Transcendence of digital expansions generated by a generalized thue-morse sequence

In this article, first we generalize the Thue-Morse sequence by means of a cyclic permutation and the k-adic expansion of non-negative integers, giving a sequence (a(n))_n=0^∞, and consider the condition that (a(n))_n=0^∞ is non-periodic. Next, we show that, if a generalized Thue-Morse sequence (a(n))_n=0^∞ is not periodic, then no subsequence of the form (a(N+nl))_n=0^∞ (where N ≥ 0 and l > 0) is periodic. We apply the combinatorial transcendence criterion established by Adamczewski, Bugeaud, Luca, and Bugeaud to find that, for a non-periodic generalized Thue-Morse sequence taking its values in {0,1,...,β-1} (where β is an integer greater than 1), the series Σ_n=0^∞ a(N+nl) β^-n-1 gives a transcendental number. Furthermore, for non-periodic generalized Thue-Morse sequences taking positive integer values, the continued fraction [0, a(N), a(N+l),..., a(N+nl), ...] gives a transcendental number.