On split families of Thue equations with linear recurrence sequences as factors

Abstract

We consider a parametrised family of Thue equations, \[ (x-G_1(n)\, y) \cdots (x-G_d(n)\, y) - y^d = \pm 1, \] which was first considered by Thomas to have an explicit set of solutions for parameters $n$ larger than some effectively computable constant. In the case where the parameter functions are polynomials belonging to an explicitly described family, this is known to be true. We consider other parameter functions, namely linear recurrence sequences, for which it is not obvious that a similar result holds, and confirm that it does for an explicitly described family of linear recurrence sequences.