### 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.

Original language | English |
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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

*Journal of Integer Sequences*,

*18*(9), [15.9.2].

**Transcendence of digital expansions generated by a generalized thue-morse sequence.** / Miyanohara, Eiji.

Research output: Contribution to journal › Article

*Journal of Integer Sequences*, vol. 18, no. 9, 15.9.2.

}

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

VL - 18

JO - Journal of Integer Sequences

JF - Journal of Integer Sequences

SN - 1530-7638

IS - 9

M1 - 15.9.2

ER -