′) We can describe the zeros of φ_{np}^{∗} − φ_{np} exactly (note, not zeros of φ_{np}). (2′) Asymptotically, as n → ∞, the number of zeros of φ_{n} near each band, N^{(n,j)},

obeys sup_{n}|\frac{n}{p} − N^{(n,j)}| < ∞, and the points on the bands closest to the zeros obey an estimate like (1.8).

(3′) z ∈ ℂ is a limit of zeros of φ_{n} (all n) if and only if z lies in supp(dμ). (4′) There are at most 2p + 2b − 1 points which are limits of zeros of φ_{mp+b} for

each b = 1, …, p and, except for these limits, zeros have no accumulation points in ℂ\bands. In Section 2, we discuss OPRL when (1.5)holds, and in Section 3, OPUC when

(1.6) holds. Each section begins with a summary of transfer matrix techniques for periodic recursion coefficients (Floquet theory).

While I am unaware of any previous work on the precise subject of Sections 2 and 3, the results are closely related to prior work of Peherstorfer [7, 8], who discusses zeros in terms of measures supported on a union of bands with a particular structure that overlaps our class of measures (see, in particular, Proposition 3.7 in [7]). For a discussion of zeros for OPUC with two bands, see [5].

Our striking result listed as (1) above (see Theorem 2.1 below) is easy to prove once one realizes it is true — indeed, we provide two elementary proofs. The referee has informed us that while there is no explicit discussion of zeros in [6], our Theorem 2.1 follows in a few lines from Corollary 2.1d in [6].

These papers also consider situations where the recursion coefficients are only almost periodic. For any finite collection of closed intervals on ℝ or closed arcs on ∂𝔻, there is a natural isospectral torus of OPRL or OPUC where the corresponding m- or F -function has minimal degree on the Riemann surface (see, e.g., [17, Section 11.8]). It would be interesting to extend the results of the current paper to that case.

2 OPRL With Periodic Jacobi Coefficients

In this section, we analyze the zeros of OPRL with Jacobi coefficients obeying (1.5). We begin with a summary of the theory of transfer matrices, discriminants, and Abelian functions associated to this situation. A reference for much of this theory is von Moerbeke [21]; a discussion of the discriminant can be found in Hochstadt [3], von Moerbeke [21], Toda [20], and Last [4]. The theory is close to the OPUC theory developed in Chapter 11 of [17].

Define the 2 × 2 matrix,

(2.1)  A_{k}(z) = \frac{1}{a_{k+1}} (\begin{matrix}z − b_{k+1} & −a_{k} \\ a_{k+1} & 0 \end{matrix})

4 B. SIMON

where

(2.2)  a_{0} ≡ a_{p}

Thus

(2.3)  det(A_{k}) = \frac{a_{k}}{a_{k+1}}

and the abstract form of (1.1)

(2.4)  zu_{n} = a_{n+1}u_{n+1} + b_{n+1}u_{n} + a_{n}u_{n−1}

is equivalent to

(2.5)  \binom{u_{n+1}}{u_{n}} = A_{n}\binom{u_{n}}{u_{n−1}}

So, in particular,

(2.6)  \binom{p_{n+1}(z)}{p_{n}(z)} = A_{n}A_{n−1} … A_{0}\binom{1}{0}

This motivates the definition of the transfer matrix,

(2.7)  T_{n}(z) = A_{n−1}(z) … A_{0}(z)

for n = 1, 2, . . .. We have, by (1.5), that

(2.8)  T_{mp+b} = T_{b}(T_{p})^{m}

suggesting that T_{p} plays a basic role. By (2.3) and (2.2),

(2.9)  det(T_{p}) = 1

A fundamental quantity is the discriminant

(2.10)  Δ(z) = Tr(T_{p}(z))

By (2.6), we have

(2.11)  T_{n}(z) = (\begin{matrix}p_{n}(z) & q_{n−1}(z) \\ p_{n−1}(z) & q_{n−2}(z) \end{matrix})

where q_{n}(z) is a polynomial of degree n that is essentially the polynomial of the second kind (the normalization is not the standard one but involves an extra a_{p}).

By (2.9) and (2.10), T_{p}(z) has eigenvalues

(2.12)  Γ_{±}(z) = \frac{Δ(z)}{2} ± \sqrt{(\frac{Δ(z)}{2})^{2} − 1}

In a moment, we will define branch cuts in such a way that on all of ℂ\cuts,

(2.13)  |Γ_{+}(z)| > |Γ_{−}(z)|

so (2.8) implies the Lyapunov exponent is given by

(2.14)  lim_{n→∞} \frac{1}{n} log ∥T_{n}(z)∥ = \frac{1}{p} log|Γ_{+}(z)| ≡ γ(z)

(2.12) means |Γ_{+}| = |Γ_{−}| if and only if Δ(z) ∈ [−2, 2], and one shows that this only happens if z is real. Moreover, if Δ(z) ∈ (−2, 2), then Δ′(x) ≠ 0. Thus, for x very negative, (−1)^{p}Δ(x) > 0 and solutions of (−1)^{p}Δ(x) = ±2 alternate as +2, −2, −2, +2, +2, −2, −2, …, which we label as

(2.15)  α_{1} < β_{1} ≤ α_{2} < β_{2} ≤ α_{3} < ⋯ < β_{p}

Since Δ(x) is a polynomial of degree p, there are p solutions of Δ(x) = 2 and of Δ(x) = −2, so 2p points {α_{j}}_{j=1}^{p} ∪ {β_{j}}_{j=1}^{p}.

The bands are [α_{1}, β_{1}], [α_{2}, β_{2}], …, [α_{p}, β_{p}] and the gaps are (β_{1}, α_{2}), (β_{2}, α_{3}), …, (β_{p−1}, α_{p+1}). If some β_{j} = α_{j+1}, we say the j-th gap is closed. Otherwise we say the gap is open.

