  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 P\'olya-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.
