On Homogeneous Combinations of Linear Recurrence Sequences

Let ( F n ) n ≥ 0 be the Fibonacci sequence given by F n + 2 = F n + 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 = F 2 n + 1 , for all n ≥ 0 . In 2012, Chaves, Marques and Togbé proved that if ( G m ) m is a linear recurrence sequence (under weak assumptions) and G n + 1 s + ⋯ + G n + ℓ s ∈ ( G m ) m , for infinitely many positive integers n , then s is bounded by an effectively computable constant depending only on ł 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 ) ∈ ( 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 .