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