ORTHOGONAL POLYNOMIALS WITH EXPONENTIALLY DECAYING RECURSION COEFFICIENTS

BARRY SIMON*

Dedicated to S. Molchanov on his 65th birthday

Abstract. We review recent results on necessary and sufficient conditions for measures on ℝ and ∂𝔻 to yield exponential decay of the recursion coefficients of the corresponding orthogonal polynomials. We include results on the relation of detailed asymptotics of the recursion coefficients to detailed analyticity of the measures. We present an analog of Carmona’s formula for OPRL. A major role is played by the Szegő and Jost functions.

1. Introduction: Szegő and Jost Functions

In broad strokes, spectral theory concerns the connection between the coefficients in differential or difference equations and the spectral measures associated to those equations. The process of going from coefficients to the measures is the direct problem, and the other direction is the inverse spectral problem. The gems of spectral theory are ones that set up one-one correspondences between classes of measures and coefficients with some properties. Examples are Verblunsky’s form of Szegő’s theorem [25] and the Killip-Simon theorem [12]. In this paper, our goal is to describe (mainly) recent results involving such gems for orthogonal polynomials whose recursion coefficients decay exponentially. These are technically simpler systems than the L^{2} results just quoted but have more involved details.

The two classes we discuss are orthogonal polynomials on the real line (OPRL) and on the unit circle (OPUC). For the OPRL case, we have a probability measure, dρ, on ℝ of bounded but infinite support whose orthonormal polynomials, p_{n}(x), obey

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

Date: March 2, 2006.

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

Submitted to the Proceedings of S. Molchanov’s 65th Birthday Conference.

with b_{n} ∈ ℝ and a_{n} ∈ (0, ∞) and called Jacobi parameters. {a_{n}, b_{n}}_{n=1}^{∞} is a description of dρ in that there is a one-one correspondence between bounded sets of such Jacobi parameters and such dρ’s. For background discussion of OPRL, see [4, 9, 18, 24].

For the OPUC case, 𝔻 = {z ∈ ℂ | |z| < 1}, and dμ is a probability measure on ∂𝔻 whose support is not a finite set. The orthonormal polynomials, φ_{n}(z), obey the Szegő recursion relation:

zφ_{n}(z) = ρ_{n}φ_{n+1}(z) + ᾱ_{n}φ_{n}^{∗}(z)  (1.2)
φ_{n}^{∗}(z) = z^{n} \overline{φ_{n}(1/z̄)}  (1.3)
ρ_{n} = (1 − |α_{n}|^{2})^{1/2}  (1.4)

with α_{n} ∈ 𝔻 and called Verblunsky coefficients. {α_{n}}_{n=0}^{∞} is a description of dμ in that there is a one-one correspondence between sequences of α_{n} obeying |α_{n}| < 1 and such dμ’s. For background discussion of OPUC, see [10, 17, 18, 19, 24].

The measure theoretic side of the equivalences will be in terms of a derived object, rather than the measures themselves. For OPUC, the object is D(z), the Szegő function [18, Section 2.4]. One says the Szegő condition holds if and only if

dμ(θ) = w(θ) \frac{dθ}{2π} + dμ_{s}  (1.5)

where dμ_{s} is singular and

∫ log(w(θ)) \frac{dθ}{2π} > −∞  (1.6)

(which is known to be equivalent to ∑_{n=0}^{∞}|α_{n}|^{2} < ∞). In that case, D(z) is defined by

D(z) = exp(∫ \frac{e^{iθ} + z}{e^{iθ} − z} log(w(θ)) \frac{dθ}{4π} )  (1.7)

which obeys

φ_{n}^{∗}(z) → D(z)^{−1}  (1.8)

if |z| < 1.

D(z) does not uniquely determine dμ, but it does if dμ_{s} = 0, as it will be in our cases of interest, since

w(θ) = lim_{r↑1} |D(re^{iθ})|^{2}  (1.9)

for a.e. θ.

