Vol. LXXX, 2 (2011)
p. 213 - 220

On explicit formulae and linear recurrent sequences

R. Euler and L. H. Gallardo

Received: June 6, 2010;   Accepted: March 17, 2011

Abstract.   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 + D gn - 1
for n > 1, where D is any nonzero polynomial in R = Fq[t] A new identity is established.

Keywords:  Polynomials; euclidean division; finite fields; even characteristic. 

AMS Subject classification: Primary:  11T55, 11T06, 11B39  

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Acta Mathematica Universitatis Comenianae
ISSN 0862-9544   (Printed edition)

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Comenius University
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