CONTINUED FRACTIONS REPRESENTING CERTAIN ANALOGUES OF EXPONENTIAL ELEMENTS IN Fq((1/x))

Abstract

Let (Qi)∞ i=1 be a sequence of nonzero monic polynomials over a finite field Fq satisfying Q1···Qi|Qi+1 (i ∈ N). Let α(n)=n i=1 1/Q1···Qi and α(∞)=∞ i=1 1/Q1···Qi. It is shown that the continued fractionfor α(∞) in the function field Fq((1/x)) can be explicitly given. As an application, by choosing suitable polynomials Qi, explicit continued fraction expansion of a function field analogue of some exponential elements in Fq((1/x)) (q ≥ 2), are derived. This gives an extension of some earlier work of Thakur in 1992.