On Homogeneous Combinations of Linear Recurrence Sequences

Abstract

Let (F-n)(n >= 0) be the Fibonacci sequence given by Fn+2 = Fn+1 + F-n, for n >= 0, where F-0 = 0 and F-1 = 1. There are several interesting identities involving this sequence such as F-n(2) + F-n+1(2) = F2n+1, for all n >= 0. In 2012, Chaves, Marques and Togbe proved that if (Gm)m is a linear recurrence sequence (under weak assumptions) and G(n+1)(s) vertical bar center dot center dot center dot vertical bar G(n+l)(s)is an element of(G(m))(m), for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on l and the parameters of (G(m))(m). In this paper, we shall prove that if P(x(1), ..., x(l)) is an integer homogeneous s-degree polynomial (under weak hypotheses) and if P(G(n+1), ...,G(n+l)) is an element of(G(m))(m) for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on l, the parameters of (G(m))(m) and the coefficients of P.