Related papers: A simple continued fraction expansion for $e^n$
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…
Recently Raayoni et al. announced various conjectures on continued fractions of fundamental constants automatically generated with machine learning techniques. In this paper we prove some of their stated conjectures for Euler number $e$ and…
This note presents an especially short and direct variant of Hermite's proof of the simple continued fraction expansion e = [2,1,2,1,1,4,1,1,6,...] and explains some of the motivation behind it.
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…
In this note, we show that any continued fraction convergent of the number $e = 2.71828...$ can be derived by approximating some integral $I_{n, m} := \int_{0}^{1} x^n (1 - x)^m e^x d x$ $(n, m \in \mathbb{N})$ by 0. In addition, we present…
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…
Several continued fraction expansions for $e$ have been produced by an automated conjecture generator (ACG) called \emph{The Ramanujan Machine}. Some of these were already known, some have recently been proved and some remain unproven.…
A well known method for convergence acceleration of continued fraction $\K(a_n/b_n)$ is to use the modified approximants $S_n(\omega_n)$ in place of the classical approximants $S_n(0)$, where $\omega_n$ are close to tails $f^{(n)}$ of…
We study explicit continued fraction expansions for certain series. Some of these expansions have symmetry that generalizes some remarkable examples discovered independently by Kmosek and Shallit. Furthermore, we prove the following…
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 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…
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…
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…
In this paper we present a convergence theorem for continued fractions of the form $K_{n=1}^{\infty}a_{n}/1$. By deriving conditions on the $a_{n}$ which ensure that the odd and even parts of $K_{n=1}^{\infty}a_{n}/1$ converge, these same…
In this short note we prove two elegant generalized continued fraction formulae $$e= 2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4+\ddots}}}}$$ and $$e= 3+\cfrac{-1}{4+\cfrac{-2}{5+\cfrac{-3}{6+\cfrac{-4}{7+\ddots}}}}$$ using elementary…
In this paper we present a method to derive Eulerian continued fractions arising from a sequence of integrals. As examples, through a new derivation, we reproduce classical continued fraction expansions for the natural logarithm, the…
We prove an explicit formula for infinitely many convergents of Hurwitzian continued fractions that repeat several copies of the same constant and elements of one arithmetic progression, in a quasi-periodic fashion. The proof involves…
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…
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 explicit and asymptotic lower bounds for the quantity $|e^{s/t}-M/N|$ by studying a generalized continued fraction expansion of $e^{s/t}$. In cases $|s|\geq 3$ we improve existing results by extracting a large common factor from the…