Continued fractions with minimal remainders
Elena Zhabitskaya
考虑一个有理数的表示形式，与“中心”欧几里德算法相关联。证明了部分有理元和有界的有理数序列的极限分布函数的一个新公式。
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.
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In this paper we define "a continued fraction expansion of the exponential integral $E_{1}(x)$ at infinity", which is analogous to the regular continued fraction expansion of real numbers, and prove that this expansion gives the same continued fraction. Moreover, we give concrete representations of rational functions which are obtained by truncating the continued fraction.
We construct a class of quadratic irrationals having continued fractions of period $n\geq2$ with "small" partial quotients for which certain integer multiples have continued fractions of period $1$, $2$ or $4$ with "large" partial quotients. We then show that numbers in the period of the new continued fraction are simple functions of the numbers in the periods of the original continued fraction. We give generalizations of some of the continued fractions and show that polynomials arising from the generalizations are related to Chebyshev and Fibonacci polynomials. We then show that some of these…
On a quartic equation and two families of hyperquadratic continued fractions in power series fields
30年前，我们曾偶然地介绍过一个系数为Fp[T]的简单的四次代数方程，它在有限域Fp上的1/T的幂级数域中有解。对于每一个素数p > 3，这个解的连分数展开是显著的，并且根据p除以3的余数1或2，它具有不同的一般模式。我们描述了两个非常大的家庭的代数连分数，每个包含这些解决方案，根据类的p模3。我们可以计算这些代数连分数的无理性测度，因此，我们得到四次方程解的两个不同值，仅取决于p模3的类。
On new identities connecting Ramanujan-Göllnitz-Gordon continued fraction and Ramanujan's continued fraction of order four
本文利用q级数和theta函数理论，建立连接拉马努金-戈尔尼茨-戈登连续分数与四阶拉马努金连续分数的新恒等式，并利用拉马努金的_1ψ_1求和公式获得Lambert级数恒等式。
We give continued fraction expansions of the generating functions of Bernoulli numbers, Cauchy numbers, Euler numbers, harmonic numbers, and their generalized or related numbers. In particular, we focus on explicit forms of the convergents of these continued fraction expansions. Linear fractional transformations of such continued fractions are also discussed. We show more continued fraction expansion for different numbers and types, in particular, on Cauchy numbers.
Some conjectural continued fractions
本文给出了几个用各种算法求出的代数连分数。
We prove that the formula giving the exact value of the irrationality exponent of regular continued fractions remains valid for semi-regular continued fractions satisfyiong certain conditions.
Multidimensional continued fraction and rational approximation
将经典连分式推广到研究多重形式洛朗级数的有理逼近问题，称之为m-连分数.证明了m-连分数的逼近式收敛于多形式洛朗级数，并且是它的最佳有理逼近，反之，对于任何多形式洛朗级数引入了一种称为m-CF变换的算法来得到它的m-连分数展开式;此外，严格m-连分数是附加了某些条件的m-连分数，和多重形式洛朗级数是1-1对应的。结果表明，m-连分数可以用来研究多序列综合问题。
We show, for each $q$-continued fraction $G(q)$ in a certain class of continued fractions, that there is an uncountable set of points on the unit circle at which $G(q)$ diverges in the general sense. This class includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fraction. We discuss the implications of our theorems for the general convergence of other $q$-continued fractions, for example the Göllnitz-Gordon continued fraction, on the unit circle.
In this paper, we investigate the monotone property of the continued fractions $G(m,λ)$ as a function of $m$ and $λ$. In particular, we obtain new inequality for the relative continued fractions.
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 necessarily positive) converge. Here we present a simple test that determines whether an integer continued fraction converges or diverges. In addition, for convergent continued fractions the test specifies whether the limit is rational or irrational. An attractive way to visualise integer continued fractions is to model them as paths on the Farey graph, which is…
In this work we prove a criterion for an algebraic continued fraction to have a proper palindromic symmetry in dimension $4$. As a multidimensional generalization of continued fractions, we consider Klein polyhedra.
