Hyperquadratic continued fractions and automatic sequences

TL;DR

This paper explores a connection between algebraic continued fractions in power series fields over finite fields and automatic sequences, illustrating the relationship with examples and a counterexample.

Contribution

It establishes a novel correspondence between algebraic continued fractions and automatic sequences over finite fields, supported by explicit examples.

Findings

Identifies a correspondence between algebraic continued fractions and automatic sequences.

Provides three illustrative examples demonstrating this connection.

Includes a counterexample showing limitations of the correspondence.

Abstract

The aim of this note is to show the existence of a correspondance between certain algebraic continued fractions in fields of power series over a finite field and automatic sequences in the same finite field. this connection is illustrated by three families of examples and a counterexample.