On a version of the Problem B of Mahler involving derivatives

Abstract

In 1902, Paul St\"ackel constructed an analytic function $f(z)$ in a neighborhood of the origin, which was transcendental, and with the property that both $f(z)$ and its inverse, as well as its derivatives, assumed algebraic values at all algebraic points in this neighborhood. Inspired by this result, Mahler in 1976 questioned the existence of an transcendental entire function $f(z)$ such that $f(\overline{\mathbb{Q}})$ and $f^{-1}(\overline{\mathbb{Q}})$ are subsets of $\overline{\mathbb{Q}}.$ This problem was solved by Marques and Moreira in 2017. As St\"acklel's result involved derivatives, it is natural to question whether we have an analogous result for transcendental entire functions involving derivatives. In this article, we show that there are an uncountable amount of such functions.