For OPRL, the object is the Jost function. The situation is not as clean as the OPUC case in that there are not simple necessary and sufficient conditions for existence in terms of the measure. There are necessary and sufficient conditions in terms of the Jacobi parameters (see [5] and [19, Section 13.9]) but not for the measure. However, there are sufficient conditions for existence that suffice for us here. Suppose

dρ(x) = f(x) dx + dρ_{s}(x)  (1.10)

where f is supported on [−2, 2], and outside this set, the singular part, dρ_{s}, has only pure points {E_{j}^{±}}_{j=0}^{N_{±}} with

E_{1}^{−} < E_{2}^{−} < ⋯ < −2 < 2 < ⋯ < E_{2}^{+} < E_{1}^{+}  (1.11)

and suppose

∑_{j,±} (|E_{j}^{±}| − 2)^{1/2} < ∞  (1.12)

and that

∫_{−2}^{2} log(f(x))(4 − x^{2})^{−1/2} dx > −∞  (1.13)

Then (originally in Peherstorfer–Yuditskii [15]; see also Simon–Zlatoš [23] and [19, Theorem 13.8.9]) there is an analytic function u on 𝔻 so that its zeros are precisely those points z_{j}^{±} in 𝔻 given by

z_{j}^{±} + (z_{j}^{±})^{−1} = E_{j}^{±}  (1.14)

and if B is the Blaschke product (convergent by (1.12))

B(z) = ∏_{j=±1} \frac{(z − z_{j}^{±})}{1 − z̄_{j}^{±}z}  (1.15)

then Bu^{−1} ∈ H^{2} and the boundary values of u obey

|u(e^{iθ})|^{2} Im M(e^{iθ}) = sin θ  (1.16)

where

M(z) = ∫ \frac{dρ(x)}{z + z^{−1} − x}  (1.17)

(so Im M(e^{iθ}) is related to f(2 cos θ)).

These properties determine u uniquely. Unlike the OPUC case, u does not determine dρ even if dρ_{s} ↾ [−2, 2] = 0 for u only determines f and the localization of the pure points of dρ on ℝ\[−2, 2]. To recover dρ, we also need to know the weights of the pure points; equivalently, the residues of the poles of M at the z_{j}^{±}.

The theme of this review is that detailed results on exponential decay of recursion coefficients are equivalent to analyticity results on D^{−1} in the OPUC case and u in the OPRL case. That exponential decay implies analyticity has been in the physics literature for Schrödinger operators for over fifty years. The subtle aspect is the strict equivalence — an idea that appeared first in Nevai–Totik [14].

In Section 2, we discuss some aspects of finite range potentials, and in Section 3, following Nevai–Totik [14] and Damanik–Simon [6], the initial equivalence. In Section 4, following Simon [20, 21], we discuss detailed exponential asymptotics and meromorphic S and u.

I would like to thank J. Christiansen, L. Golinskii, P. Nevai, and V. Totik for useful comments.

Stas Molchanov is a leading figure in spectral theory. It is a pleasure to present this birthday bouquet to him.

2. Finite Range

In this section, we present new results on approximation by finite range “potentials.” We begin with an OPRL analog of Carmona’s result [3] on boundary condition averaging for Schrödinger operators. We will also see that Bernstein–Szegő measures for OPUC can be viewed through the Carmona lens. Carmona’s proof relies on computing derivatives of Prüfer variables — our proof here is spectral averaging making the relation to [22] transparent.

Let J be the semi-infinite Jacobi matrix

J = (\begin{matrix}b_{1} & & & a_{1} & & 0 & & … \\ a_{1} & & & b_{2} & & a_{2} & & … \\ 0 & & & a_{2} & & b_{3} & & … \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . \end{matrix})  (2.1)

associated to an OPRL with measure dρ; J_{n;F}, the matrix obtained from the top n rows and n left columns; and J_{n}^{b}, the matrix J_{n;F} with b_{n} replaced by b_{n} + b. Here b ∈ ℂ. Notice that if Im b ≤ 0, then spec(J_{n}^{b}) ⊂ ℂ\ℂ_{+}, so

m_{n}^{(b)}(z) = ⟨δ_{0}, (J_{n}^{(b)} − z)^{−1}δ_{0}⟩

is analytic for b ∈ \overline{ℂ}_{−} and z fixed in ℂ_{+}.