If we remove the bands from ℂ, Γ_{±}(z) are single-valued analytic functions and (2.13) holds. Moreover, Γ_{+} has an analytic continuation to the Riemann surface, S, of genus ℓ ≤ p − 1 where ℓ is the number of open gaps. S is defined by the function [(z − α_{1})(z − β_{p}) ∏_{open gaps}(z − β_{j})(z − α_{j+1})]^{1/2}. Γ_{−} is precisely the analytic continuation of Γ_{+} to the second sheet.

The Dirichlet data are partially those x’s where

(2.16)  T_{p}(x)\binom{1}{0} = c_{x}\binom{1}{0}

that is, points where the 21 matrix element of T_{p} vanishes. It can be seen that the Dirichlet data x’s occur, one to each gap, that is, x_{1}, …, x_{p−1} with β_{j} ≤ x_{j} ≤ α_{j+1}. If x is at an edge of a gap, then c_{j} ≡ c_{x_{j}} is ±1. Otherwise |c_{j}| ≠ 1. If |c_{j}| > 1, we add the sign σ_{j} = −1 to x_{j}, and if |c_{j}| < 1, we add the sign σ_{j} = +1 to x_{j}. Thus the values of Dirichlet data for each open gap are two copies of [α_{j}, β_{j}] glued at the ends, that is, a circle. The set of Dirichlet data is thus an ℓ-dimensional torus. It is a fundamental result [21] that the map from a’s and b’s to Dirichlet data sets up a one-one correspondence to all a’s and b’s with a given Δ, that is, the set of a’s and b’s with a given Δ is an ℓ-dimensional torus.

The m-function (1.7) associated to dρ has a meromorphic continuation to the Riemann surface, S, with poles precisely at the points x_{j} on the principal sheet if σ_{j} = +1 and on the bottom sheet if σ_{j} = −1. ρ has point mass precisely at those x_{j} ∈ (β_{j}, α_{j+1}) with σ_{j} = +1. It has absolutely continuous support exactly the union of the bands, and has no singular part other than the possible point masses in the gaps.

Finally, in the review, we note that the potential theoretic equilibrium measure dk for the set of bands has several critical properties:

(1) If k(x) = ∫_{α_{1}}^{x} dk, then

(2.17)  k(β_{j}) = k(α_{j+1}) = \frac{j}{p}

k can be related explicitly to Δ by

Δ(x) = 2 cos(pπ(1 − k(x)))

(2) The Thouless formula holds:

(2.18)  γ(z) = ∫ log|z − x| dk(x) + log C_{B}

where γ is given by (2.14) and C_{B} is the (logarithmic) capacity of B.

(3) The (logarithmic) capacity of the bands is given by

(2.19)  C_{B} = ( ∏_{j=1}^{p} a_{j})^{−1}

(4) Γ_{+} is the complex Green’s function for ℂ\bands, that is,

(2.20)  Γ_{+}(z) = C_{B} exp(p ∫ log(z − x) dk(x))

That completes the review of periodic OPRL. We now turn to the study of the zeros. We begin by describing exactly (not just asymptotically!) the zeros of P_{mp−1}:

Theorem 2.1. The zeros of P_{mp−1}(x) are exactly

(i) The p − 1 Dirichlet data points {x_{j}}_{j=1}^{p−1}.

(ii) The (m − 1)p points \{x_{k,q}^{(m)}\}\begin{matrix}k=1,...,p \\ q=1,...,m−1 \end{matrix} where

(2.21)  k(x_{k,q}^{(m)}) = \frac{k − 1}{p} + \frac{q}{mp}

Remarks. 1. The points of (2.21) can be described as follows. Break each band [α_{j}, β_{j}] into m pieces of equal size in equilibrium measure. The x_{k,q}^{(m)} are the interior break points.

2. If a gap is closed, we include its position in the “Dirichlet points” of (i).

3. Generically, there are not zeros at the band edges, that is, (2.21) has q = 1, …, m − 1 but not q = 0 or q = m. But it can happen that one or more of the Dirichlet data points is at an α_{j+1} or a β_{j}.

4. This immediately implies that once one proves that the density of zeros exists, that it is given by dk.

5. It is remarkable that this result is new, given that it is so elegant and its proof so simple! I think this is because the OP community most often focuses on measures and doesn’t think so much about the recursion parameters and the Schrödinger operator community doesn’t usually think of zeros of P_{n}.

6. We will give two proofs of this result below. We have been informed by the referee that it can also be proven using Corollary 2.1d in [6].

Example 2.2. Let b_{n} ≡ 0, a_{n} ≡ \frac{1}{2} which has period p = 1. It is well-known in this case that the P_{n} are essentially Chebyshev polynomials of the second kind, that is,

(2.22)  P_{n}(cos θ) = \frac{1}{2^{n}} \frac{sin(n + 1)θ}{sin θ}

Thus P_{m−1} has zeros at points where

(2.23)  θ = \frac{jπ}{m}  j = 1, …, m − 1

(the zeros at θ = 0 and θ = π are cancelled by the sin(θ)). k(x) = π − arccos(x) and (2.23) is (2.21). We see that Theorem 2.1 generalizes the obvious result on the zeros of the Chebyshev polynomials of the second kind. □

First Proof of Theorem 2.1. By (2.11), zeros of P_{mp−1} are precisely points where the 12 matrix element of T_{mp} vanishes, that is, points where \binom{1}{0} is an eigenvector of T_{mp}. That is, zeros of P_{mp−1} are Dirichlet points for this period mp problem.

When (1.5) holds, we can view the a’s and b’s as periodic of period mp. There are closed gaps where T_{mp}(z) = ±1, that is, interior points to the original bands where (Γ_{±})^{m} = 1, that is, points where (2.21) holds. Thus, the Dirichlet data for T_{mp} are exactly the points claimed. □

Theorem 2.1 immediately implies point (2) from the introduction.

