Boris Adamczewski, Yann Bugeaud
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 continued fraction expansions, including expansions with unbounded partial quotients. Their proofs heavily depend on the Schmidt Subspace Theorem.
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In this paper we introduce a generalization of palindromic continued fractions as studied by Adamczewski and Bugeaud. We refer to these generalized palindromes as $m$-palindromes, where $m$ ranges over the positive integers. We provide a simple transcendence criterion for $m$-palindromes, extending and slightly refining an analogous result of Adamczewski and Bugeaud. We also provide methods for constructing examples of $m$-palindromes. Such examples allow us to illustrate our transcendence criterion and to explore the relationship between $m$-palindromes and stammering continued fractions, ano…
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 doubly palindromic. In this setting, doubly palindromic means that each sequence is a repeated concatenation of two palindromes and a compatibility condition between the lengths of these palindromes is satisfied.
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.
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 ones, that correspond to the multiplication by units of the corresponding number field, and so called palindromic ones. The main result of the paper is a criterion for a two-dimensional continued fraction to have a palindromic symmetry. This criterion is analogous to the criterion for the period of a quadratic irrationality to be symmetric.
We study the periods of Markov sequences, which are derived from the continued fraction expression of elements in the Markov spectrum. This spectrum is the set of minimal values of indefinite binary quadratic forms that are specially normalised. We show that the periods of these sequences are palindromic after a number of circular shifts, the number of shifts being given by Stern's diatomic sequence.
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.
The Thue-Morse sequence is generalized to the $TM_m$ sequences and two equivalent definitions are given. This generalization leads to transcendental numbers and has Queffélec's theorem on Thue-Morse continued fractions as a special case. It is also shown that the theorem of Adamczewski and Bugeaud for palindromic continued fractions cannot be applied in this case.
Tribonacci-Lucas numbers that are palindromic concatenations of two distinct repdigits
本文利用 Baker 理论和连分数约化方法，完整确定了所有可表示为两个不同重复数字串回文拼接的 Tribonacci-Lucas 数。
Transcendence of $p$-adic continued fractions and a quantitative $p$-adic Roth theorem
本文研究了p进数列的超越性，证明回文和拟周期性p进数列收敛于超越数或二次无理数，并提供了一个量化版的p进Roth定理。
This paper is a sequel to our previous work in which we found a combinatorial realization of continued fractions as quotients of the number of perfect matchings of snake graphs. We show how this realization reflects the convergents of the continued fractions as well as the Euclidean division algorithm. We apply our findings to establish results on sums of squares, palindromic continued fractions, Markov numbers and other statements in elementary number theory.
In this paper we recast the Serret theorem about a characterization of palindromic continued fractions in the context of polynomial continued fractions. Then, using the relation between symmetric tridiagonal matrices and polynomial continued fractions we give a quick exposition of the mathematical aspect of the perfect quantum state transfer problem.
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.
Transcendence of continued fractions over function fields and a quantitative version of Uchiyama's theorem
本文在特征零的域上，改进拟周期连分数的Maillet-Baker判据，证明函数域内山定理的定量版本，给出代数幂级数异常好有理逼近的明确界，还得到收敛分母增长的上界，且证明回文连分数的二分性。
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 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.
Several conjectural continued fractions found with the help of various algorithms are published in this paper.
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.
The classical continued fraction is generalized for studying the rational approximation problem on multi-formal Laurent series in this paper, the construction is called m-continued fraction. It is proved that the approximants of an m-continued fraction converge to a multi-formal Laurent series, and are best rational approximations to it; conversely for any multi-formal Laurent series an algorithm called m-CF transform is introduced to obtain its m-continued fraction expansions; moreover, strict m-continued fractions, which are m-continued fractions imposed with some additional conditions, and …
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…
We establish a new transcendence criterion of $p$-adic continued fractions which are called Ruban continued fractions. By this result, we give explicit transcendental Ruban continued fractions with bounded $p$-adic absolute value of partial quotients. This is $p$-adic analogy of Baker's result. We also prove that $p$-adic analogy of Lagrange Theorem for Ruban continued fractions is not true.
