Algebraic automatic continued fractions in characteristic $2$ II

TL;DR

This paper introduces two families of automatic sequences in characteristic 2 that define algebraic continued fractions, expanding understanding of their algebraic properties and relationships.

Contribution

It identifies and characterizes two specific families of automatic sequences, including the period-doubling and Thue-Morse sequences, related to algebraic continued fractions in characteristic 2.

Findings

Period-doubling sequence belongs to family P

Thue-Morse sequence belongs to family G

Family G includes all iterated sums of sequences from P

Abstract

We present two families of automatic sequences that define algebraic continued fractions in characteristic $2$. The period-doubling sequence belongs to the first first family $\mathcal{P}$; and its sum, the Thue-Morse sequence, belongs to the second family $\mathcal{G}$. The family $\mathcal{G}$ contains all the iterated sums of sequences from the $\mathcal{P}$ and more.