Theorem 2.3. Let P_{n}(x) be a family of OPRL associated to a set of Jacobi parameters obeying (1.5). Let (α_{j}, β_{j}) be a single band and let N^{(n,j)} be the number of zeros of P_{n} in that band. Then

(2.24)  |N^{(mp+b,j)} − (m − 1)| ≤ min(b + 1, p − b)

for −1 ≤ b ≤ p − 1. In particular,

(2.25)  |N^{(n,j)} − \frac{n}{p}| ≤ 1 + \frac{p}{2}

Proof. By a variational principle for any n, n′,

(2.26)  |N^{(n,j)} − N^{(n′,j)}| ≤ |n − n′|

(2.24) is immediate from Theorem 2.1 if we take n′ = mp − 1 and n′ = mp + (p − 1). (2.25) follows from (2.24) given that min(b + 1, p − b) ≤ p/2. □

Remark. Because of possibilities of Dirichlet data zeros at α_{j} and/or β_{j}, we need (α_{j}, β_{j}) in defining N^{(n,j)}. It is more natural to use [α_{j}, β_{j}]. If one does that, (2.24) becomes 2 + min(b + 1, p − b) and (2.25), 3 + \frac{p}{2}.

To go beyond these results and prove clock behavior for the zeros of p_{mp+b} (b ≢ −1 mod p), we need to analyze the structure of p_{n} in terms of Γ_{+}, Γ_{−}. For z not a branch point (or closed gap), Γ_{+} ≠ Γ_{−}. Γ_{+} is well-defined on ℂ\bands since |Γ_{+}| > |Γ_{−}|. On the bands, |Γ_{+}| = |Γ_{−}| and, indeed, the boundary values on the two sides of a band are distinct. But Γ_{+} is analytic on ℂ\bands, so for such z, we can define P_{±} by

(2.27)  T_{p}(z) = Γ_{+}P_{+} + Γ_{−}P_{−}

where P_{+}, P_{−} are 2 × 2 rank one projections obeying

(2.28)  P_{+}^{2} = P_{+}  P_{−}^{2} = P_{−}  P_{+}P_{−} = P_{−}P_{+} = 0

and

(2.29)  P_{+} + P_{−} = 1

It follows from (2.27) and (2.29) that

(2.30)  P_{+} = \frac{T_{p}(z) − Γ_{−}(z)1}{Γ_{+} − Γ_{−}}
(2.31)  P_{−} = \frac{T_{p}(z) − Γ_{+}(z)1}{Γ_{−} − Γ_{+}}

which, in particular, shows that P_{+} is a meromorphic function on S whose secondsheet values are just P_{−}.

Define

(2.32)  a(z) = ⟨\binom{0}{1}, P_{+}(z)\binom{1}{0}⟩

(2.33)  b(z) = ⟨\binom{1}{0}, P_{+}(z)\binom{1}{0}⟩

so (2.29) implies

(2.34)  ⟨\binom{0}{1}, P_{−}(z)\binom{1}{0}⟩ = −a(z)

(2.35)  ⟨\binom{1}{0}, P_{−}(z)\binom{1}{0}⟩ = 1 − b(z)

Under most circumstances, a(z) has a pole at band edges where Γ_{+} − Γ_{−} → 0. For later purpose, we note that ⟨\binom{0}{1}, (T_{p}(z) − Γ_{−}1)\binom{1}{0}⟩ = ⟨\binom{0}{1}, T_{p}(z)\binom{1}{0}⟩ has a finite limit at such points. Later we will be looking at

a(z)(Γ_{+}^{m} − Γ_{−}^{m}) = ⟨\binom{0}{1}, T_{p}(z)\binom{1}{0}⟩ \frac{Γ_{+}^{m} − Γ_{−}^{m}}{Γ_{+} − Γ_{−}}

→ ⟨\binom{0}{1}, T_{p}(z)\binom{1}{0}⟩mΓ_{+}^{m−1}

if Γ_{+} − Γ_{−} → 0. This is zero if and only if ⟨\binom{0}{1}, T_{p}(z)\binom{1}{0}⟩ = 0, that is, if and only if the edge of the band is a Dirichlet data point.

(2.27) and (2.28) imply

(2.36)  T_{mp}(z) = T_{p}(z)^{m} = Γ_{+}^{m}P_{+} + Γ_{−}^{m}P_{−}

so

(2.37) T_{mp}(z)\binom{0}{1} = [a(z)(Γ_{+}^{m} − Γ_{−}^{m})]\binom{0}{1} + [b(z)Γ_{+}^{m} + (1 − b(z))Γ_{−}^{m})]\binom{1}{0}

Thus, by (2.25) for b ≥ 0,

P_{mp+b−1} = ⟨\binom{0}{1}, T_{b}T_{mp}\binom{1}{0}⟩ (2.38)

= [(Γ_{+}^{m} − Γ_{−}^{m})a(z)]q_{b−2}(z) + [b(z)Γ_{+}^{m} + (1 − b(z))Γ_{−}^{m})]p_{b−1}(z) (2.39)

where

(2.40)  q_{−2}(z) ≡ 1  q_{−1}(z) ≡ 0

Remark. See Peherstorfer [6, Theorem 3.1] for another formula for P_{mp+b−1}.

Second Proof of Theorem 2.1. For b = 0, p_{b−1} ≡ 0 and q_{b−2} = 1, so

(2.41)  p_{mp−1}(z) = (Γ_{+}^{m} − Γ_{−}^{m})a(z)

Its zeros are thus points where a(z) = 0 or where Γ_{+}^{m} = Γ_{−}^{m}, except that at branch points, a(z) can have a pole which can cancel a zero of Γ_{+}^{m} − Γ_{−}^{m}.

a(z) = 0 if and only if \binom{1}{0} is an eigenvector of T_{p}(z), that is, exactly at the Dirichlet data points.

