On Explicit Formulae and Linear Recurrent Sequences

We notice that some recent explicit results about linear recurrent sequences over a ring R with 1 were already obtained by Agou in a 1971 paper by considering the euclidean division of polynomials over R. In this paper we study an application of these results to the case when R = Fq [t] and q is even, completing Agou’s work. Moreover, for even q we prove that there is an infinity of indices i such that gi = 0 for the linear recurrent, Fibonacci-like, sequence defined by g0 = 0, g1 = 1, and gn+1 = gn + ∆gn−1 for n > 1, where ∆ is any nonzero polynomial in R = Fq [t]. A new identity is established.