Hyperquadratic continued fractions over a finite field of odd characteristic with partial quotients of degree 1

TL;DR

This paper generalizes known examples of algebraic power series over finite fields with continued fraction expansions having all partial quotients of degree one, revealing a large family of such series in odd characteristic fields.

Contribution

It introduces a broad family of algebraic power series over finite fields with continued fractions having degree-one partial quotients, extending previous isolated examples.

Findings

Includes known examples within a large family of continued fractions.

Applies to finite fields of odd characteristic.

Shows these series are algebraic with specific continued fraction structures.

Abstract

In 1986, some examples of algebraic, and nonquadratic, power series over a finite prime field, having a continued fraction expansion with partial quotients all of degree one, were discovered by W. Mills and D. Robbins. In this note we show how these few examples are included in a very large family of continued fractions for certain algebraic power series over an arbitrary finite field of odd characteristic.