Γ_{+}^{m} = Γ_{−}^{m} is equivalent to Γ_{+}^{2m} = 1 since Γ_{−} = Γ_{+}^{−1}. This implies |Γ_{+}| = |Γ_{−}|, so can only happen on the bands. On the bands, by (2.20),

(2.42)  Γ_{+}(x) = exp(πip k(x))

and Γ_{+}^{2m} = 1 if and only if

(2.43)  mp k(x) ∈ ℤ

that is, if (2.21) holds for some q = 0, …, m. But at q = 0 or q = m, a(z) has a pole that cancels the zero of Γ_{+}^{m} − Γ_{−}^{m}, so the zeros of p_{mp−1} are precisely given by (i) and (ii) of Theorem 2.1. □

We can use (2.39) to analyze zeros of p_{mp+b−1} for large m. We begin with the region away from the bands:

Theorem 2.4. Let z ∈ ℂ\bands and let b be fixed. Then

(2.44)  lim_{m→∞} Γ_{+}(z)^{−m}p_{mp+b−1}(z) = a(z)q_{b−2}(z) + b(z)p_{b−1}(z)

In particular, if the right side of (2.44) is called j_{b}(z), then

(1) If j_{b}(z_{0}) ≠ 0, then p_{mp+b−1}(z) is nonvanishing near z_{0} for m large.

(2) If j_{b}(z_{0}) = 0, then p_{mp+b−1}(z) has a zero (k zeros if z has a k-th order zero at z_{0}) near z_{0} for m large.

(3) There are at most 2p + 2b − 3 points in ℂ\bands where j_{b}(z_{0}) is zero.

Proof. (2.44) is immediate from (2.39) and |Γ_{−}/Γ_{+}| < 1. (1) and (2) then follow by Hurwitz’s theorem if we show that j_{b}(z) is not identically zero.

By (2.1) and (2.7) near z = ∞,

T_{p}(z) = ( ∏_{j=1}^{p} a_{j})^{−1}z^{p} (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) + O(z^{p−1})

which implies Γ_{+} = (∏_{j=1}^{p} a_{j})^{−1}z^{p} + O(z^{p−1}) and Γ_{−}(z) = O(z^{−p}). It follows that a(z) → 0 as z → ∞ and b(z) → 1. Thus, since p_{b−1} has degree b − 1, (2.39) shows that as z → ∞ on the main sheet, j_{b}(z_{0}) has a pole of order b − 1.

On the other sheet, P_{+} changes to P_{−}, so a(z) → 0 and b(z) → 0 on the other sheet. It follows that j_{b}(z) has a pole at ∞ of degree at most b − 2. j_{b} also has poles of degree at most 1 at each branch point. Thus, j_{b}(z) as a function on S has total degree at most 2p+(b−1)+(b−2) = 2p+2b−3 which bounds the number of zeros. □

Finally, we turn to zeros on the bands. A major role will be played by the function on the right side of (2.44) (j is for “Jost” since this acts in many ways like a Jost function):

(2.45)  j_{b}(z) = a(z)q_{b−2}(z) + b(z)p_{b−1}(z)

Lemma 2.5. j_{b} is nonvanishing on the interior of the bands.

Remark. By j_{b}(x) for x real, we mean (2.45)with a defined via lim_{ε↓0} a(x+iε) since P_{±} are only defined off ℂ\bands.

Proof. As already mentioned, the boundary values obey

(2.46)  lim_{ε↓0} P_{+}(x + iε) = lim_{ε↓0} P_{−}(x − iε)

(by the two-sheeted nature of P_{+} and P_{−}). Thus, by (2.32)–(2.35),

(2.47)  a(x + i0) = −a(x − i0)
(2.48)  b(x + i0) = 1 − b(x − i0)

Moreover, since T_{p} and Γ_{±} are real on ℝ\bands, a(z) and b(z) are real on ℝ\bands (by (2.26)). Thus

(2.49)  a(x + i0) = \overline{a(x − i0)}
(2.50)  b(x + i0) = \overline{b(x − i0)}

The last four equations imply for x in the bands

(2.51)  Re(a(x + i0)) = 0
(2.52)  Re(b(x + i0)) = \frac{1}{2}

p and q are real on ℝ, so

(2.53)  Re(j_{b}(x)) = \frac{1}{2} p_{b−1}(x)

Thus, if j_{b}(x_{0}) = 0 on the bands, p_{b−1}(x_{0}) = 0.

As we have seen, a(z) = 0 only at the Dirichlet points and so not in the bands. If p_{b−1}(x_{0}) = 0 = j_{b}(x_{0}), then since a(x_{0}) ≠ 0, we also have q_{b−2}(x_{0}) = 0. By (2.11), if p_{b−1}(x_{0}) = q_{b−2}(x_{0}), then det(T_{b}(x_{0})) = 0, which is false. We conclude via proof by contradiction that j_{b}(x) has no zeros. □

Theorem 2.6. For each b and each band j, there is an integer D_{b,j} so, for sufficiently large m, the number of zeros N_{b,j}(m) of p_{mp+b−1} is either m − D_{b,j} or m − D_{b,j} + 1. In particular,

(2.54)  sup_{n,j} |\frac{n}{p} − N^{(n,j)}| < ∞

Moreover, (1.8) holds.

Remark. This result is closely related to Proposition 3.7 in [7].

Proof. By (2.39), (2.46), (2.47), and (2.48), we have

(2.55)  p_{mp+b−1}(x) = j_{b}(x)Γ_{+}(x)^{m} + \overline{j_{b}(x)} \overline{Γ_{+}(x)}^{m}

on the bands. By the lemma, j_{b}(x) is nonvanishing inside band j, so

(2.56)  j_{b}(x) = |j_{b}(x)|e^{iγ_{b}(x)}

where γ_{b} is continuous — indeed, real analytic — and by a simple argument, γ_{b} and γ′_{b} have limits as x ↓ α_{j} or x ↑ β_{j}.

By (2.42), (2.55) becomes

(2.57)  p_{mp+b−1}(x) = 2|j_{b}(x)| cos(πmp k(x) + γ_{b}(x))

Define D_{b,j} to be the negative of the integral part of [γ_{b}(β_{j}) − γ_{b}(α_{j})]/π. Since sup_{bands}|γ′_{b}(x)| < ∞, there is, for large m, at most one solution of πmp k(x) + γ_{b}(x) = πℓ for each ℓ. Given this, it is immediate that the number of zeros is m − D_{b,j} or m − D_{b,j} + 1.

