A remarquable family of recurrent sequences

Abstract

Abstract. A Remarquable Family of Recurrent Sequences.Let u(n) be a recurrent sequence of rational integers, i.e., u(n + s) + a s_ l u(n + s -l) + ... + aou(n) = O, n >10, aieZ; i = 0,..., s -1. The polynomial P(x) = x "~ + a S i x'~ + "'" + ao is the companion or the characteristic polynomial of the recurrence. It is known that if none of the ratios of the roots of P is a root of unity, then the set A = {n,u(n) = 0} is finite. A recent result of F. Beukers shows that ifs = 3, then the set A has at most 6 elements and there exists, up to trivial transformations, only one recurrence of order 3 with 6 zeros, found by J. Berstel. In this paper, we construct for each s, s/> 2 a recurrent s 2 s sequence of order s, with at least + -1 zeroes, which generalize Berstel's sequence. 2 2

Linear Equations in Variables which Lie in a Multiplicative Group

“…Moreover, in the generic case, these will be nondegenerate solutions. Therefore A(n, 1) ≥ n. Bavencoffe and Bézivin [1] have given a more sophisticated example which even shows that A(n, 1) ≥ c n 2 where c is an absolute constant.…”