Extension of a theorem of Duffin and Schaeffer

Let <i>r₁</i>,..., <i>r_s</i>: Z_n≥0 → C be linearly recurrent sequences whose associated eigenvalues have arguments in πQ and let <i>F(z)</i> := Σ_n ≥ 0 <i>f(n)zn</i>, where <i>f(n) ∈ {r1(n),..., rs(n)}</i> for each <i>n</i> ≥ 0. We prove that if <i>F(z)</i> 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 <i>f(n)</i> takes on values of finitely many polynomials.