Transcendental $p$-adic continued fractions
建立了$p$ -进连分数的一个新的超越准则，称之为Ruban连分数.利用这个结果，我们给出了部分分式的绝对值有界的$p$ -adic超鲁班连分数.这是Baker结果的$p$ -adic类比。本文还证明了鲁班连分数的Lagrange定理的$p$ -adic类比是不成立的.
Corrections to "On continued fractions of given period"
更正带来的一篇文章弗里森对连分数的一个给定的时期。
Generalization of Ramanujan's Continued Fractions for Even Order
该研究借助拉马努金连分数恒等式推导偶数阶广义连分数，建立theta函数恒等式，以76阶连分数为例得到分拆理论恒等式等，推广了拉马努金连分数。
Generalized Continued Logarithms and Related Continued Fractions
我们继续研究由比尔戈恩和J. Borwein等研究介绍的算法。al..在提供了一个概述的类型I和类型II推广的二进制连续矩阵介绍Borwein等。例如，我们专注于一个新的推广到任意整数基$b$。我们证明了所有的III型连续图都是收敛的，并且所有的有理数都有有限型III型连续图。与简单连分数一样，我们证明了对于几乎每个实数，连续对数项都遵循一个特定的分布。我们还将Khinchine常数从简单连分数推广到连分数，并证明了这些对数Khinchine常数具有初等封闭形式。最后，我们证明了简单连分数是我们的连续图的极限情况，并简要地考虑了我们如何推广过去的连续图。
We state and prove three general formulas allowing to transform formal finite sums into formal continued fractions and apply them to generalize certain expansions in continued fractions given by Hone and Varona.
Lagrange's Theorem for continued fractions on the Heisenberg group
我们证明了一个类似的拉格朗日定理的连分数的海森堡群：最终周期性的连分数扩展的点是那些满足特定类型的二次型，反之亦然。
For $α_0 = \left[a_0, a_1, \ldots\right]$ an infinite continued fraction and $σ$ a linear fractional transformation, we study the continued fraction expansion of $σ(α_0)$ and its convergents. We provide the continued fraction expansion of $σ(α_0)$ for four general families of continued fractions and when $\left|\det σ\right| = 2$. We also find nonlinear recurrence relations among the convergents of $σ(α_0)$ which allow us to highlight relations between convergents of $α_0$ and $σ(α_0)$. Finally, we apply our results to some special and well-studied continued fractions, like Hurwitzian and Taso…
This work is devoted to the proof of the statement about the existence of palindromic continued fractions in an arbitrary dimension. In addition, it is proved the criterion that an algebraic continued fraction has proper cyclic palindromic symmetry in the case $n = 4$. Klein polyhedra are considered as a multidimensional generalization of continued fractions.
In this paper we prove in detail a criterion for an algebraic continued fraction to have a proper palindromic symmetry in dimension $4$. We also present a new proof of the criterion for an algebraic continued fraction to have a proper cyclic palindromic symmetry in dimension $4$. As a multidimensional generalization of continued fractions, we consider Klein polyhedra.
The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization arising from an algorithm due to Jacobi, have been poorly investigated in this sense, up to now. In this paper, we propose a combinatorial interpretation of the convergents of multidimensional continued fractions in terms of counting some particular tilings, generalizing some results that hold for classical continued fractions.