If dρ is a determinate moment problem, then J is essentially selfadjoint on finite sequences [16], so

m_{n}^{(b)}(z) → m(z) ≡ ∫ \frac{dρ(x)}{x − z}  (2.2)

for any b. Thus, if dν_{n} is defined by

m̃_{n}(z) ≡ \frac{1}{π} ∫_{−∞}^{∞} m_{n}^{(b)}(z) \frac{db}{1 + b^{2}}  (2.3)
= ∫ \frac{dν^{(n)}(x)}{x − z}  (2.4)

OPS WITH EXPONENTIAL DECAY 5

then

dν^{(n)} → dρ  (2.5)

weakly. dν^{(n)} is thus the average over b of the pure point spectral measures of J_{n}^{(b)}.

Theorem 2.1. If p_{n}(x) are the orthonormal OPRL, then

dν^{(n)}(x) = \frac{dx}{π(a_{n}^{2}p_{n}^{2}(x) + p_{n−1}^{2}(x))}  (2.6)

In particular, the right-hand side of (2.6) converges weakly to dρ. More is true, for Gaussian quadrature implies that if m_{n}^{(b)}(z) = ∫ dρ_{n}^{(b)}(x)(x − z)^{−1}, then ∫ x^{ℓ} dρ_{n}^{(b)}(x) = ∫ x^{ℓ} dρ(x) for ℓ ≤ 2n − 2, and thus,

∫ x^{ℓ} dν^{(n)}(x) = ∫ x^{ℓ} dρ ℓ = 0, …, 2n − 2  (2.7)

Of course, dν^{(n)} does not have all moments finite; indeed, ∫ |x|^{ℓ} dν^{(n)} = ∞ for ℓ ≥ 2n − 1.

Proof of Theorem 2.1. It is well known (see [18, Section 1.2]) that

det(z − J_{n;F}) = P_{n}(z)  (2.8)

the monic OPRL, and if J_{n;F}^{(1)} is the matrix obtained by removing the top row and leftmost column (i.e., 11 minor), then

det(z − J_{n;F}^{(1)} ) = Q_{n}(z)  (2.9)

the monic second kind polynomial of degree n − 1.

By expanding det(z − J_{n}^{(b)}) in minors, we see

det(z − J_{n}^{(b)}) = P_{n}(z) − bP_{n−1}(z)  (2.10)
= (a_{1} … a_{n−1})(a_{n}p_{n}(z) − bp_{n−1}(z))  (2.11)

and thus,

m_{n}^{(b)}(z) = −\frac{(a_{n}q_{n}(z) − bq_{n−1}(z))}{(a_{n}p_{n}(z) − bp_{n−1}(z))}  (2.12)

As noted above, if Im b ≤ 0, m_{n}^{(b)}(z) has its poles in Im z ≤ 0 and thus, if Im z > 0, m_{n}^{(b)}(z) is analytic in Im b ≤ 0. Thus, we can close the contour in the lower half-plane and find for Im z > 0,

m̃_{n}(z) = m_{n}^{(b=−i)}(z)
= −\frac{(a_{n}q_{n}(z) + iq_{n−1}(b))}{(a_{n}p_{n}(z) + ip_{n−1}(z))}  (2.13)

Thus, m̃_{n} is analytic on \overline{ℂ}_{+}, so

dν_{n}(x) = π^{−1} Im m̃_{n}(x) dx

Since p_{n}, p_{n−1}, q_{n}, q_{n−1} are real on ℝ,

Im m̃_{n}(x) = \frac{a_{n}(p_{n−1}(x)q_{n}(x) − p_{n}(x)q_{n−1}(x))}{(a_{n}^{2}p_{n}(x)^{2} + p_{n−1}(x)^{2})}  (2.14)

which is (2.8) by a standard Wronskian calculation (see (1.2.51) of [18]). □

By this same calculation, one can recover Carmona’s formula for the Schrödinger operator case.

One can ask about the analog of this for OPUC. Given a nontrivial measure, dμ, on ∂𝔻 and ω = e^{iθ} ∈ ∂𝔻, we define dμ_{n}^{(ω)} to be the trivial measure with Verblunsky coefficients

α_{j} = α_{j}(dμ) j = 0, …, n − 1
α_{n} = ω

Then dμ_{n}^{(ω)} is the measure with n + 1 pure points at the zeros of the paraorthogonal polynomial (POPUC),

Φ_{n+1}^{(ω)}(z) = zΦ_{n}(z) − ω̄Φ_{n}^{∗}(z)  (2.15)

Theorem 2.2. dμ_{n} ≡ ∫ \frac{dθ}{2π} dμ_{n}(e^{iθ}) is the Bernstein–Szegő measure

dμ_{n} = \frac{dθ}{2π|φ_{n}(e^{iθ})|^{2}}  (2.16)

