Related papers: The Balkans Continued Fraction
In some recent papers, the authors considered regular continued fractions of the form \[ [a_{0};\underbrace{a,...,a}_{m}, \underbrace{a^{2},...,a^{2}}_{m}, \underbrace{a^{3},...,a^{3}}_{m}, ... ], \] where $a_{0} \geq 0$, $a \geq 2$ and $m…
A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector…
We introduce a $q$-analog of the higher continued fractions introduced by the last three authors in a previous work (together with Gregg Musiker), which are simultaneously a generalization of the $q$-rational numbers of Morier-Genoud and…
We study the dynamics of a family of continued fraction maps parametrized by the unit interval. This family contains as special instances the Gauss continued fraction map and the Fibonacci map. We determine the transfer operators of these…
Special kinds of continued fractions have been proved to converge to transcendental real numbers by means of the celebrated Subspace Theorem. In this paper we study the analogous $p$--adic problem. More specifically, we deal with Browkin…
In this Letter, we introduce a notion of local fraction for experiments taking place against arbitrary static causal backgrounds -- greatly generalising previous results on no-signalling scenarios -- and we explicitly formulate a linear…
In this short note, we give a proof, conditional on the Generalized Riemann Hypothesis, that there exist numbers x which are normal with respect to the continued fraction expansion but not to any base b expansion. This partially answers a…
Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of $p$-adic numbers whose continued fractions arise from…
We introduce a combinatorial enumeration problem that is solved using generalized Catalan numbers. We also study generalizations of the Cycle Lemma beyond the computation of the generalized Catalan numbers.
We build, for real quadratic fields, infinitely many periodic continuous fractions uniformly bounded, with a seemingly better bound than the known ones. We do that using continuous fraction expansions with the same shape as those of real…
In this paper we describe the group of symmetries of a two-dimensional continued fraction. As a multidimensional generalization of continued fractions we consider Klein polyhedra. We distinguish two types of symmetries: the Dirichlet-type…
In this paper we show how to apply various techniques and theorems (including Pincherle's theorem, an extension of Euler's formula equating infinite series and continued fractions, an extension of the corresponding transformation that…
The aim of this paper is to establish new inequalities for the Euler-Mascheroni by the continued fraction method.
This paper investigates integer multiplication of continued fractions using geometric structures. In particular, this paper shows that integer multiplication of a continued fraction can be represented by replacing one triangulation of an…
We give a construction of a real number that is normal to all integer bases and continued fraction normal. The computation of the first n digits of its continued fraction expansion performs in the order of n^4 mathematical operations. The…
We describe a database of 1779 continued fractions with polynomial coefficients, of which more than 1500 are new, both for interesting constants and for transcendental functions, and provide the database inside the \TeX\ source of the…
A relationship between continued fractions and Weyl groupoids of Cartan schemes of rank two is found. This allows to decide easily if a given Cartan scheme of rank two admits a finite root system. We obtain obstructions and sharp bounds for…
Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive integer coefficients converges; however, it is unknown precisely which continued fractions with integer coefficients (not…
In this work, we study a continued fractions theory for the topological completion of the field of Puiseux series. As usual, we prove that any element in the completion can be developed as a unique continued fractions, whose coefficients…
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…