Related papers: Hyperquadratic continued fractions and automatic s…
We demonstrate that discrete m-functions with eventually periodic continued fraction coefficients have an algebraic relationship to their second solution if and only if the periodic part of the sequence of continued fraction coefficients is…
This is a translation of Euler's Latin paper "De fractionibus continuis observationes" into English. In this paper Euler describes his theory of continued fractions. He teaches, how to transform series into continued fractions, solves the…
A contiguous relation for complementry pairs of very well poised balanced ${}_{10}\phi_9$ basic hypergeometric functions is used to derive an explict expression for the associated continued fraction. This generalizes the continued fraction…
In some particular cases we give criteria for morphic sequences to be almost periodic (=uniformly recurrent). Namely, we deal with fixed points of non-erasing morphisms and with automatic sequences. In both cases a polynomial-time algorithm…
The main result of this paper is that if E is a field extension of finite odd degree over a real field Q, and if E is a repeated radical extension of Q, then every intermediate field is also a repeated radical extension of Q. This paper…
Frames for Hilbert spaces are interesting for mathematicians but also important for applications e.g. in signal analysis and in physics. Both in mathematics and physics it is natural to consider a full scale of spaces, and not only a single…
There is a theory of continued fractions for Laurent series in x^{-1} with coefficients in a field F. This theory bears a close analogy with classical continued fractions for real numbers with Laurent series playing the role of real numbers…
This paper establishes new bridges between number theory and modern harmonic analysis, namely between the class of complex functions, which contains zeta functions of arithmetic schemes and closed with respect to product and quotient, and…
This note presents a discussion of the algebraic and combinatorial aspects of the theory of pure O-sequences. Various instances where pure O-sequences appear are described. Several open problems that deserve further investigation are also…
We consider two classes of $q$-continued fraction whose odd and even parts are limit 1-periodic for $|q|>1$, and give theorems which guarantee the convergence of the continued fraction, or of its odd- and even parts, at points outside the…
It is known that the continued fraction expansion of a real number is periodic if and only if the number is a quadratic irrational. In an attempt to generalize this phenomenon to other settings, Jun-Ichi Tamura and Shin-Ichi Yasutomi have…
In this paper we present some results related to the problem of finding periodic representations for algebraic numbers. In particular, we analyze the problem for cubic irrationalities. We show an interesting relationship between 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…
We prove that algebraic isomorphisms between limit algebras are automatically continuous, and consider consequences of this result. In particular, we give partial solutions to a conjecture of Power [Limit Algebras, Longman, 1992, Notes to…
It is desirable that a given continued fraction algorithm is simple in the sense that the possible representations can be characterized in an easy way. In this context the so-called finite range condition plays a prominent role. We show…
This work deals with scalar field theories and supersymmetric quantum mechanics. The investigation is inspired by a recent result, which shows how to use the reconstruction mechanism to describe two distinct field theories from the very…
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…
Let $p$ be a prime number and $K$ be a field with embeddings into $\mathbb{R}$ and $\mathbb{Q}_p$. We propose an algorithm that generates continued fraction expansions converging in $\mathbb{Q}_p$ and is expected to simultaneously converge…
We begin by defining general hypergeometric functions over finite fields and obtaining a finite field analogue of a classical symmetry in their complex counterparts. We give a geometric proof for the symmetry by constructing isomorphisms…
We study the structure of an algebraically closed field with extra function resembling the classical exponentiation on complex numbers.