Proof. If ψ_{n} are the second kind polynomials, Geronimus’ formula for F (z) (see [18, Theorem 3.2.4]) implies (F (z; dμ) = ∫ \frac{e^{iθ}+z}{e^{iθ}−z}dμ(θ))

F (z; dμ_{n}^{(ω)}) = \frac{ψ_{n}^{∗}(z) − ωzψ_{n}(z)}{φ_{n}^{∗}(z) − ωzφ_{n}(z)}  (2.17)

Averaging ω over \frac{dθ}{2π} gives the value at ω = 0 since this function is analytic in ω for z fixed in 𝔻. It follows that

F (z; dμ_{n}) = \frac{ψ_{n}^{∗}(z)}{φ_{n}^{∗}(z)}  (2.18)

and yields (2.16) by (3.2.35) of [18]. □

The Bernstein–Szegő approximation also has the property of being the measure associated to extending the α’s up to n to be free beyond n (i.e., α_{j} = 0 for j ≥ n). One can ask about the analogous approximation for OPRL. We will get the function S_{n} used by Dombrowski–Nevai Let J_{ℓ} be the Jacobi matrix with parameters

a_{n}(J_{ℓ}) = \begin{cases}a_{n}(J) & n = 1, …, ℓ − 1 \\ 1 & n ≥ ℓ \end{cases}  (2.19)

b_{n}(J_{ℓ}) = \begin{cases}b_{n}(J) & n = 1, …, ℓ \\ 0 & n ≥ ℓ \end{cases}  (2.20)

According to Theorem 13.6.1 (with a_{ℓ} replaced by 1), its Jost function is (x = z + 1/z)

g_{ℓ}(z) = z^{ℓ}(p_{ℓ}(z + \frac{1}{z}) − zp_{ℓ−1}(z + \frac{1}{z}))  (2.21)

Define S_{ℓ}(x) by

S_{ℓ}(z + \frac{1}{z}) ≡ g_{ℓ}(z)g_{ℓ}(\frac{1}{z})  (2.22)

Then, by (2.21),

S_{ℓ}(x) = p_{ℓ}(x)^{2} + p_{ℓ−1}(x)^{2} − xp_{ℓ}(x)p_{ℓ−1}(x)  (2.23)

Taking into account the different normalization (for us, “free” is a_{k} = 1; for them, a_{k} = \frac{1}{2}), this is the function S_{ℓ}(x) of Dombrowski–Nevai [8]. The approximating measure has a.c. part related to dx/|g_{ℓ}(z)|^{2} on [−2, 2] which is dx/S_{ℓ}(x). The eigenvalues of J_{ℓ} are zeros of S_{ℓ}(x) but not all zeros since S_{ℓ} also vanishes if g_{ℓ}(1/z) = 0, that is, at antibound state and resonance energies.

For most purposes, (2.8) is a more useful representation than the one associated to S_{ℓ}.

3. Necessary and Sufficient Conditions on Exponential Decay

The starting point of the recent results on exponential decay is the following result of Nevai–Totik for OPUC:

Theorem 3.1 ([14]). Let dμ be a nontrivial probability measure on ∂𝔻 and R > 1. Then the following are equivalent: (a)

lim sup_{n→∞} |α_{n}(dμ)|^{1/n} ≤ R^{−1}  (3.1)

(b) dμ_{s} = 0, the Szegő condition (1.6) holds, and D(z)^{−1} has an analytic continuation to {z | |z| < R}.

Remark. Since R^{−1} < 1, (3.1) is an expression of exponential decay.

The proof is easy. If (3.1) holds, Szegő recursion first implies inductively that for |z| = 1,

|Φ_{n+1}(e^{iθ})| ≤ (1 + |α_{n}|)|Φ_{n}(e^{iθ})|  (3.2)

so

sup_{n,|z|≤1} |Φ_{n}^{∗}(z)| = sup_{n,θ} |Φ_{n}^{∗}(e^{iθ})| (by the maximum principle)

≤ ∏_{j=0}^{∞}(1 + |α_{j}|) ≡ C < ∞  (3.3)

and thus, for |z| > 1,

|Φ_{n}(z)| ≤ C|z|^{n}  (3.4)

Iterating

Φ_{n+1}^{∗}(z) = Φ_{n}^{∗}(z) − α_{n}zΦ_{n}(z)  (3.5)

we get

Φ_{n}^{∗}(z) = 1 − ∑_{j=0}^{n−1} α_{j}zΦ_{j}(z)  (3.6)

(3.1), (3.4), and (3.6) imply that for any ε > 0,

sup_{n,|z|<R−ε} |Φ_{n}^{∗}(z)| < ∞

which implies that φ_{n}^{∗}(z) has a limit for |z| < R. This limit defines the analytic continuation of D(z)^{−1}.

