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

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In this article, first we generalize the Thue-Morse sequence (a(n))ₙ₌₀∞ (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∞ (where N ≥ 0 and l >0) of the generalized Thue-Morse sequences are not periodic. Finally we apply the criterion of [ABL], [Bu1] on transcendental numbers, to find that, for a non periodic generalized Thue-Morse sequences taking the values on {0,1,,β-1}(where β is an integer greater than 1), the series ∑ₙ₌₀∞ a(N+nl) β⁻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),,a(N+nl),] gives a transcendental number, too.