Related papers: Continued fractions, determinant expressions, and …
This paper aims to introduce high school students to the intriguing world of continued fractions, a mathematical concept that provides a unique representation of numbers. The study focuses on the exploration and development of the…
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 prove some new modular identities for the Rogers\textendash Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers\textendash Ramanujan continued fraction, then…
Continued fractions have a long history in number theory, especially in the area of Diophantine approximation. The aim of this expository paper is to survey the main results on the theory of $p$--adic continued fractions, i.e. continued…
Proper continued fractions are generalized continued fractions with positive integer numerators $a_i$ and integer denominators with $b_i\geq a_i$. In this paper we study the strength of approximation of irrational numbers to their…
In this paper, we will first summarize known results concerning continued fractions. Then we will limit our consideration to continued fractions of quadratic numbers. The second author described periods and sometimes precise form 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…
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…
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…
In this paper, we introduce the polynomial continued fraction, a close relative of the well-known simple continued fraction expansions which are widely used in number theory and in general. While they may not possess all the intriguing…
We investigate some properties of the higher continued fractions defined recently by Musiker, Ovenhouse, Schiffler, and Zhang. We prove that the maps defining the higher continued fractions are increasing continuous functions on the…
We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial…
We describe various properties of continued fraction expansions of complex numbers in terms of Gaussian integers. Numerous distinct such expansions are possible for a complex number. They can be arrived at through various algorithms, as…
In this paper we recall some results and some criteria on the convergence of matrix continued fractions. The aim of this paper is to give some properties and results of continued fractions with matrix arguments. Then we give continued…
We use the method of generating functions to find the limit of a $q$-continued fraction, with 4 parameters, as a ratio of certain $q$-series. We then use this result to give new proofs of several known continued fraction identities,…
We provide a generalization of continued fractions to the Heisenberg group. We prove an explicit estimate on the rate of convergence of the infinite continued fraction and several surprising analogs of classical formulas about continued…
We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To…
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…
We study a natural extension to complex numbers of the standard continued fractions. The basic algorithm is due to Lagrange and Gauss, though it seems to have gone mostly unnoticed as a way to create continued fractions. The new…
The aim of this note is to show the existence of a correspondance between certain algebraic continued fractions in fields of power series over a finite field and automatic sequences in the same finite field. this connection is illustrated…