Related papers: On certain recurrent and automatic sequences in fi…
The aim of this note is to show the existence of a correspondance between certain algebraic continued fractions in fields of power series over a finite field and automatic sequences in the same finite field. this connection is illustrated…
Morphic sequences form a natural class of infinite sequences, extending the well-studied class of automatic sequences. Where automatic sequences are known to have several equivalent characterizations and the class of automatic sequences is…
We present family of automatic sequences that define algebraic continued fractions in characteristic 2. This family is constructed from ultimately period words and contains the period-doubling sequence.
Automatic sequences have many properties that other sequences (in particular, non-uniformly morphic sequences) do not necessarily share. In this paper we survey a number of different methods that can be used to prove that a given sequence…
There exists a particular subset of algebraic power series over a finite field which, for different reasons, can be compared to the subset of quadratic real numbers. The continued fraction expansion for these elements, called…
In the following pages we discuss infinite sequences defined on a finite alphabet, and more specially those which are generated by finite automata. We have divided our paper into seven parts which are more or less self-contained. Needless…
Motivated by a question of van der Poorten about the existence of infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen…
The first part of this note is a short introduction on continued fraction expansions for certain algebraic power series. In the last part, as an illustration, we present a family of algebraic continued fractions of degree 4, including a toy…
We discuss the form of certain algebraic continued fractions in the field of power series over $F_p$, where p is an odd prime number. This leads to give explicit continued fractions in these fields, satisfying an explicit algebraic equation…
In this note, we describe a family of particular algebraic, and nonquadratic, power series over an arbitrary finite field of characteristic 2, having a continued fraction expansion with all partial quotients of degree one. The main purpose…
The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ in a suitable numeration system. In this paper, instead of considering automatic sequences built on a numeration…
Let L be an infinite regular language on a totally ordered alphabet (A,<). Feeding a finite deterministic automaton (with output) with the words of L enumerated lexicographically with respect to < leads to an infinite sequence over the…
We prove that if $y=\sum_{n=0}^\infty{\bf a}(n)x^n\in\mathbb{F}_q[[x]]$ is an algebraic power series of degree $d$, height $h$, and genus $g$, then the sequence ${\bf a}$ is generated by an automaton with at most $q^{h+d+g-1}$ states, up to…
The notion of almost periodicity nontrivially generalizes the notion of periodicity. Strongly almost periodic sequences (=uniformly recurrent infinite words) first appeared in the field of symbolic dynamics, but then turned out to be…
Abstract numeration systems encode natural numbers using radix ordered words of an infinite regular language and linear recurrence sequences play a key role in their valuation. Sequence automata, which are deterministic finite automata with…
By replacing the letters to polynomials in F_2[t], an infinite word, over a finite alphabet, can be seen as the sequence of partial quotients of a continued fraction in F_2((1/t)). Here is described a family of such infinite words,…
The paper studies different variants of almost periodicity notion. We introduce the class of eventually strongly almost periodic sequences where some suffix is strongly almost periodic (=uniformly recurrent). The class of almost periodic…
We prove decidability results on the existence of constant subsequences of uniformly recurrent morphic sequences along arithmetic progressions. We use spectral properties of the subshifts they generate to give a first algorithm deciding…
In this paper we use the framework of automatic sequences to study combinatorial sequences modulo prime powers. Given a sequence whose generating function is the diagonal of a rational power series, we provide a method, based on work of…
We study automatic sequences and automatic systems generated by general constant length (nonprimitive) substitutions. While an automatic system is typically uncountable, the set of automatic sequences is countable, implying that most…