Transcendence of digital expansions and continued fractions generated by a cyclic permutation and $k$-adic expansion

TL;DR

This paper generalizes the Thue-Morse sequence using cyclic permutations and $k$-adic expansions, establishing conditions for non-periodicity and demonstrating that certain series and continued fractions derived from these sequences are transcendental.

Contribution

It introduces a new class of generalized Thue-Morse sequences and provides criteria for their non-periodicity, proving that associated series and continued fractions are transcendental.

Findings

Non-periodic generalized Thue-Morse sequences exist under specific conditions.

Series constructed from these sequences are transcendental numbers.

Continued fractions from these sequences are also transcendental.

Abstract

In this article, first we generalize the Thue-Morse sequence $(a(n))_{n=0}^\infty$ (the generalized Thue-Morse sequences) by a cyclic permutation and $k$ -adic expansion of natural numbers, and consider the necessary-sufficient condition that it is non-periodic. Moreover we will show that, if the generalized Thue-Morse sequence is not periodic, then all equally spaced subsequences $(a(N+nl))_{n=0}^\infty$ (where $N \ge 0$ and $l >0$) of the generalized Thue-Morse sequences are not periodic. Finally we apply the criterion of [ABL], [Bu$1$] on transcendental numbers, to find that , for a non periodic generalized Thue-Morse sequences taking the values on $\{0,1,\cdots,\beta-1\}$(where $\beta$ is an integer greater than $1$), the series $\sum_{n=0}^\infty a(N+nl) {\beta}^{-n-1}$ gives a transcendental number, and further that for non periodic generalized Thue-Morse sequences taking the values on positive integers, the continued fraction $[0:a(N), a(N+l),\cdots,a(N+nl ), \cdots]$ gives a transcendental number, too.