Finally, (1.8) is immediate from (2.57). Given that γ is C^{1}, we even get that

{□ \atop (2.58)} k(x_{ℓ+1}^{(n,j)}) − k(x_{ℓ}^{(n,j)}) = \frac{1}{n} + O(\frac{1}{n^{2}} )

As for point (3) from the introduction, the proof of Theorem 2.4 shows that if z_{0} is not in the bands and is a limit of zeros of p_{mp+b−1}(z), then p_{mp+b−1}(z_{0}) goes to zero exponentially (as Γ_{−}^{m}). If this is true for each b, then ∑_{n=0}^{∞}|p_{n}(z_{0})|^{2} < ∞, which means z_{0} is in the pure point spectrum of dμ. Since the bands are also in the spectrum, we have

Proposition 2.7. z_{0} ∈ ℂ is a limit of zeros of p_{n}(z) (all n) if and only if z_{0} ∈ supp(dμ).

Remark. This also follows from a result of Denisov-Simon [1], but their argument, which applies more generally, is more subtle.

3 OPUC With Periodic Verblunsky Coefficients

In this section, we analyze the zeros of OPUC with Verblunsky coefficients obeying (1.6). We begin with a summary of the transfer matrices, discriminants, and Abelian functions in this situation. These ideas, while an obvious analog of the OPRL situation, seem not to have been studied before their appearance in [17], which is the reference for more details. Many of the consequences of these ideas were found earlier in work of Peherstorfer and Steinbauer [9, 10, 11, 12, 13, 14, 15].

Throughout, we will suppose that p is even. If (α_{0}, …, α_{p−1}, α_{p}, …) is a sequence with odd period, (β_{0}, β_{1}, …) = (α_{0}, 0, α_{1}, 0, α_{2}, …) has even period and

(3.1)  Φ_{2n}(z, {β_{j}}) = Φ_{n}(z^{2}, {α_{j}})

so results for the even p case immediately imply results for the odd p.

Define the 2 × 2 matrix

(3.2)  A_{k}(z) = \frac{1}{ρ_{k}} (\begin{matrix}z & −ᾱ_{k} \\ −zα_{k} & 1 \end{matrix})

where ρ_{k} is given by (1.4). Then

det(A_{k}(z)) = z

(1.2) and its^{∗} are equivalent to

(3.3)  \binom{φ_{n+1}}{φ_{n+1}^{∗}} = A_{n}(z)\binom{φ_{n}}{φ_{n}^{∗}}

The second kind polynomials, ψ_{n}(z), are the OPUC with Verblunsky coefficients {−α_{j}}_{j=0}^{∞}. Then it is easy to see that

(3.4)  \binom{ψ_{n+1}}{−ψ_{n+1}^{∗}} = A_{n}(z)\binom{ψ_{n}}{−ψ_{n}^{∗}}

with A given by (3.2).

We thus define

(3.5)  T_{n}(z) = A_{n−1}(z) … A_{0}(z)

By (1.6), we have

(3.6)  T_{mp+b} = T_{b}(T_{p})^{m}

(3.3) and (3.4) imply that

(3.7)  \binom{φ_{n}}{φ_{n}^{∗}} = T_{n}\binom{1}{1}

(3.8)  \binom{ψ_{n}}{−ψ_{n}^{∗}} = T_{n}\binom{1}{−1}

so that

(3.9)  T_{n}(z) = \frac{1}{2} (\begin{matrix}φ_{n}(z) + ψ_{n}(z) & φ_{n}(z) − ψ_{n}(z) \\ φ_{n}^{∗}(z) − ψ_{n}^{∗}(z) & φ_{n}^{∗}(z) + ψ_{n}^{∗}(z) \end{matrix})

The discriminant is defined by

(3.10)  Δ(z) = z^{−p/2} Tr(T_{p}(z))

The z^{−p/2} factor (recall p is even) is there because det(z^{−p/2}T_{p}(z)) = 1, so z^{−p/2}T_{p}(z) has eigenvalues Γ_{±}(z) given by (2.12). Δ(z) is real on ∂𝔻 so

(3.11)  Δ(z) = \overline{Δ(1/z̄)}

Δ(z) ∈ (−2, 2) only if z = e^{iθ} and there are p roots, each of Tr(T_{p}(z)) ∓ 2z^{p/2} = 0, that is, p solutions of Δ(z) = ±2. These alternate on the circle at points +2, −2, −2, +2, +2, −2, −2, …, so we pick

(3.12)  0 ≤ x_{1} < y_{1} ≤ x_{2} < y_{2} ≤ ⋯ < y_{p} ≤ 2π

where e^{ix_{j}}, e^{iy_{j}} are solutions of Δ(z) = ±2.

The bands

(3.13)  B_{j} = {e^{iθ} | x_{j} ≤ θ ≤ y_{j}}

are precisely the points where Δ(z) ∈ [−2, 2]. In between are the gaps

(3.14)  G_{j}{e^{iθ} | y_{j} < θ < x_{j+1}}

where x_{p+1} = x_{1} + 2π. Some gaps can be closed, that is, G_{j} is empty (i.e., y_{j} = x_{j+1}).

We also see that on ℂ\bands, |Γ_{+}| > |Γ_{−}|, so the Lyapunov exponent is given by

(3.15)  lim_{n→∞} \frac{1}{n} log ∥T_{n}(z)∥ = \frac{1}{2} log|z| + \frac{1}{p} log|Γ_{+}(z)| ≡ γ(z)

If we remove the bands from ℂ, (2.13) holds. Moreover, Γ_{+}(z) has an analytic continuation to the Riemann surface, S, of [∏_{open gaps}(z−e^{iy_{j+1}})(z−e^{ix_{j}})]^{1/2}. The genus of S, ℓ ≤ p − 1, where ℓ + 1 is the number of open gaps. (In some sense, the OPRL case, where the genus ℓ is the number of gaps, has ℓ + 1 gaps also, but one gap is ℝ\[α_{1}, β_{p}] which includes infinity.) Γ_{−} is the analytic continuation of Γ_{+} to the second sheet.

