A problem around Mahler functions

Boris Adamczewski (1) , Jason P. Bell (2)

1 ICJ - Institut Camille Jordan
2 Department of Pure Mathematics [Waterloo]

Let $K$ be a field of characteristic zero and $k$ and $l$ be two multiplicatively independent positive integers. We prove the following result that was conjectured by Loxton and van der Poorten during the Eighties: a power series $F(z)\in K[[z]]$ satisfies both a $k$- and a $l$-Mahler type functional equation if and only if it is a rational function.

Mots clés: Mahler functions; functional equations; Mahler's method; automata; Cobham's theorem; transcendence

Domaines: Théorie des nombres [math.NT]; Combinatoire [math.CO]

Boris Adamczewski, Jason P. Bell. A problem around Mahler functions. 2012. ⟨hal-00798303⟩