Corrections are brought to an article of Friesen on continued fractions of a given period.
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。
We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on a new generalization to an arbitrary integer base $b$. We show that all of our so-called type III continued logarithms converge and all rational numbers have finite type III continued logarithms. As with simple continued fractions, we show that the continued logarithm terms, for almost every real number, follow a specific distribution. We also generalize Khinc…
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.
We prove an analog of Lagrange's Theorem for continued fractions on the Heisenberg group: points with an eventually periodic continued fraction expansion are those that satisfy a particular type of quadratic form, and vice-versa.
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…
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 …
We give the first transcendence results for the Rosen continued fractions. Introduced over half a century ago, these fractions expand real numbers in terms of certain algebraic numbers.
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 equates infinite products and continued fractions, extensions and contractions of continued fractions and the Bauer-Muir transformation) to derive infinite families of in-equivalent polynomial continued fractions in which each continued fraction has the same limit. This allows us, for example, to construct infinite families of polynomial continued fractions for f…
Hurwitz (1887) defined a continued fraction algorithm for complex numbers which is better behaved in many respects than a more "natural" extension of the classical continued fraction algorithm to the complex plane would be. Although the Hurwitz complex continued fraction algorithm is not "reducible" to another complex continued fraction algorithm, over the reals the story is different. In this note we make clear the relation between the restriction of Hurwitz's algorithm to the real numbers and the classical continued fraction algorithm. As an application we reprove the main result of Choudhur…
This paper is devoted to a detailed exposition of geometry of continued fractions. We pay particular interest to the case of quadratic irrationalities and use the technique described to prove a criterion for the continued fraction of a quadratic surd to have a symmetric period.
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…
We will provide a family of continued fractions for which there is no correspondence between dynamic and approximation pairs, leading to an anomaly in their corresponding Spaces of Jager Pairs.
We study Ramanujan's cubic continued fraction and explicit evaluations of theta-functions
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。
We study some properties of the function $μ_α(t)$ associated with the Minkowski diagonal continued fraction for real $α$.
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.
We introduce here a general framework for studying continued fraction expansions for complex numbers and establish some results on the convergence of the corresponding sequence of convergents. For continued fraction expansions with partial quotients in a discrete subring of $\mathbb C$ an analogue of the classical Lagrange theorem, characterising quadratic surds as numbers with eventually periodic continued fraction expansions, is proved. Monotonicity and exponential growth are established for the absolute values of the denominators of the convergents for a class of continued fraction algorith…
Bounded continued fractions and quadratic orders in $\Q(\sqrt{127})$
本文研究Q(√127)中部分商≤4的有界连分数的周期与乘子序，证明相关精确公式，结合Hensley定理得到二次域允许无穷多此类连分数的数量下界。
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.
We introduced a new continued fraction expansions in our previous paper. For these expansions, we show formulae of probability about incomplete quotients. Furthermore, we prove the existence of invariant measures with respect to the continued fraction transformations associated with expansions.
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 construction works by defining successive refinements of appropriate subintervals to achieve, in the limit, simple normality to all integer bases and continued fraction normality. The main diffculty is to control the length of these subintervals. To achieve this we adapt and combine known metric theorems on continued fractions and on expansions in integers ba…
We provide several results on the diophantine properties of continued fractions on the Heisenberg group, many of which are analogous to classical results for real continued fractions. In particular, we show an analog of Khinchin's theorem and that convergents are also best approximants up to a constant factor of the denominator.
Given an odd prime number p, we describe a continued fraction in the field F(p) of power series in 1/T with coefficients in the finite field F_p, where T is a formal indeterminate. This continued fraction satisfies an algebraic equation of a particular type, with coefficients in F_p[T] which are explicitely given. We observe the close connection with other algebraic continued fractions studied thirty years ago by Mills and Robbins.
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 example considered about thirty years ago in a pioneer work in this area.
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.
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