Une famille remarquable de suites recurrentes lineaires

Abstract

Letu(n) be a recurrent sequence of rational integers, i.e.,u(n+s)+a _s−1 u(n+s−1)+...+a ₀ u(n)=0,n≥0,a _i∈ℤ,i=0,...,s−1. The polynomialP(x)=x ^s +a _s−1x^s +...+a ₀ is the companion or the characteristic polynomial of the recurrence. It is known that if none of the ratios of the roots ofP is a root of unity, then the setA={n,u(n)=0} is finite. A recent result of F. Beukers shows that ifs=3, then the setA 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 eachs, s≥2 a recurrent sequence of orders, with at leasts ²/2+s/2−1 zeroes, which generalize Berstel's sequence.