Extension of a theorem of Duffin and Schaeffer

Let r₁,..., r_s: Z_n≥0 → C be linearly recurrent sequences whose associated eigenvalues have arguments in πQ and let F(z) := Σ_n ≥ 0 f(n)zn, where f(n) ∈ {r1(n),..., rs(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.