Journal of Integer Sequences, Vol. 18 (2015), Article 15.9.2

Transcendence of Digital Expansions Generated by a Generalized Thue-Morse Sequence

Eiji Miyanohara
Graduate School of Fundamental Science and Engineering
Waseda University
3-4-1 Okubo, Shinjuku
Tokyo 169-8555


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 Σn=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.

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Received August 9 2014; revised version received October 10 2014; April 1 2015; July 21 2015; July 31 2015. Published in Journal of Integer Sequences, July 31 2015.

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