# Tagged Questions

The tag has no usage guidance.

2k views

### How does this relationship between the Catalan numbers and SU(2) generalize?

This is a question, or really more like a cloud of questions, I wanted to ask awhile ago based on this SBS post and this post I wrote inspired by it, except that Math Overflow didn't exist then. As ...
3k views

### Showing e is transcendental using its continued fraction expansion

Can the transcendence of e be shown using its continued fraction expansion e = [2;1,2,1,1,4,1,1,6,...]?
1k views

### Open problems in continued fractions theory

I propose to collect here open problems from the theory of continued fractions. Any types of continued fractions are welcome.
1k views

### The Riemann's Zeta Function represented as a continued fraction and a question of convergence.

The Riemann's zeta function can be expressed as a continued fraction as follows \begin{align*} \zeta(z)=\newcommand{\bigk}{\mathop{\Huge\vcenter{\hbox{K}}}}\left(1-\bigk_{k=1}^{\infty }\frac{-e^{-2\...
217 views

### Confusion with practically implementing rational approximations

Writing a program visualizing Ford circles I've encountered a seemingly purely programmatic puzzle but then gradually realized there are some mathematical aspects of it which I don't understand. Let ...
420 views

### Do simple, multi-dimensional generalizations of this continued fraction formula exist?

Background Let $x\in [0,1)\setminus \mathbb{Q}$ have regular continued fraction expansion given by $$x = [a_1, a_2, a_3, \dots] = \cfrac{1}{a_1+\cfrac{1}{a_2+\dots}}, \qquad a_i \in \mathbb{N}.$$ ...
760 views

### Is it possile for all real algebraic numbers to have continued fractions with bounded partial quotients ?

Roth's theorem states that for every real algebraic $\alpha$ and $\epsilon>0$, there is a $c>0$ such that $$|\alpha -\frac{p}{q}| > \frac{c}{q^{2+\epsilon}}.$$ Lang's conjecture strengthened ...
1k views

### Some unpublished notes of Hofstadter

I'm looking for some unpublished notes called "Eta Lore," which are apparently related to a talk Douglas Hofstadter first gave at the Stanford Math Club in 1963. I know these notes exist because they'...
It is known that at $x=1$, the following continued fraction represents $\frac{4}{\pi}$ and can be approximated rapidly using Gauss' Arithmetic Geometric mean. $$C(x) = x + \frac{1^{2}}{2x + \frac{3^{... 1answer 818 views ### Searching for an inhomogeneous diophantine approximation algorithm Given two nonzero real numbers x and y such that y/x is irrational, a real number z to be approximated, and a tolerance \epsilon, what is an algorithm that will provide coprime integers a ... 2answers 1k views ### Applications of periodic continued fractions Some answers from Applications of finite continued fractions in fact are Applications of periodic continued fractions. I think that it should be separate question. What can you add to the following ... 1answer 299 views ### Algorithm to determine sign of a polynomial I've been working with a collaborator (Arek Goetz) on a dynamics problem involving piecewise isometries (a map T on a domain X (say a subset of the plane) such that X is divided into a finite ... 1answer 130 views ### Mean value of a function associated with continued fractions Suppose that an irrational x in (0,1) has convergents c(k,x), and let$$d(x) = \sum_{k=0}^{\infty} \mid x - c(k,x)\mid.$$What is the mean value of d? 1answer 144 views ### Relations between modular functions of certain q-continued fractions Given the j-function j:=j(\tau), and q=e^{2\pi i\tau} = \exp(2\pi i\tau) where, for convenience, we set \tau=\sqrt{-n}. I. \frac{A_2(q)}{A_1(q)} = \text{q-cfrac}:\; Icosahedral group$$\...
It is well known that some transcendental numbers like e.g. rational multiples of $e^{2/n}$ with $n\in\mathbb N$, when written as regular continued fractions (R.C.F.), yield what can be called a ...