2014/04/16 by Eiji Miyanohara, Miyanohara, Eiji
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1404.4153
18pages add two references and remove the section 6
arxiv created 2014/05/07 · arxiv updated 2014/05/08
In this article, first we generalize the Thue-Morse sequence (a(n))n=0^∞ (the generalized Thue-Morse sequences) by a cyclic permutation and k -adic expansion of natural numbers, and consider the necessary-sufficient condition that it is non-periodic. Moreover we will show that, if the generalized Thue-Morse sequence is not periodic, then all equally spaced subsequences (a(N+nl))n=0^∞ (where N ≥ 0 and l >0) of the generalized Thue-Morse sequences are not periodic. Finally we apply the criterion of [ABL], [Bu1] on transcendental numbers, to find that , for a non periodic generalized Thue-Morse sequences taking the values on \0,1,⋯,β-1\(where β is an integer greater than 1), the series ∑n=0^∞ a(N+nl) β-n-1 gives a transcendental number, and further that for non periodic generalized Thue-Morse sequences taking the values on positive integers, the continued fraction [0:a(N), a(N+l),⋯,a(N+nl ), ⋯] gives a transcendental number, too.