Let \(\sum a_nx^n\in \bar{\mathbb {Q}}[[x]]\) be the power series representation of a rational function and let \(f:\ \{0,1,\ldots \}\rightarrow \bar{\mathbb {Q}}\) be a so-called almost quasi-polynomial. Under a necessary stability condition, we prove that \(\sum f(n)a_nx^n\) satisfies the Pólya–Carlson dichotomy: it is either a rational function or it cannot be extended analytically to a strictly larger domain than its disk of convergence. This latter property is much stronger than being transcendental. The first application and motivation of our result is the solution of a conjecture by Byszewski–Cornelissen. This gives a complete understanding of the analytic continuation behavior of the Artin–Mazur zeta function associated to a dynamical system on an abelian variety. Further applications include the solution of a conjecture by Bell–Miles–Ward and a significant case of an open problem by Royals–Ward.
