Extension of a Theorem of Duffin and Schaeffer

Michael Coons School of Mathematical and Physical Sciences The University of Newcastle Callaghan, NSW 2308 Australia

                Michael.Coons@newcastle.edu.au

Abstract Let r_{1}, …, r_{s} : ℤ_{n≥0} → ℂ be linearly recurrent sequences whose associated eigenvalues have arguments in πℚ and let F (z) := ∑_{n≥0} f(n)z^{n}, where f(n) ∈ {r_{1}(n), …, r_{s}(n)} for each n ≥ 0. We prove that if F (z) is bounded in a sector of its disk of convergence, then it is a rational function. This extends a very recent result of Tang and Wang, who gave the analogous result when the sequence f(n) takes on values of finitely many polynomials.

1 Introduction

In 1945, Duffin and Schaeffer [2] proved that if a power series whose coefficients are taken from a finite set is bounded in a sector of the unit disk, then it is already a rational function. In this short note, we prove the following extension.

Theorem 1. Let r_{1}, …, r_{s} : ℤ_{n≥0} → ℂ be linearly recurrent sequences whose associated eigenvalues have arguments in πℚ and let F (z) := ∑_{n≥0} f(n)z^{n}, where f(n) ∈ {r_{1}(n), …, r_{s}(n)} for each n ≥ 0. If F (z) is bounded in a sector of its disk of convergence, then it is a rational function.

Our result, follows from a very recent result of Tang and Wang [3, Theorem 1.3], who used a method developed by Borwein, Erdélyi, and Littmann [1] to show that if p_{1}(z), …, p_{s}(z) ∈ ℂ[z] are polynomials and F (z) := ∑_{n≥0} f(n)z^{n}, where f(n) ∈ {p_{1}(n), …, p_{s}(n)} for each n ≥ 0, is bounded in a sector of the unit disk, then it is a rational function.

2 Proof of Theorem 1

Proof of Theorem 1. Let r_{1}, …, r_{s} : ℤ_{n≥0} → ℂ be linearly recurrent sequences whose associated eigenvalues have arguments in πℚ and f(n) ∈ {r_{1}(n), …, r_{s}(n)} for each n ≥ 0. Then for each i = 1, …, s, there exist an integer ℓ, ℓ × 1 vectors u_{i} and v_{i} with complex entries, and an ℓ × ℓ matrix A with complex entries such that r_{i}(n) = u_{i}^{T} A_{i}^{n}v_{i}. Recall that r_{i}(n) can be written in the eigenvalue expansion

r_{i}(n) = ∑_{j=0}^{k} α_{j}n^{m_{j}−1}λ_{j}^{n},

where λ_{1}, …, λ_{k} are the eigenvalues of A_{i}, which we have assumed have arguments in πℚ, m_{j} is the multiplicity of λ_{j}, and α_{1}, …, α_{k} ∈ ℂ.

Similarly, we can consider a “common” eigenvalue expansion of the linear recurrences r_{1}, …, r_{s}. In particular, let

ρ_{1} < ρ_{2} < ⋯ < ρ_{D}  (1)

be the distinct moduli of the collection of eigenvalues of A_{1}, …, A_{s}. Then there exist integers D and d and a finite collection of complex numbers {α_{n,i,m,t}}_{n≥0,1≤i≤s,1≤m≤D,0≤d}, many of which may be zero, such that

r_{i}(n) = ∑_{m=1}^{D} ∑_{t=0}^{d} α_{n,i,m,t}n^{t}ρ_{m}^{n} (i = 1, 2, …, s, and n ≥ 0).  (2)

We note that the collection {α_{n,i,m,t}}_{n≥0,1≤i≤s,1≤m≤D,0≤d} is finite precisely because all of the eigenvalues associated to the linear recurrences have arguments that are rational multiples of π (this assumption in the statement of the theorem is crucial for this proof to work).

Now consider F (z) := ∑_{n≥0} f(n)z^{n} satisfying the assumptions of the theorem. Suppose that F (z) is bounded in a sector of its disk of convergence, that F (z) is irrational, and that D is the minimal such number so that the previous two properties hold. We will prove the theorem by showing that such a minimal D cannot exist.

