Generalized Thue-Morse Continued Fractions

TL;DR

This paper generalizes the Thue-Morse sequence to TM_m sequences, explores their continued fractions, and extends transcendence results, showing that some existing theorems do not apply in this broader context.

Contribution

It introduces a new generalization of Thue-Morse sequences and extends known transcendence results to these sequences, highlighting limitations of existing theorems.

Findings

Generalization of Thue-Morse sequences to TM_m sequences

Extension of Queffélec's theorem to TM_m continued fractions

Demonstration that Adamczewski and Bugeaud's theorem does not apply here

Abstract

The Thue-Morse sequence is generalized to the $TM_m$ sequences and two equivalent definitions are given. This generalization leads to transcendental numbers and has Queff\'elec's theorem on Thue-Morse continued fractions as a special case. It is also shown that the theorem of Adamczewski and Bugeaud for palindromic continued fractions cannot be applied in this case.