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

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.

18pages add two references and remove the section 6

References in corpus (1)

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