For the other direction, one can use either of two similar-looking but distinct formulae relating D to α_{n}. One can use a formula of Geronimus [10] and Freud [9] as Nevai–Totik [14] do (it requires dμ_{s} = 0)

α_{n} = −κ_{∞} ∫ \overline{Φ_{n+1}(e^{iθ})} D(e^{iθ})^{−1}dμ(θ)  (3.7)

or the following formula of Simon [20] derived from iterated Szegő recursion:

α_{n} = −κ_{∞}^{−1}κ_{n}^{2} ∫ \overline{Φ_{n}(e^{iθ})} [D(e^{iθ})^{−1} − D(0)^{−1}]e^{−iθ} dμ(θ)  (3.8)

In these formulae,

κ_{n} = ∏_{j=0}^{n−1}(1 − |α_{j}|^{2})^{−1/2} κ_{∞} = lim_{n→∞} κ_{n}

To get exponential decay of α_{n} from (3.7) or (3.8), one uses ∫ \overline{Φ_{n}(e^{iθ})}e^{−ijθ} dμ(θ) = 0 for j < n and the Taylor series for D^{−1} to see that α_{n} is bounded by the tail of the Taylor series of D(z)^{−1} which, of course, decays exponentially if D(z)^{−1} is analytic in |z| < R.

For OPRL, the analogs of Theorem 3.1 are due to Damanik–Simon [6]. The result is simpler if there are no bound states or resonances where

Definition. We say a measure dρ on ℝ has no bound states or resonances if

dρ(x) = f(x) dx + dρ_{s}  (3.9)

where

supp(dρ) ⊂ [−2, 2]  (3.10)

and

∫ (4 − x^{2})^{−1}f(x) dx < ∞  (3.11)

Theorem 3.2 ([6]). Let R > 1. Suppose dρ has no bound states or resonances. Then u(z) has an analytic continuation to {z | |z| < R} if and only if

lim sup[|a_{n}(dρ) − 1| + |b_{n}(dρ)|]^{1/2n} ≤ R^{−1}  (3.12)

[6] has several proofs, but the simplest one is in [21]. When (3.11) holds, there is a measure dμ on ∂𝔻 given by

dμ(θ) = w(θ) \frac{dθ}{2π} + dμ_{s}  (3.13)

where

w(arccos(\frac{x}{2})) = c(4 − x^{2})^{−1/2}f(x)  (3.14)

for suitable c and dμ_{s}. The Verblunsky coefficients α_{n} for dμ and Jacobi parameters for dρ are related by ([2, 11]; [19, Section 13.2])

b_{n+1} = α_{2n} − α_{2n+2} − α_{2n+1}(α_{2n} + α_{2n+2})  (3.15)
a_{n+1}^{2} − 1 = α_{2n+1} − α_{2n+3} − α_{2n+2}^{2}(1 − α_{2n+3})(1 + α_{2n+1}) − α_{2n+3}α_{2n+1} (3.16)

and the Jost function for dρ and Szegő function for dμ by

u(z) = (1 − |α_{0}|^{2})(1 − α_{1})D(z)^{−1}  (3.17)

From this, it is easy to derive Theorem 3.2 from Theorem 3.1.

