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.