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 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.
| Original language | English |
|---|---|
| Article number | 15.9.2 |
| Journal | Journal of Integer Sequences |
| Volume | 18 |
| Issue number | 9 |
| Publication status | Published - 2015 |
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Keywords
- Digit counting
- Stammering sequence
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
Cite this
Transcendence of digital expansions generated by a generalized thue-morse sequence. / Miyanohara, Eiji.
In: Journal of Integer Sequences, Vol. 18, No. 9, 15.9.2, 2015.Research output: Contribution to journal › Article
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TY - JOUR
T1 - Transcendence of digital expansions generated by a generalized thue-morse sequence
AU - Miyanohara, Eiji
PY - 2015
Y1 - 2015
N2 - 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.
AB - 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.
KW - Digit counting
KW - Stammering sequence
UR - http://www.scopus.com/inward/record.url?scp=84938946935&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84938946935&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:84938946935
VL - 18
JO - Journal of Integer Sequences
JF - Journal of Integer Sequences
SN - 1530-7638
IS - 9
M1 - 15.9.2
ER -