On the algebraicity of Thue-Morse and period-doubling continued fractions

TL;DR

This paper explores whether certain continued fractions derived from Thue-Morse and period-doubling sequences are algebraic or transcendental, using automata-based methods to prove specific cases of these conjectures.

Contribution

It introduces the Guess'n'Prove method leveraging automata structure to address conjectures on algebraicity and transcendence of these special continued fractions.

Findings

Proposed several conjectures on algebraicity and transcendence.

Developed the Guess'n'Prove method for special case proofs.

Provided partial proofs for specific conjectures.

Abstract

We put forward several general conjectures concerning the algebraicity or transcendence of continued fractions and Stieltjes continued fractions defined by the Thue-Morse and period-doubling sequences in characteristic $2$. We present our Guess'n'Prove method, in which we exploit the structure of automata, for proving some of our conjectures in special cases.