To this end, set Ψ(z) := F (z/ρ_{D}), so that the coefficients ψ(n) of Ψ(z) satisfy ψ(n) = f(n)ρ_{D}^{−n}, Ψ(z) is bounded in a sector of the unit disk, and Ψ(z) is irrational. Using (2) we have that

ψ(n) = α_{n,i_{n},D,d} · n^{d} + α_{n,i_{n},D,d−1} · n^{d−1} + ⋯ + α_{n,i_{n},D,0} + r(n),  (3)

where for each n,

r(n) ∈ {r′_{1}(n), …, r′_{s}(n)} with r′_{i}(n) = ∑_{m=1}^{D−1} ∑_{t=0}^{d} α_{n,i,m,t}n^{t}(ρ_{m}/ρ_{D})^{n},  (4)

and where we have used the notation i_{n} to indicate the value of i ∈ {1, …, s} for which f(n) = r_{i}(n). By (1), (4), and the minimality of D, we have that the function R(z) = ∑_{n≥0} r(n)z^{n} is bounded throughout the unit disk (as it has radius of convergence at least ρ_{D}/ρ_{D−1} > 1) and is rational. Thus by (3), the function

Ψ(z) − R(z) = ∑_{n≥0} (α_{n,i_{n},D,d} · n^{d} + α_{n,i_{n},D,d−1} · n^{d−1} + ⋯ + α_{n,i_{n},D,0}) z^{n}

is bounded in a sector of the unit disk and irrational. But by Tang and Wang’s result, the function Ψ(z) − R(z) is rational. This contradiction proves the theorem.

3 A question

The careful reader will notice that while Tang and Wang’s result holds for polynomials in ℂ[z], the linear recurrences in our theorem, while also being complex-valued, must have associated eigenvalues with arguments in πℚ. This is because if we let the eigenvalues be more general complex numbers, the arguments of the powers of those eigenvalues may not take only finitely many values. Our proof works as long as this finiteness condition holds.

Because of the necessity of this finiteness condition, Theorem 1 is essentially the best possible result that can be proved from the methods of Borwein, Erdélyi, and Littmann [1], and Tang and Wang [3]. An interesting generalization would be a statement as in Theorem 1 for general complex linear recurrences. To apply the same basic theory, one would need to answer the following question in the affirmative.

Question 2. Let a : ℤ_{≥0} → ℂ be given by

a(n) = ∑_{i=0}^{M} b_{i,n}ζ_{i}^{n},

where b_{i,n}, ζ_{i} ∈ B, B is finite set of complex numbers, and ζ_{i} lies on the unit circle for each i. Suppose that for each ε > 0 there exist a positive integer q and complex numbers γ_{0}, …, γ_{q−1} such that for all n ≥ 0,

|γ_{0}a(n) + ⋯ + γ_{q−1}a(n + q − 1) + a(n + q)| < ε.

Is it true that the sequence a is linearly recurrent?

4 Acknowledgment

We thank the referee and editor—their great care made this a better (and correct!) paper. This paper was written while visiting the Alfréd Rényi Institute of the Hungarian Academy of Sciences; we thank the Institute and its members for their kindness and support.

References

[1] P. Borwein, T. Erdélyi, and F. Littmann, Polynomials with coefficients from a finite set, Trans. Amer. Math. Soc. 360 (2008), 5145–5154.

[2] R. J. Duffin and A. C. Schaeffer, Power series with bounded coefficients, Amer. J. Math. 67 (1945), 141–154.

[3] S. Tang and C. Wang, Rational-transcendental dichotomy of power series with a restriction on coefficients, Monatsh. Math. (2017), published online 24 May 2017.

2010 Mathematics Subject Classification: Primary 30B30 Secondary 11B37, 11J91. Keywords: Power series, rational function, Duffin-Schaeffer theorem.

Received June 8 2017; revised versions received August 9 2017; September 5 2017; September 11 2017. Published in Journal of Integer Sequences, September 15 2017.

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