Ramanujan recorded four reciprocity formulas for the Roger-Ramanujan continued fraction. Two reciprocity formulas each are also associated with the Ramanujan--Göllnitz--Gordon continued fraction and a level-13 analog of the Roger-Ramanujan continued fraction. We show that all eight reciprocity formulas are related to a pair of quadratic equations. The solution to the first equation generalizes the golden ratio and is used to set the value of a coefficient in the second equation; and the solution to the second equation gives a pair of values for a continued fraction. We relate the coefficients …
Transcendence with Rosen continued fractions
给出了Rosen连分数的第一个超越性结果。半个多世纪前引入的这些分数用特定的代数数来扩展实数。
Real Numbers With Polynomial Continued Fraction Expansions
在本文中，我们将展示如何应用各种技术和定理（包括平歇尔定理，欧拉公式的推广，使无穷级数和连分数相等，相应变换的推广，使无穷乘积和连分数相等，扩展和压缩的连分数和鲍尔-缪尔变换），以获得无限的家庭，等价的多项式连分数，其中每个连分数具有相同的极限。例如，这允许我们为著名的常数如$π$和$e$、$ζ{(k)}$（对于每个正整数$k\geq 2$）、各种以整数参数求值的特殊函数和各种代数数构造多项式连分数的无限族。我们还提出了几个问题的性质的一组实数有一个多项式连分数的扩展。
The Hurwitz continued fraction expansion as applied to real numbers
Hurwitz（1887）为复数定义了一种连分数算法，这种算法在许多方面比经典的连分数算法在复平面上的更“自然”的扩展更好。虽然Hurwitz复连分数算法不能“简化”为另一个复连分数算法，但实际上情况是不同的。本文阐明了Hurwitz算法对实数的限制与经典的连分数算法之间的关系。作为应用，我们重新证明了Choudhuri和Dani（2015）的主要结果。
Palindromes and periodic continued fractions
本文对连分数几何作了较详细的论述。我们特别感兴趣的情况下，二次无理数，并使用所描述的技术来证明一个标准的连分数的二次surd有一个对称的周期。
Multidimensional continued fractions generalize classical continued fractions with the aim of providing periodic representations of algebraic irrationalities by means of integer sequences. However, there does not exist any algorithm that provides a periodic multidimensional continued fraction when algebraic irrationalities are given as inputs. In this paper, we provide a characterization for periodicity of Jacobi--Perron algorithm by means of linear recurrence sequences. In particular, we prove that partial quotients of a multidimensional continued fraction are periodic if and only if numerato…
Polynomial Continued Fractions for Algebraic Numbers
本文证明任何实三次代数数可展开为多项式系数的连分数，并提及若干推广。
We present a general introduction to continued fractions, with special consideration to the function fields case. These notes were prepared for a summer class given this year in Beijing at Beihang university.
We show how to obtain infinitely many continued fractions for certain Z-linear combinations of zeta and L values. The methods are completely elementary.
We give a concise introduction to the theory of continuants and show how Perron used them in his proof of Tietze theorem on the convergence of infinite semi-regular continued fractions, as well as for the study of the convergence of purely periodic continued fractions.
Continued fractions have been introduced in the field of $p$--adic numbers $\mathbb{Q}_p$ by several authors. However, a standard definition is still missing since all the proposed algorithms are not able to replicate all the properties of continued fractions in $\mathbb{R}$. In particular, an analogue of the Lagrange's Theorem is not yet proved for any attempt of generalizing continued fractions in $\mathbb{Q}_p$. Thus, it is worth to study the definition of new algorithms for $p$--adic continued fractions. The main condition that a new method needs to fulfill is the convergence in $\mathbb Q…
Approximation pathologies for certain continued fractions
我们将提供一个家庭的连分数，有没有对应的动态和近似对，导致一个异常，在其相应的空间的雅格对。
On Ramanujan's cubic continued fraction and explicit evaluations of theta-functions
研究了Ramanujan三次连分数和θ函数的显式求值
In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of $q$-continued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each $q$-continued fraction, $G(q)$, in this class, that there is an uncountable set of points, $Y_{G}$, on the unit circle such that if $y \in Y_{G}$ then $G(y)$ does not converge to a finite va…
Transcendence criteria for multidimensional continued fractions
本文针对 Jacobi 引入的多维连分数建立超越性判据，证明某些 Liouville 型和拟周期多维连分数是超越的，并通过周期多维连分数产生的三次无理数高度上界完成拟周期情形的证明。
Some Identities of Ramanujan's q-Continued Fractions of Order Fourteen and Twenty-Eight, and Vanishing Coefficients
本文从Ramanujan的一般连分数恒等式推导出十四阶和二十八阶连分数，建立theta函数恒等式，并推导色分区恒等式及消失系数结果。
On Minkowski diagonal continued fraction
研究了实$α$的Minkowski对角连分数函数$μ_α(t)$的一些性质.
