On almost-periodic sequences (Q1127967)

A sequence of complex numbers \(u_n\) is said to be algebraic if the generating power series \(f(z)= \sum^\infty_{n=0} u_nz^n\) is algebraic over \(\mathbb{C}(z)\). The author characterizes those algebraic sequences that are 1) bounded (by means of the singularities of \(f(z))\), 2) almost-periodic. Then he generalizes his results to differentially finite sequences (that means: \(f(z)\) satisfies a linear differential equation with polynomial coefficients). In the special case of linear recurrent sequences, these results were proved by \textit{L. Lucht} [Arch. Math. 68, 22-26 (1997; Zbl 0869.11012)].