On certain recurrent and automatic sequences in finite fields

TL;DR

This paper explores the automaticity of certain algebraic and recurrent sequences over finite fields, extending previous work to establish new criteria and families of automatic sequences in arbitrary finite fields.

Contribution

It generalizes the connection between automatic sequences and algebraic power series over finite fields, introducing new criteria and families of automatic sequences.

Findings

Proves automaticity of a specific family of sequences in finite fields of characteristic 2.

Provides a generalized criterion for automatic sequences in finite fields.

Introduces new families of automatic sequences in arbitrary finite fields.

Abstract

In this work we extend our study on a link between automaticity and certain algebraic power series over finite fields. Our starting point is a family of sequences in a finite field of characteristic $2$, recently introduced by the first author in connection with algebraic continued fractions. By including it in a large family of recurrent sequences in an arbitrary finite field, we prove its automaticity. Then we give a criterion on automatic sequences, generalizing a previous result and this allows us to present new families of automatic sequences in an arbitrary finite field.