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 fraction

for α(∞) in the function field Fq((1/x)) can be explicitly given. As an

application, by choosing suitable polynomials Qi, explicit continued frac

tion 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.