A function field variant of Pillai's problem

Abstract

In this paper, we consider a variant of Pillai's problem over function fields F in one variable over C. For given simple linear recurrence sequences Gn and Hm, defined over F and satisfying some weak conditions, we will prove that the equation Gn−Hm=f has only finitely many solutions (n,m)∈N2 for any non-zero f∈F, which can be effectively bounded. Furthermore, we prove that under suitable assumptions there are only finitely many effectively computable f with more than one representation of the form Gn−Hm.