If the odd and even parts of a continued fraction converge to different values, the continued fraction may or may not converge in the general sense. We prove a theorem which settles the question of general convergence for a wide class of such continued fractions. We apply this theorem to two general classes of $q$ continued fraction to show, that if $G(q)$ is one of these continued fractions and $|q|>1$, then either $G(q)$ converges or does not converge in the general sense. We also show that if the odd and even parts of the continued fraction $K_{n=1}^{\infty}a_{n}/1$ converge to different va…
Legendre found that the continued fraction expansion of $\sqrt N$ having odd period leads directly to an explicit representation of $N$ as the sum of two squares. Similarly, it is shown here that the continued fraction expansion of $\sqrt N$ having even period directly produces a factor of a composite $N$. Shanks' infrastructural method is then revisited, and some consequences of its application to factorization by means of the continued fraction expansion of $\sqrt N$ are derived.
Continued fraction expansions for complex numbers - a general approach
本文介绍了研究复数的连分数展开式的一般框架，并建立了相应的收敛序列的收敛性的一些结果。对于$\mathbb C$的离散子环中具有部分同元的连分数展开式，证明了经典Lagrange定理的一个类似定理，即二次根号是具有最终周期连分数展开式的数.对Eisenstein整数环上一类带部分幂元的连分数算法的收敛性证明了其收敛因子的绝对值的单调性和指数增长性。
In this article, we will discover some new generalized identity regarding continued fractions. We will connect the results to Fibonacci numbers and Lucas numbers. For all the proof, we will use induction.
On the Metrical Theory of a Generalized Continued Fraction Expansion
我们在前面的文章中引入了一种新的连分数展开式。对于这些展开式，我们给出了关于不完全连续项的概率公式。此外，我们还证明了与展开式相关的连分数变换的不变测度的存在性。
On absolutely normal and continued fraction normal numbers
给出了一个正规于所有整数基且连分数正规的实数的构造。其连分数展开式的前n位数的计算按照n^4次数学运算的顺序执行。该构造通过定义适当子区间的连续细化来实现，在极限下，对所有整数基的简单正态性和连分数正态性。主要困难是控制这些子区间的长度。为了实现这一目标，我们适应和结合已知的度量定理连分数和整数基础上的扩展。
Diophantine properties of continued fractions on the Heisenberg group
给出了海森堡群上连分数的丢番图性质的几个结果，其中许多结果与实连分数的经典结果类似.特别是，我们展示了一个类似的欣钦定理，并convertizes也是最佳逼近的分母的一个常数因子。
On a particular hyperquadratic continued fraction in F(p) with p>2
给定一个奇素数p，刻画了系数在有限域F_p中的1/T中的幂级数在域F（p）中的一个连分数，其中T是形式不定式.这个连分数满足一个特殊类型的代数方程，其系数在F_p[T]中是明确给定的。我们观察到密切的联系与其他代数连分数研究三十年前米尔斯和罗宾斯。
On hyperquadratic continued fractions in power series fields over a finite field
本文的第一部分简要介绍了某些代数幂级数的连分数展开式。在最后一部分，作为一个例子，我们提出了一个家庭的代数连分数的程度4，包括一个玩具的例子，认为大约30年前在这方面的先驱工作。
We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aimed to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan's continued fractions, $θ$-expansions, $N$-continued fractions and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.
