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

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].

由轮换和$k$ -adic展开生成的数字展开式和连分数的超越性

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

本文首先利用自然数的轮换和$k$-adic展开对Thue-Morse序列$(a(n))_{n=0}^\infty$（广义Thue-Morse序列）进行了推广，并给出了它是非周期序列的充要条件。此外，我们还证明了，如果广义Thue-Morse序列不是周期的，那么广义Thue-Morse序列的所有等间距序列$(a(N+nl))_{n=0}^\infty$（其中$N \ge 0$和$l >0$）都不是周期的.最后，我们应用[ABL]，[Bu $1$ ]关于超越数的判据，发现对于取值于$\{0,1,\cdots,β-1\}$上的非周期广义Thue-Morse序列，（其中$β$是大于$1$的整数），级数$\sum_{n=0}^\infty a(N+nl) β^{-n-1}$给出超越数，并且进一步地，对于取值于正整数的非周期广义Thue-Morse序列，连分数$[0:a(N), a(N+l),\cdots,a(N+nl ), \cdots]$也给出一个超越数。

Study of the transcendence of a family of generalized continued fractions

We study a family of generalized continued fractions, which are defined by a pair of substitution sequences in a finite alphabet. We prove that they are stammering sequences, in the sense of Adamczewski and Bugeaud. We also prove that this family consists of transcendental numbers which are not Liouvillian. We explore the partial quotients of their regular continued fraction expansions, arriving at no conclusion concerning their boundedness.

多元马勒函数的超越性及其值之间的代数关系

Transcendence of multivariate Mahler functions and algebraic relations between their values

研究多元马勒函数的函数性质，证明其在特定区域的解析性与亚纯性及有理 - 超越二分法，加强相关提升定理，能计算函数值间代数关系，还证明了正则点情况下的多元下降定理。

Mahler's method in several variables I: The theory of regular singular systems

This is the first part of a work devoted to the study of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence. We prove two main results concerning systems that are regular singular at the origin. Our interest in Mahler's method comes from the possible applications of these results to old problems involving automata theory and which concern the expansion of both natural numbers and real numbers in integer bases. In particular, problems which involve finite automata and base change. Such applications are studied in Part II of this work.

On digital sequences associated with Pascal's triangle

We consider the sequence of integers whose $n$th term has base-$p$ expansion given by the $n$th row of Pascal's triangle modulo $p$ (where $p$ is a prime number). We first present and generalize well-known relations concerning this sequence. Then, with the great help of Sloane's On-Line Encyclopedia of Integer Sequences, we show that it appears naturally as a subsequence of a $2$-regular sequence. Its study provides interesting relations and surprisingly involves odious and evil numbers, Nim-sum and even Gray codes. Furthermore, we examine similar sequences emerging from prime numbers involvin…