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

Graduate School of Fundamental Science and Engineering

Waseda University

3-4-1 Okubo, Shinjuku

Tokyo 169-8555

Japan

**Abstract:**

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.

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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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