We study Möbius transformations (also known as linear fractional transformations) of quadratic numbers. We construct explicit upper and lower bounds on the period of the continued fraction expansion of a transformed number as a function of the period of the continued fraction expansion of the original number. We provide examples that show that the bound is sharp.
Generalized Thue-Morse Continued Fractions
将Thue-Morse序列推广到$TM_m$序列，并给出了两个等价定义.这种推广导致了超越数，并有Queffélec定理的Thue-Morse连分数作为一个特殊情况。本文还证明了Adamczewski和Bugeaud关于回文连分数的定理在这种情况下不能应用。
We consider continued fractions with partial quotients in the ring of integers of a quadratic number field $K$ and we prove a generalization to such continued fractions of the classical theorem of Lagrange. A particular example of these continued fractions is the $β$-continued fraction introduced by Bernat. As a corollary of our theorem we show that for any quadratic Perron number $β$, the $β$-continued fraction expansion of elements in $\mathbb Q(β)$ is either finite of eventually periodic. The same holds for $β$ being a square root of an integer. We also show that for certain 4 quadratic Per…
Continued fractions with odd partial quotients
考虑将有理数表示为连分数，与“奇”欧几里德算法相关联。本文证明了部分有理元和有界的有理数序列的极限分布函数的若干性质
A Generalized Family of Multidimensional Continued Fractions: TRIP Maps
大多数著名的多维连分数，包括Mönkemeyer映射和三角形映射，都是通过重复细分三角形来生成的。本文通过对这些三角形在每次剖分前后的顶点进行置换，构造了一族多维连分数。通过合成这类映射，我们得到了更大的一类多维连分数。这些算法包括Brun、Parry-Daniels和Güting。我们给标准时，多维连分数关联序列的唯一点，这使我们能够确定相应的多维连分数的周期性对应于对实数是三次无理数在同一个号码字段。
Continued fraction expansions of algebraic numbers
本文建立了两个代数数的连分数展开式的独立性。粗略地说，如果两个无理代数数的连分数展开式有相同的长子字，则这两个连分数展开式有相同的尾。如果两个展开式具有镜像对称的长子字，则这两个代数数都是二次的。应用上述结果，我们证明了一个类似于Roth定理的关于代数数逼近的定理.
The Möbius transformation of continued fractions with bounded upper and lower partial quotients
令$h$：$x\mapsto \frac{ax+b}{cx+d} $为具有整数项的非退化莫比乌斯变换。利用$x$的连分式的上、下界得到了$h(x)$的连分式的一个上、下界，推广了Stambul \cite{S2}的一个结果.
Pell equation: A generalization of continued fraction and Chakravala algorithms using the LLL-algorithm
本文推广了求解佩尔方程的连分数和Chakravala算法，采用LLL算法处理秩2格，提出新方法。
On the period of the continued fraction expansion of ${\sqrt {2^{2n+1}+1}}$
本文证明了当$n$通过奇正整数趋于无穷大时，${\sqrt {2^{n}+1}}$的连分数展开式的周期趋于无穷大.
Adolf Hurwitz proposed in 1887 a continued fraction algorithm for complex numbers: Hurwitz continued fractions (HCF). Among other similarities between HCF and regular continued fractions, quadratic irrational numbers over $\mathbb{Q}(i)$ are precisely those with periodic HCF expansions. In this paper, we give some necessary as well as some sufficient conditions for pure periodicity of HCF. Then, we characterize badly approximable complex numbers in terms of HCF. Finally, we prove a slightly weaker complex analogue of a theorem by Y. Bugeaud on the transcendence of certain continued fractions.
We prove the convergence of a wide class of continued fractions, including generalized continued fractions over quaternions and octonions. Fractional points in these systems are not bounded away from the unit sphere, so that the iteration map is not uniformly expanding. We bypass this problem by analyzing digit sequences for points that converge to the unit sphere under iteration, expanding on previous methods of Dani-Nogueira.
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