Hyperquadratic power series in F3((T−1)) with partial quotients of degree 1

Abstract

In this note, we describe a large family of nonquadratic continued fractions in the field F3((T−1)) of power series over the finite field F3. These continued fractions are remarkable for two reasons: first, they satisfy an algebraic equation with coefficients in F3[T] given explicitly, and, second, all the partial quotients in the expansion are polynomials of degree 1. In 1986, in a basic article in this area of research, Mills and Robbins (J. Number Theory 23:388–404, 1986) gave the first example of an element belonging to this family.