Related papers: Integer continued fractions for complex numbers
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…
The continued fraction mapping maps a number in the interval $[0,1)$ to the sequence of its partial quotients. When restricted to the set of irrationals, which is a subspace of the Euclidean space $\mathbb{R}$, the continued fraction…
Consider the representation of a rational number in the form, associated with "centered" Euclidean algorithm. We prove a new formula for the limit distribution function for sequences of rationals with bounded sum of partial quotients.
This paper focuses on the equivalent expression of fractional integrals/derivatives with an infinite series. A universal framework for fractional Taylor series is developed by expanding an analytic function at the initial instant or the…
I present and discuss an extremely simple algorithm for expanding a formal power series as a continued fraction. This algorithm, which goes back to Euler (1746) and Viscovatov (1805), deserves to be better known. I also discuss the…
Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to…
We examine a pair of dynamical systems on the plane induced by a pair of spanning trees in the Cayley graph of the Super-Apollonian group of Graham, Lagarias, Mallows, Wilks and Yan. The dynamical systems compute Gaussian rational…
Continued fractions whose elements are polynomial sequences have been carefully studied mostly in the cases where the degree of the numerator polynomial is less than or equal to two and the degree of the denominator polynomial is less than…
In the present work, we investigate real numbers whose sequence of partial quotients enjoys some combinatorial properties involving the notion of palindrome. We provide three new transendence criteria, that apply to a broad class of…
In this paper we extend some set theoretic concepts of numerical semigroups for arbitrary sub-semigroups of natural numbers. Then we characterized gapsets which leads to a more efficient computational approach towards numerical semigroups…
We find a generating function expressed as a continued fraction that enumerates ordered trees by the number of vertices at different levels. Several Catalan problems are mapped to an ordered-tree problem and their generating functions also…
Large and moderate deviation principles are proved for Engel continued fractions, a new type of continued fraction expansion with non-decreasing partial quotients in number theory.
In a recent paper (Asci \textit{et al.}, 2008) it has been shown that certain random continued fractions have a density which is a product of a beta density and a hypergeometric function $_{2}F_{1}$. In the present paper we fully exploit a…
We show how to obtain infinitely many continued fractions for certain Z-linear combinations of zeta and L values. The methods are completely elementary.
In this paper we study in detail a family of continued fraction expansions of any number in the unit closed interval $[0,1]$ whose digits are differences of consecutive non-positive integer powers of an integer $m \geq 2$. For the…
We discuss continued fractions on real quadratic number fields of class number 1. If the field has the property of being 2-stage euclidean, a generalization of the euclidean algorithm can be used to compute these continued fractions.…
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…
This paper is an attempt to apply the tools of supergeometry to arithmetic. Supergeometric objects are defined over supercommutative rings of coefficients, and we consider an integral ring with exactly two odd variables. In this case the…
For uniformly chosen random $\alpha \in [0,1]$, it is known the probability the $n^{\rm th}$ digit of the continued-fraction expansion, $[\alpha]_n$ converges to the Gauss-Kuzmin distribution $\mathbb{P}([\alpha]_n = k) \approx \log_2 (1 +…
It is well known that central simple algebras are split by suitable finite Galois extensions of their centers. A counterpart of this result was studied by Juan and Magid in the set up of differential matrix algebras, wherein Picard-Vessiot…