The Dirichlet data are partly those points in ∂𝔻, z_{j}, where

(3.16)  T_{p}(z)\binom{1}{1} = c_{z}\binom{1}{1}

It can be shown there is one such z_{j} in each gap (including closed gaps) for the p roots of φ_{p}(z) − φ_{p}^{∗}(z). We let c_{j} = c_{z_{j}}. If z_{j} is at a gap edge, |c_{j}| = 1; otherwise |c_{j}| ≠ 1. If |c_{j}| > 1, we add sign −1 to z_{j} and place the Dirichlet point on the lower sheet of S at point z_{j}. If |c_{j}| < 1, we add sign +1 and put the Dirichlet point on the initial sheet. +1 points correspond to pure points in dμ.

As in the OPRL case, the set of possible Dirichlet data points is a torus, but now of dimension ℓ + 1. This torus parametrizes those μ with periodic α’s and discriminant Δ.

The F -function, (1.9), has a meromorphic contribution to S with poles precisely at the Dirichlet data points.

The potential theoretic equilibrium measures dk for the bands have several critical properties:

(1) If k(e^{iθ_{0}}) = k({e^{iθ} | x_{1} < θ < θ_{0}}), then

(3.17)  k(e^{iy_{j}}) = k(e^{ix_{j+1}}) = \frac{j}{p}

(2) The Thouless formula holds:

(3.18)  γ(z) = ∫ log|z − e^{iθ}| dk(e^{iθ}) + log C_{B}

where γ is given by (3.15) and C_{B} is the capacity of the bands.

(3) We have

(3.19)  C_{B} = ∏_{j=0}^{p−1}(1 − |α_{j}|^{2})^{1/2}

(4)

(3.20)  Γ_{+}(z) = C_{B}z^{−p/2} exp(p ∫ log(z − e^{iθ}) dk(e^{iθ}))

This completes the review of periodic OPUC. The analog of Theorem 2.1 does not involve Φ_{n} but Φ_{n} − Φ_{n}^{∗}:

Theorem 3.1. The zeros of Φ_{mp}(z) − Φ_{mp}^{∗}(z) are at the following points:

(i) the p Dirichlet data z_{j}’s in each gap of the period p problem. (ii) the (m − 1)p points where

(3.21)  k(e^{iθ}) = \frac{k − 1}{p} + \frac{q}{mp}

k = 1, …, p; q = 1, …, m − 1.

Proof. As noted (and proven several ways in [17, Chapter 11]), for a period mp problem, Φ_{mp} − Φ_{mp}^{∗} has its zeros, one in each gap. The gaps of the mp problem are the gaps of the original problem plus a closed gap at each point where (3.21) holds. There is a zero in each closed gap and at each point where (3.16)holds since then T_{mp}(z)\binom{1}{1} = c_{j}^{m}\binom{1}{1}. □

We now turn to the analysis of zeros of φ_{mp+b}(z), b = 0, 1, …, p − 1; m = 0, 1, 2, . . .. The analog of (2.38) is, by (3.7),

(3.22)  φ_{mp+b} = ⟨\binom{1}{0}, T_{b}(T_{p})^{m}\binom{1}{1}⟩

As in Section 2, we write, for z ∈ ℂ\bands:

(3.23)  z^{−p/2}T_{p}(z) = Γ_{+}(z)P_{+}(z) + Γ_{−}(z)P_{−}(z)

where P_{±} are 2×2 matrices which are complementary projections, that is, (2.28)/(2.29) hold. (2.30)/(2.31) are replaced by

(3.24)  P_{+} = \frac{z^{−p/2}T_{p}(z) − Γ_{−}(z)1}{Γ_{+} − Γ_{−}}
(3.25)  P_{−} = \frac{z^{−p/2}T_{p}(z) − Γ_{+}(z)1}{Γ_{−} − Γ_{+}}

So, in particular, P_{±} have meromorphic continuations to S, and P_{+} continued to the other sheet is P_{−}.

Define

(3.26)  a(z) = \frac{1}{2} ⟨\binom{1}{1}, P_{+}\binom{1}{1}⟩

(3.27)  b(z) = \frac{1}{2} ⟨\binom{1}{−1}, P_{+}\binom{1}{1}⟩

so that, by (2.29),

(3.28)  \frac{1}{2} ⟨\binom{1}{1}, P_{−}\binom{1}{1}⟩ = 1 − a(z)

(3.29)  \frac{1}{2} ⟨\binom{1}{−1}, P_{−}\binom{1}{1}⟩ = −b(z)

Thus, since \frac{1}{\sqrt{2}} \binom{1}{1}, \frac{1}{\sqrt{2}} \binom{1}{−1} are an orthonormal basis,

\begin{aligned}z^{−mp/2}T_{mp}\binom{1}{1} & = Γ_{+}^{m}[a(z)\binom{1}{1} + b(z)\binom{1}{−1}] \\ & + Γ_{−}^{m}[(1 − a(z))\binom{1}{1} − b(z)\binom{1}{−1}] \end{aligned} ; (3.30)

Therefore, by (3.7), (3.8), and (3.22),

\begin{aligned}φ_{mp+b}(z) & = φ_{b}(z)[a(z)z^{mp/2}Γ_{+}^{m} + (1 − a)z^{mp/2}Γ_{−}^{m}] \\ & + ψ_{b}(z)[b(z)z^{mp/2}Γ_{+}^{m} − b(z)z^{mp/2}Γ_{−}^{m}] \end{aligned} ; (3.31)

We thus define

(3.32)  j_{b}(z) = a(z)φ_{b}(z) + b(z)ψ_{b}(z)

and we have, since |Γ_{+}| > |Γ_{−}| on ℂ\bands:

Theorem 3.2. For z ∈ ℂ\bands,

(3.33)  lim_{m→∞} z^{−mp/2}Γ_{+}^{−m}φ_{mp+b}(z) = j_{b}(z)

In addition, j_{b} is nonvanishing near z = ∞.

