Transcendence of digital expansions generated by a generalized Thue-Morse sequence

Miyanohara, Eiji

Journal of integer sequences, Tome 18 (2015) no. 9

Résumé

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 $(a(n))_{n=0}^{\infty }$, and consider the condition that $(a(n))_{n=0}^{\infty }$ is non-periodic. Next, we show that, if a generalized Thue-Morse sequence $(a(n))_{n=0}^{\infty }$ is not periodic, then no subsequence of the form $(a(N+nl))_{n=0}^{\infty }$ (where $N \ge 0$ 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 ${0,1,\dots ,\beta -1}$ (where $\beta $ is an integer greater than 1), the series $\Sigma _{n=0}^{\infty } a(N+nl) \beta ^{-n-1}$ gives a transcendental number. Furthermore, for non-periodic generalized Thue-Morse sequences taking positive integer values, the continued fraction $[0, a(N), a(N+l),\dots , a(N+nl), \dots ]$ gives a transcendental number.

Détail

Classification : 11A63, 11J99
Keywords: digit counting, stammering sequence