Study on the non-periodicity of the generalized Thue-Morse sequences generated by cyclic permutations

TL;DR

This paper generalizes the Thue-Morse sequence using cyclic permutations and p-adic systems, establishing conditions for non-periodicity and exploring implications for transcendental number construction.

Contribution

It introduces a new generalization of Thue-Morse sequences and provides necessary and sufficient conditions for their non-periodicity, linking sequence properties to transcendence results.

Findings

Generalized Thue-Morse sequences are non-periodic under certain conditions.

All equally spaced subsequences of non-periodic sequences are also non-periodic.

Application of combinatorial transcendence results yields many transcendental numbers.

Abstract

First we generalize the Thue-Morse sequence (the generalized Thue-Morse sequences) by a cyclic permutations and p-adic system, and consider the necessary-sufficient condition that it is non-periodic. Moreover if the generalized Thue-Morse sequence is not periodic, then all equally spaced subsequences of the generalized Thue-Morse sequences are not periodic. Finally we apply recently combinatorial transcendence results to the generalized non-periodic Thue-Morse sequence, we can found many transcendental numbers.