In particular, if z_{0} ∉ bands and j_{b}(z_{0}) ≠ 0, then for some ε > 0 and M, we have φ_{mp+b}(z_{0}) ≠ 0 if |z − z_{0}| < ε and m ≥ M. If z_{0} ∉ bands and j_{b}(z_{0}) has a zero of order k, then for some ε > 0 and all m large, φ_{mp+b}(z) has precisely k zeros (counting multiplicity). The number of z_{0} in ℂ\bands with j_{b}(z_{0}) = 0 is at most 2p + 2b − 1.

Remark. Asymptotics related to (3.33) can be found in [15, Proposition 3.1].

Proof. As noted, (3.31) and |Γ_{+}| > |Γ_{−}| imply (3.33). To analyze j_{b}(z) near z = ∞, we proceed as follows: We have, by (3.2) and (3.5), that as |z| → ∞,

(3.34) T_{p}(z) = z^{p}( ∏_{j=0}^{p−1} ρ_{j}^{−1})[(\begin{matrix}1 & 0 \\ −α_{p−1} & 0 \end{matrix}) … (\begin{matrix}1 & 0 \\ −α_{0} & 0 \end{matrix})] + O(z^{p−1})

= z^{p}( ∏_{j=1}^{p−1} ρ_{j}^{−1}) (\begin{matrix}1 & 0 \\ −α_{p−1} & 0 \end{matrix}) + O(z^{p−1}) (3.35)

from which it follows that

(3.36)  P_{+} = (\begin{matrix}1 & 0 \\ −α_{p−1} & 0 \end{matrix}) + O(z^{−1})
(3.37)  P_{−} = (\begin{matrix}0 & 0 \\ α_{p−1} & 1 \end{matrix}) + O(z^{−1})

and

(3.38)  a(z) = \frac{1}{2} (1 − α_{p−1}) + O(z^{−1})
(3.39)  b(z) = \frac{1}{2} (1 + α_{p−1}) + O(z^{−1})

We have

(3.40)  φ_{b}(z) = ( ∏_{j=0}^{p−1} ρ_{j}^{−1})z^{b} + O(z^{b−1})

(3.41)  ψ_{b}(z) = ( ∏_{j=0}^{p−1} ρ_{j}^{−1})z^{b} + O(z^{b−1})

from which we see that

(3.42)  j_{b}(z) = ( ∏_{j=0}^{p−1} ρ_{j}^{−1})z^{b} + O(z^{b−1})

since (3.38)/(3.39)imply a(z)+b(z) = 1+O(z^{−b}). In particular, j_{b}(z) is not zero near ∞, so j_{b} is not identically zero, and the assertion about locations of zeros of φ_{mp+b}(z) follows from Hurwitz’s theorem.

Since 1 − a(z) − b(z) = O(z^{−1}), (3.28)/(3.29) imply that, on the second sheet, the analytic continuation of j_{b}(z) near ∞ is O(z^{b−1}). It follows that j_{b} has a pole of order b at ∞ on the main sheet (regular if b = 0) and a pole of order at most b − 1 (a zero if b = 0 and is regular if b = 1) at ∞ on the second sheet. j_{b} also can have at most 2p simple poles at the 2p branch points.

It follows that the degree of j_{b} as a meromorphic function on S is at most 2p + 2b − 1 (if b = 0, 2p). Thus the number of zeros is at most 2p + 2b − 1 if b ≠ 0. If b = 0, there are at most 2p + 2b zeros. But since then one is at ∞ on the second sheet, the number of zeros on finite points is at most 2p + 2b − 1. □

Next, we note that

Theorem 3.3. Let {α_{n}} be periodic and not at all 0. z_{0} is a limit of zeros of φ_{n}(z) (i.e., there exist z_{n} with φ_{n}(z_{n}) = 0 and z_{n} → z_{0} if and only if z_{0} lies in the support of dμ).

Remark. α_{n} = 0 has 0 as a limit point of zeros at φ_{n}(z) = z^{n}, so one needs some additional condition on the α’s to assure this result.

Proof. By Theorems 8.1.11 and 8.1.12 of [16], if z_{0} ∈ supp(dμ), then it is a limit point of zeros. For the other direction, suppose z_{0} ∉ bands and is a limit point of zeros. By Theorem 3.2, j_{b}(z_{0}) = 0 for each b = 0, 1, …, p − 1, so by (3.31), φ_{mp+b}(z) ∼ C(Γ_{−}z_{0}^{p/2})^{m} which, since |z_{0}| ≤ 1 and |Γ_{−}| < 1, implies that φ_{n}(z_{0}) goes to zero exponentially.

Since α_{n} is not identically zero, some α_{j}, j ∈ {0, 1, …, p − 1} is nonzero. Thus, by Szegő recursion for φ_{j},

φ_{mp+j}^{∗}(z_{0}) = α_{j}^{−1}[z_{0}φ_{mp}(z_{0}) − ρ_{j}φ_{mp+1}(z_{0})]

goes to zero exponentially in m.

Since α_{n} is periodic, sup_{n}|α_{n}| < 1, and so, sup_{n} ρ_{n}^{−1} < ∞. Since

φ_{mp+j+1}^{∗}(z_{0}) = ρ_{j+1}^{−1}(φ_{mp+j}^{∗}(z_{0}) − α_{j}φ_{mp+j}(z_{0}))

we see φ_{mp+j+1}^{∗}(z_{0}) decays exponentially and so, by induction, φ_{n}^{∗}(z_{0}) decays exponentially. By the Christoffel-Darboux formula (see [16, eqn. (2.2.70)]), |φ_{n}^{∗}(z_{0})|^{2} ≥ 1 − |z_{0}|^{2}, so the decay implies |z_{0}| = 1. But if z_{0} ∈ ∂𝔻 and ∑_{n}|φ_{n}(z_{0})|^{2} < ∞, then μ({z_{0}}) > 0 (see [16, Theorem 2.7.3]).

