We study a family of generalized continued fractions, which are defined by a pair of substitution sequences in a finite alphabet. We prove that they are stammering sequences, in the sense of Adamczewski and Bugeaud. We also prove that this family consists of transcendental numbers which are not Liouvillian. We explore the partial quotients of their regular continued fraction expansions, arriving at no conclusion concerning their boundedness.
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.
We extend a result of Hančl, Kolouch and Nair on the irrationality and transcendence of continued fractions. We show that for a sequence $\{α_n\}$ of algebraic integers of bounded degree, each attaining the maximum absolute value among their conjugates and satisfying certain growth conditions, the condition $$ \limsup_{n \rightarrow \infty} \vertα_n\vert^{\frac{1}{Dd^{n-1} \prod_{i=1}^{n-2}(Dd^i + 1)}} = \infty $$ implies that the continued fraction $α= [0;α_1, α_2, \dots]$ is not an algebraic number of degree less than or equal to $D$.
We study the values of the recently introduced involution J (jimm) of the real line, which is equivariant with the action of the group PGL(2,Z). We test our conjecture that this involution sends algebraic numbers of degree at least three to transcendental values. We also deduce some theoretical results concerning the continued fraction statistics of the generic values of this involution and compare them with the experimental results.
Euler gave recipes for converting alternating series of two types, I and II, into equivalent continued fractions, i.e., ones whose convergents equal the partial sums. A condition we prove for irrationality of a continued fraction then allows easy proofs that $e,\sin1$, and the primorial constant are irrational. Our main result is that, if a series of type II is equivalent to a simple continued fraction, then the sum is transcendental and its irrationality measure exceeds $2$. We construct all $\aleph_0^{\aleph_0}=\mathfrak{c}$ such series and recover the transcendence of the Davison--Shallit a…
Let $θ$ and $ρ$ be real numbers with $0 \le θ, ρ< 1$ and $θ$ irrational. We show that the Hecke-Mahler series $$ F_{θ, ρ} (z_1, z_2) = \sum_{k_1 \ge 1} \, \sum_{k_2 = 1}^{\lfloor k_1 θ+ ρ\rfloor} \, z_1^{k_1} z_2^{k_2}, $$ where $\lfloor \cdot \rfloor$ denotes the integer part function, takes transcendental values at any algebraic point $(β, α)$ with $0 < |β|, |βα^θ| < 1$. This extends earlier results of Mahler (1929) and Loxton and van der Poorten (1977), who settled the case $ρ=0$. Furthermore, for positive integers $b$ and $a$, with $b \ge 2$ and $a$ congruent to $1$ modulo $b-1$, we give t…
In this article, first we generalize the Thue-Morse sequence $(a(n))_{n=0}^\infty$ (the generalized Thue-Morse sequences) by a cyclic permutation and $k$ -adic expansion of natural numbers, and consider the necessary-sufficient condition that it is non-periodic. Moreover we will show that, if the generalized Thue-Morse sequence is not periodic, then all equally spaced subsequences $(a(N+nl))_{n=0}^\infty$ (where $N \ge 0$ and $l >0$) of the generalized Thue-Morse sequences are not periodic. Finally we apply the criterion of [ABL], [Bu$1$] on transcendental numbers, to find that , for a non per…
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…
A singular point on a plane conic defined over $\mathbb{Q}$ is a transcendental point of the curve which admits very good rational approximations, uniformly in terms of the height. Extremal numbers and Sturmian continued fractions are abscissa of such points on the parabola $y=x^2$. In this paper we provide a measure of transcendence for singular points on conics defined over $\mathbb{Q}$ which, in these two cases, improves on the measure obtained by Adamczewski et Bugeaud. The main tool is a quantitative version of Schmidt subspace theorem due to Evertse.
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.
We use the Schmidt Subspace Theorem to establish the transcendence of a class of quasi-periodic continued fractions. This improves earlier works of Maillet and of A. Baker. We also improve an old result of Davenport and Roth on the rate of increase of the denominators of the convergents to any real algebraic number.
We establish new combinatorial transcendence criteria for continued fraction expansions. Let $α= [0; a_1, a_2,...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients $(a_{\ell})_{\ell \ge 1}$ of $α$ cannot be generated by a finite automaton, and that the complexity function of $(a_{\ell})_{\ell \ge 1}$ cannot increase too slowly.
We give 50 digits values of the simple continued fractions whose denominators are formed from a) prime numbers, b) twin primes, c) generalized $d$-twins, d) primes of the form $m^2+n^4$, e)primes of the form $m^2+1$, f) Mersenne primes and g) primorial primes. All these continued fractions belong to the set of measure zero of exceptions to the theorems of Khinchin and Levy. We claim that all these continued fractions are transcendental numbers. Next we propose the conjecture which indicates the way to deduce the transcendence of some continued fractions from transcendence of another ones.
We give continued fraction algorithms for a particular class of Fuchsian triangle groups. In particular, we give an explicit form of each such group that is a subgroup of the Hilbert modular group of its trace field and provide an interval map that is piecewise linear fractional, given in terms of group elements. Using natural extensions, we find an ergodic invariant measure for the interval map. We also study diophantine properties of approximation in terms of the continued fractions; and furthermore show that these continued fractions are appropriate to obtain transcendence results.
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.
It is widely believed that the continued fraction expansion of every irrational algebraic number $α$ either is eventually periodic (and we know that this is the case if and only if $α$ is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine. A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions. The main purpose of the present work is to present new families of transcendental continued fractions with bounded partial quotients. Our results are de…
We put forward several general conjectures concerning the algebraicity or transcendence of continued fractions and Stieltjes continued fractions defined by the Thue-Morse and period-doubling sequences in characteristic $2$. We present our Guess'n'Prove method, in which we exploit the structure of automata, for proving some of our conjectures in special cases.
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…
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.
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…
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.
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…
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 …
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…
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.
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…
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…
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.
