ON PERIODIC CONTINUED FRACTIONS OVER Fq((X−1))

Abstract

Let Fq be a field with q elements of characteristic p and Fq((X−1)) be the field of formal power series over Fq. Let f be a quadratic formal power series of continued fraction expansion [b0; b1, . . ., bs, a1, . . ., at], we denote by t = Per(f) the period length of the partial quotients of f. The aim of this paper is to study the continued fraction expansion of Af where A is a polynomial ∈ Fq[X]. In particular we study the asymptotic behavior of the functions S(N, n) = sup deg A=N sup f∈Λn Per (Af) and R(N) = sup n≥1 S(N, n) n , where Λn is the set of quadratic formal power series of period n in Fq((X−1)).