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

Eiji Miyanohara

    Research output: Contribution to journalArticle

    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 languageEnglish
    Article number15.9.2
    JournalJournal of Integer Sequences
    Volume18
    Issue number9
    Publication statusPublished - 2015

    Fingerprint

    Thue-Morse Sequence
    Transcendence
    Transcendental number
    Integer
    Continued fraction
    Subsequence
    Permutation
    Non-negative
    Generalise
    Series

    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 journalArticle

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    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.

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