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

Eiji Miyanohara*


    研究成果: Article査読


    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.

    ジャーナルJournal of Integer Sequences
    出版ステータスPublished - 2015

    ASJC Scopus subject areas

    • 離散数学と組合せ数学


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