Related papers: Generalized Palindromic Continued Fractions
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 Queff\'elec's theorem on Thue-Morse continued fractions as a special case.…
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…
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…
It is widely believed that the continued fraction expansion of every irrational algebraic number $\alpha$ either is eventually periodic (and we know that this is the case if and only if $\alpha$ is a quadratic irrational), or it contains…
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 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…
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…
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…
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…
Classical results on Diophantine approximation, such as Roth's theorem, provide the most effective techniques for proving the transcendence of special kinds of continued fractions. Multidimensional continued fractions are a generalization…
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…
In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions with deterministic terms are…
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.
In this paper, we investigate the monotone property of the continued fractions $G(m,\lambda)$ as a function of $m$ and $\lambda$. In particular, we obtain new inequality for the relative continued fractions.
In this paper we define a new type of continued fraction expansion for a real number $x \in I_m:=[0,m-1], m\in N_+, m\geq 2$: \[x = \frac{m^{-b_1(x)}}{\displaystyle 1+\frac{m^{-b_2(x)}}{1+\ddots}}:=[b_1(x), b_2(x), ...]_m. \] Then, we…
We study the generalized continued fraction expansions of complex numbers in term of elements from Euclidean subrings, especially Gaussian or Eisenstein integers, in a general framework as pursued in [3] and [1]. We introduce a common…
The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These…
For integers $m \geq 2$, we study divergent continued fractions whose numerators and denominators in each of the $m$ arithmetic progressions modulo $m$ converge. Special cases give, among other things, an infinite sequence of divergence…
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…
In this paper, we improve some transcendence results for $p$--adic continued fractions. In particular, we prove that palindromic and quasi--periodic $p$--adic continued fractions converge either to transcendental numbers or quadratic…