Thus if z_{0} is a limit of zeros, either z_{0} ∈ bands or μ({z_{0}}) > 0, that is, z_{0} ∈ supp(dμ). □

Finally, in our analysis of periodic OPUC, we turn to zeros near to the bands. We define j̃_{b} on ℂ\bands so that (3.31) becomes

(3.43)  φ_{mp+b}(z) = j_{b}(z)z^{mp/2}Γ_{+}^{m} + j̃_{b}(z)z^{mp/2}Γ_{−}^{m}

While φ_{mp+b}(z) is continuous across the bands, j_{b}, j̃_{b}, and Γ_{±} are not. In fact, Γ_{+} (resp. j_{b}) continued across a band becomes Γ_{−} (resp. j̃_{b}). We define all four objects at e^{iθ} ∈ ∂𝔻 as limits as r ↑ 1 of the values at re^{iθ}.

Proposition 3.4. (i) In the bands,

(3.44)  e^{iθp/2}Γ_{+}(e^{iθ}) = exp(−iπp k(θ))

(ii) At no point in the bands do both j_{b}(e^{iθ}) and j̃_{b}(e^{iθ}) vanish. (iii) j̃_{b} is everywhere nonvanishing on the interiors of the bands.

Proof. (i) This follows from (3.20). There is an issue of checking that it is exp(−iπp k(θ)), not exp(iπp k(θ)). To confirm this, note that

\frac{∂}{∂θ} Im log(exp(−iπp k(θ))) ≤ 0

and mainly < 0. Since ∂|Γ_{+}|/∂r ≤ 0 at r = 1, this is consistent with (3.44) and the Cauchy-Riemann equations.

(ii) follows from (3.43) and the fact that φ_{n}(z) is nonvanishing on ∂𝔻.

(iii) Continue (3.43) through the cut. Since φ_{m} is entire, the continuation onto the “second sheet” is also φ_{m}. Γ_{±} get interchanged by crossing the cut. Let us use j_{b,2}, j̃_{b,2} for the continuation to the second sheet (of course, j_{b,2} is j̃_{b} on the second sheet, but that will not concern us).

By this (3.43) continued, φ_{mp+b}(z) = 0 if and only if

(3.45)  (\frac{Γ_{−}(z)}{Γ_{+}(z)})^{m} = −\frac{j̃_{b,2}(z)}{j_{b,2}(z)}

If j̃_{b}(z_{0}) = 0 for z_{0} ∈ ∂𝔻, then |j̃_{b,2}(rz_{0})/j_{b,2}(rz_{0})| goes from 0 to a nonzero value as r increases. On the other hand, since |Γ_{−}/Γ_{+}| < 1 on ℂ\bands, for m large, |Γ_{−}(rz_{0})/Γ_{+}(rz_{0})|^{m} goes from 1 to a very small value as r increases. It follows that for m large,

|\frac{Γ_{−}(z)}{Γ_{+}(z)}|^{m} = |\frac{j̃_{b,2}(z)}{j_{b,2}(z)}|

has a solution r_{m}z_{0} with r_{m} > 1 and r_{m} → 1. As in [18], we can change the phase slightly to ensure (3.45) holds for some point, z_{m}, near r_{m}z_{0} with |z_{m}| > 1. Since φ has no zero in ℂ\𝔻, this is a contradiction. □

Remark. This proof shows that in the bands |j̃_{b}(e^{iθ})| > |j_{b}(e^{iθ})|.

(3.43) says we want to solve

(3.46)  (\frac{Γ_{−}(z)}{Γ_{+}(z)})^{m} = g_{b}(z)

to find zeros of φ_{mp+b}(z). We have, by the remark, that |g_{b}(θ)| < 1. Definition. We call z_{0} ∈ bands a singular point of order k if j_{b}(z_{0}) = 0 and the zero is of order k.

We do not know if there are singular points in any example! If so, they should be nongeneric. We define the functions

(3.47)  g̃_{b}(θ) = −\frac{j_{b}(e^{iθ})}{j̃_{b}(e^{iθ})}

and

(3.48)  g_{b}(z) = −\frac{j_{b}(z)}{j̃_{b}(z)}

For e^{iθ} in the interior of a band minus the singular points, let A(θ) be given by

(3.49)  \frac{g̃_{b}(θ)}{\overline{g̃_{b}(θ)}} = exp(2iA(θ))

with A continuous away from the singular points.

The analysis of a similar equation to (3.46) in [18] shows that:

(a) The solutions of (3.46) near |z| = 1 lie in sectors where

(3.50)  2πp k(θ) = A(θ) + \frac{2πj}{m} + O(\frac{1}{m log m})

with exactly one solution in each such sector.

(b) The magnitudes of the solutions obey

(3.51)  |z| = 1 − O(\frac{log m}{m})

(c) Successive zeros z_{k+1}, z_{k} obey

(3.52)  k(arg(z_{k+1})) − k(arg z_{k}) = \frac{1}{mp} + O(\frac{1}{m log m})

and

(3.53)  |\frac{z_{k+1}}{z_{k}} | = 1 + O(\frac{1}{m log m})

(d) All estimates in (a)–(c) are uniform on a band.

(e) Away from singular points, all O(1/m log m) errors can be replaced by O(1/m^{2}) and O(log m/m) in (3.51) by O(1/m). If there are no singular points, these error bounds are uniform over a band. It is easy to see that the total variation of A in each interval between singular

points (or band endpoints) is finite, so (3.50) and the fact that k varies by 1/p over a band say that the number of solutions in a band differs from m by a finite amount. This implies

Theorem 3.5. Let N^{(n,j)} be the number of zeros, z_{ℓ}, of φ_{n}(z) that obey

(a) arg z_{ℓ} ∈ band j (b)

(3.54)  (1 − |z_{ℓ}|) ≤ n^{−1/2}

Then

(a) sup_{n,j}|N^{(n,j)} − \frac{n}{p}| < ∞

(b) For n large, all such zeros have

(3.55)  (1 − |z_{0}|) ≤ C \frac{log n}{n}

and if there are no singular points, we can replace log n/n in (3.55) by 1/n.

Acknowledgment. This work was partially supported by NSF grant DMS-0140592. It is a pleasure to thank Chuck Newman and Percy Deift for the hospitality of the Courant Institute where some of this work was done.

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