Integral zeros of a polynomial with linear recurrences as coefficients

Abstract

Let K be a number field, S a finite set of places of K, and O S be the ring of S-integers. Moreover, letbe a polynomial in Z having simple linear recurrences of integers evaluated at n as coefficients. Assuming some technical conditions we give a description of the zeros (n, z) ∈ N × O S of the above polynomial. We also give a result in the spirit of Hilbert irreducibility for such polynomials.