To understand the situation when J has bound states, we note the analytic continuation of (1.16) says

u(z)u(\frac{1}{z})[M(z) − M(\frac{1}{z})] = z − z^{−1}  (3.18)

(this uses also u, M real on ℝ). Recall that if z_{0} ∈ 𝔻 is such that z_{0} + z_{0}^{−1} is an eigenvalue of J, then u(z_{0}) = 0. An argument shows that if |z_{0}| > R^{−1} and |a_{n} − 1| + |b_{n}| ≤ CR^{−2n}, then u(z_{0}^{−1}) ≠ 0 and M(1/z) is regular at z_{0}. Thus, (3.18) implies a relation between u′(z_{0}), u(1/z_{0}), and the residue of the pole of M(z) at z_{0}. This leads to

Definition. Suppose u is analytic in {z | |z| < R} for some R > 1 and z_{0} ∈ 𝔻 with u(z_{0}) = 0 and |z_{0}| > R^{−1}. We say the weight of the point mass at z_{0} + z_{0}^{−1} is canonical if

lim_{z→z_{0}} (z − z_{0})M(z_{0}) = (z_{0} − z_{0}^{−1})[u′(z_{0})u(\frac{1}{z_{0}} )]  (3.19)

Theorem 3.3 ([6]). Fix R > 1. Then (3.13) holds if and only if

(i) u(z) has an analytic continuation to {z | |z| < R}. (ii) The point mass at each z_{0} ∈ 𝔻 with |z_{0}| > R^{−1} and u(z_{0}) = 0 is a

canonical weight.

If u is entire and has m zeros in 𝔻, the set of measures with that u has dimension m − 1. A single point on this space has decay at rate faster than any exponential. Similarly, if u is a polynomial, {a_{n} −1, b_{n}} has finite support if and only if all weights are canonical.

4. Detailed Asymptotics

Let S be defined by

S(z) = − ∑_{j=0}^{∞} α_{j−1}z^{j}  (4.1)

where α_{−1} = −1. Of course, when D exists, both D(z)^{−1} and S(z) are analytic near z = 0. Theorem 3.1 can be rephrased.

Theorem 4.1. The Taylor series of D(z)^{−1} and S(z) have the same radius of convergence.

Barrios, López, and Saff [1] extend this to show S(z) is meromorphic in {z | |z| < R + ε} with a single simple pole at z = R if and only if D(z)^{−1} is meromorphic in a similar region. This condition on S is, of course, equivalent to

α_{n} = CR^{−n} + O(R^{−n(1+δ)})  (4.2)

which is how they phrased their result. To go further, it is useful to define

r(z) = \overline{D(1/z̄)} D(z)^{−1}  (4.3)

which is analytic in {z | 1 − ε < |z| < R} if (3.1) holds. Simon [18] proved that r(z) − S(z) is analytic in {z | 1 − ε < |z| < R^{2}} when (3.1) holds, thereby generalizing [1]. The ultimate result of this genre was found independently by Deift–Ostensson [7] and Martínez-Finkelshtein et al. [13]; an alternate proof was then found by Simon [20].

Theorem 4.2. If (3.1) holds for some R > 1, then r(z) − S(z) is analytic in {z | 1 − ε < |z| < R^{3}}.

This is optimal in that there are examples [13, 20] where S (and r) have a simple pole at z = R but S − r has a pole at z = R^{3}.

Motivated by this, Simon [20] proved:

Theorem 4.3. S(z) is an entire meromorphic function if and only if D(z)^{−1} is.

One can even relate the poles. Given a set S in {z | |z| > 1} which is discrete, one defines 𝔾(S) to be the set of all products z_{1} … z_{n+1}z̄_{n+2} … z̄_{2n+1} where z_{j} ∈ S. Then

Theorem 4.4 ([20]). Let S(z) be entire meromorphic and let P be the poles of D(z)^{−1} and T the set of poles of S(z). Then P ⊂ 𝔾(T) and T ⊂ 𝔾(P).

Simon [21] studies analogs of the results for OPRL. In the Jacobi case, define

B(z) = 1 − ∑_{n=0}^{∞}[b_{n+1}z^{2n+1} + (a_{n+2} − 1)z^{2n+2}]  (4.4)

The analog of Theorem 4.2 is

Theorem 4.5 ([21]). Suppose R > 1 and

lim sup_{n→∞} (|a_{n}^{2} − 1| + |b_{n}|)^{1/2n} = R^{−1}

Then (1 − z^{2})u(z) + z^{2}u(1/z)B(z) is analytic in {z | R^{−1} < |z| < R^{2}}.

As explained there, R^{2} is optimal. The analog of Theorem 4.3 is

Theorem 4.6. B(z) is an entire meromorphic function if and only if u(z) is.

The connection between poles, that is, the analog of Theorem 4.4 is complicated but appears in [21].

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