arXiv subjects

Eiji Miyanohara

Publications and source records attributed to Eiji Miyanohara.

5 recordsLinked to original sources

arXiv2021

Generalized $k$-regular sequences II:digital pattern and transcendence

In this paper, we prove that an uncountable quantity of real numbers generated by digital pattern sequences gives the transcendental number. This result gives a generalization of Main theorem in Morton and Mourant [MortM], which state that countable real numbers generated by digital pattern sequences gives the transcendental number. Our method relies on the combinatorial quantitative transcendence criterion established by Adamczewski-Bugeaud [AdB2] and the properties of generalized $k$-regular sequences, which is introduced by the author [Mi2].

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arXiv2021

Digital pattern and transcendence via generalized $k$-regular sequences

In this paper, we prove that there are uncountable many real transcendental numbers, which are generated by digital pattern sequences. This generalizes the main theorem in Morton and Mourant, which states the existence of countable many similar numbers. Our method relies on the combinatorial quantitative transcendence criterion established by Adamczewski and Bugeaud and properties of generalized k-regular sequences, which is introduced by this paper.

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arXiv2021

Generalized $k$-regular sequences III: Arithmetical properties of generalized $k$-regular series

Let $F(z)$ be a $k$-regular series in $\mathbb{Z}[[z]]$ and $b$ be an integer with $b\ge2$. Bell, Bugeaud and Coons [BelBC] proved that $F(\frac{1}{b})$ is either rational or transcendental. In [Mi], we introduce a generalized $k$-regular sequence as a unification of several kinds of important sequences including $k$-regular, $k$-additive and $k$-multiplicative sequences. In this paper, we give a generalization of the result of Bell, Bugeaud and Coons for certain generalized $k$-regular series. Especially, we show that the values of irrational generating functions of certain sum of $k$-additive sequences and certain $k$-multiplicative sequences are either rational or transcendental. Moreover, we also give a partly generalization of a result obtained by Tachiya[Ta]. Especially, we show that the values of irrational generating functions of certain $k$-additive sequences and certain $k$-multiplicative sequences give transcendental numbers.

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arXiv2015

Study on the non-periodicity of the generalized Thue-Morse sequences generated by cyclic permutations

First we generalize the Thue-Morse sequence (the generalized Thue-Morse sequences) by a cyclic permutations and p-adic system, and consider the necessary-sufficient condition that it is non-periodic. Moreover if the generalized Thue-Morse sequence is not periodic, then all equally spaced subsequences of the generalized Thue-Morse sequences are not periodic. Finally we apply recently combinatorial transcendence results to the generalized non-periodic Thue-Morse sequence, we can found many transcendental numbers.

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arXiv2014

Transcendence of digital expansions and continued fractions generated by a cyclic permutation and $k$-adic expansion

In this article, first we generalize the Thue-Morse sequence $(a(n))_{n=0}^\infty$ (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 $(a(N+nl))_{n=0}^\infty$ (where $N \ge 0$ and $l >0$) of the generalized Thue-Morse sequences are not periodic. Finally we apply the criterion of [ABL], [Bu$1$] on transcendental numbers, to find that , for a non periodic generalized Thue-Morse sequences taking the values on $\{0,1,\cdots,β-1\}$(where $β$ is an integer greater than $1$), the series $\sum_{n=0}^\infty a(N+nl) β^{-n-1}$ gives a transcendental number, and further that for non periodic generalized Thue-Morse sequences taking the values on positive integers, the continued fraction $[0:a(N), a(N+l),\cdots,a(N+nl ), \cdots]$ gives a transcendental number, too.

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