相关论文: Analytic Proof of a Quartic Continued Fraction Ide…

We prove a polynomial continued fraction identity for the constant $-\pi/4$, conjectured by the Ramanujan Machine project. The proof proceeds by explicitly solving the underlying second-order linear difference equation. We derive a…

Fundamental mathematical constants like $e$ and $\pi$ are ubiquitous in diverse fields of science, from abstract mathematics to physics, biology and chemistry. For centuries, new formulas relating fundamental constants have been scarce and…

The Ramanujan Machine project predicts new continued fraction representations of numbers expressed by important mathematical constants. Generally, the value of a continued fraction is found by reducing it to a second order linear difference…

We prove a two-parameter family of continued fraction identities for $\arctan(p/q)$, where $p$ and $q$ are positive integers with $p\le q$. For every such pair, the identity \[ \arctan\frac{p}{q} =…

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.…

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…

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…

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 $\mathbb{Q}(i)$…

In this work, we study a continued fractions theory for the topological completion of the field of Puiseux series. As usual, we prove that any element in the completion can be developed as a unique continued fractions, whose coefficients…

Inspired by the recent pioneering work, dubbed "The Ramanujan Machine" by Raayoni et al. (arXiv:1907.00205), we (automatically) [rigorously] prove some of their conjectures regarding the exact values of some specific infinite continued…

This paper is a direct continuation of "Functional analysis behind a Family of Multidimensional Continued Fractions: Part I," in which we started the exploration of the functional analysis behind the transfer operators for triangle…

Our aim is to find a complex continued fraction algorithm finding all the best Diophantine approximations to a complex number. Using the sequence of minimal vectors in a two dimensional lattice over Gaussian integers, we obtain an algorithm…

Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron in order to generalize the classical continued fractions. In this paper, we propose an introductive fundamental study about MCFs in the field of the $p$--adic…

The properties of continued fractions whose partial quotients belong to a quadratic number field K are distinct from those of classical continued fractions. Unlike classical continued fractions, it is currently impossible to identify…

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…

Let $K$ be a number field. We show that, up to allowing a finite set of denominators in the partial quotients, it is possible to define algorithms for $\mathfrak P$-adic continued fractions satisfying the finiteness property on $K$ for…

We discuss continued fractions on real quadratic number fields of class number 1. If the field has the property of being 2-stage euclidean, a generalization of the euclidean algorithm can be used to compute these continued fractions.…

Triangle partition maps form a family that includes many, if not most, well-known multidimensional continued fraction algorithms. This paper begins the exploration of the functional analysis behind the transfer operator of each of these…

We show that two notions of continued fraction normality, one where overlapping occurrences of finite patterns are counted as distinct occurrences, and another where only disjoint occurrences are counted as distinct, are identical. This…

The Gauss-Kuzmin statistics for the triangle map (a type of multidimensional continued fraction algorithm) are derived by examining the leading eigenfunction of the triangle map's transfer operator. The technical difficulty is finding the…