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9
votes
1answer
391 views

Constructing a family of convergents from continued fractions formed by a set of prime partial quotients

For a given real number $x$, the continued fraction representation $x = [a_0; a_1, a_2, \cdots]$ where $(a_n)_{n \geq 0}$is defined by setting $x = \alpha_0$, then $a_i = \lfloor \alpha_i \rfloor$, ...
6
votes
1answer
222 views

Rogers-Ramanujan continued fraction $R(e^{-2 \pi \sqrt 5})$

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$ It is easy to evaluate $R(e^{-2 \pi/ \sqrt 5})$ using the Dedekind eta function identity $\eta(-\frac{...
4
votes
2answers
141 views

Expected Cardinality of the First n Coefficients of a Continued Fraction

Is there anything known about the asymptotic expected growth of $card\{a_0,a_1,...,a_n\}$, where the $a_i$ are the first $n$ coefficient of a continued fraction $[a_0; a_1, ... ]$?
6
votes
2answers
320 views

Motivation for Hirzebruch-Jung Modified Euclidean Algorithm

Let $a,b \in \mathbb{N} \ \ s.t. \ \ a > b$ have $\gcd(a,b) =1$. We can define the Hirzebruch-Jung modified euclidean algorithm as follows: Let $e_i \in \mathbb{N} >2$, and $ r_k \in \mathbb{N}$...
14
votes
2answers
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Some unpublished notes of Hofstadter

I'm looking for some unpublished notes called "Eta Lore," which are apparently related to a talk Douglas Hofstadter first gave at the Stanford Math Club in 1963.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'...
11
votes
0answers
523 views

Connection between Infinite continued fractions, elliptic integrals and AGM

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^{...
8
votes
3answers
287 views

Combinatorial aspects of continued fractions

Recently, I got interested in the study of the combinatorial aspects of continued fractions. Precisely, I read of the following lemma of Flajolet (see here): Lemma. It holds $$\sum_{\omega} \nu(\...
2
votes
0answers
207 views

Identity with Ramanujan's generalized continued fraction

Let $F(x,q)=\sum_{n\geq 0}x^n\dfrac{q^{n^2}}{(q)_n}$, where $(q)_n=(1-q)(1-q^2)\dots(1-q^n)$. Then: $$H(x,q)=\frac{F(-xq,q)}{F(-x,q)}=\dfrac{1}{1-\dfrac{qx}{1-\dfrac{q^2x}{1-\dots}}}$$ is the ...
4
votes
1answer
128 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 $d$?
0
votes
0answers
63 views

identity which include continued fraction

In this encyclopedia, http://encyclopedia-of-equation.webnode.jp/including-continued-fraction/ I found this identity. $$\sum_{n=1}^{k-1} \frac{2n}{(n^2+r^2)^2}+ \frac{1}{ (k-\frac{1}{2})^2+\frac{1+...
16
votes
2answers
436 views

“Fractally self-similar” numbers

This is another question about visualization of Ford circles, the previous one being Confusion with practically implementing rational approximations. Here is an output of zooming into Ford circles at $...
10
votes
1answer
216 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 ...
18
votes
3answers
1k views

Is there any pattern to the continued fraction of $\sqrt[3]{2}$? [closed]

Is there any pattern to the continued fraction of $\sqrt[3]{2}$ ? Wolfram Alpha returns for cube root of 2: $\sqrt[3]{2}=$ [1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, ...
11
votes
7answers
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.
0
votes
0answers
65 views

uniqueness of $p$-regular infinite continued fraction expansion

I'm not sure what's the accepted terminology is in this regard, but --- following M. Kojima --- let us call a $p$-regular infinite continued fraction $\big[a_0, a_1, a_2, \dots \big]$ an expansion of ...
2
votes
0answers
140 views

Continued Fraction: Please prove $\frac{1}{e \gamma (x+1,1)}=x+\frac{1}{x+1+\frac{2}{x+2+\frac{3}{x+3+\frac{4}{\dots}}}}$ [closed]

I have been playing around with Mathematica and continued fractions and I noticed something. ContinuedFractionK[n, n + x, {n, 1, Infinity}] ==-x + 1/(E Gamma[1 + x] - E Gamma[1 + x, 1])==-x + 1/(E ...
6
votes
1answer
207 views

Young-Fibonacci lattice and purely periodic continued fractions

The Fibonacci lattice $\mathcal{F}$ is the poset of all finite words consisting of 1's and 2's where a word $v$ covers a word $u$ if $v$ is obtained from $v$ by either (a) inserting a 1 in $u$ prior ...
7
votes
0answers
78 views

A conjectured q-continued fraction related to the Göllnitz-Gordon partition identities

Given $q=e^{2i\pi\tau}$ with $|q|\lt1$, define the well-known Göllnitz-Gordon identities $$A(q)=\sum_{n=0}^\infty \frac{q^{n(n+1)}(-q;q^2)_n}{(q^2;q^2)_n}=\prod_{n=1}^\infty \frac{1}{(1-q^{8n-3})(1-...
5
votes
0answers
101 views

On existence of rapid Arithmetic geometric procedure?

We know that $\pi$ can be computed by Arithmetic Geometric mean using Gauss-Legendre procedure which does provide fastest convergence rate as well with a guarantee of $2^n$ bits of $\pi$ at $n$th ...
10
votes
3answers
467 views

Continued Fractions from Digit Streams

let $x=\sum_{i=1}^{\infty}\delta_i2^{-i},\ \delta_i\in\{0,1\}$. Is there an algorithm that converts the sequence $(\delta_0,\ \delta_1,\ ...)$ of the binary digits of $x$ to the sequence $[a_0;a_1,\...
5
votes
0answers
252 views

Are these two $q$-continued fractions equivalent?

In this MSE post, Nicco Mnisi defined a particular $q$-continued fraction of order $12$. More generally, define the cfrac found in Ramanujan's Notebooks, Vol III, Chap. 16, page 24, where $|ab|<1$ ...
1
vote
0answers
149 views

Continued fractions and modular forms

Let $q=e^{2\pi it}$. If $u(t)$ is Ramanujan's octic continued fraction, is it true that the generator of the octahedral group can be expressed as a continued fraction of the form $$ (u(2t))^2=\frac{2q^...
4
votes
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 $$\...
2
votes
2answers
318 views

Brun's algorithm

Does anyone have an exact reference for the weak convergence (convergence in angle) of Brun's subtractive multi-dimensional continued fractions algorithm (in all dimensions)? I have been given ...
4
votes
0answers
86 views

Binary Quadratic Forms with coefficients in $F_q[T]$

I aim to study the binary forms $ax^2 + bxy + cy^2 = (a,b,c)$ where $a,b,c \in {F_q}[T]$ (charasteristic of $F_q$ not 2) in particular those such that the discriminant $D = b^2 - 4ac \in F_q[T]$ has ...
26
votes
4answers
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 ...
8
votes
2answers
398 views

Last term of repeating continued fraction expansion

Once again, working with stable vector bundles on $\mathbb{P}^2$ I have run into a question that is really out of my area. (Thanks to everybody who helped out with my last question!) Let $D>9$ be ...
6
votes
1answer
202 views

An upper bound for the length of the continued fraction expansion of $\sqrt d$

Let $d\ge 2$ and let $$ \sqrt d =[a_0; \overline{a_1,\dots, a_\ell, 2a_0}] $$ be its continued fraction expansion. Clearly, if $d=n^2+1$, then $\ell=0$, which gives the lower bound for $\ell$. ...
24
votes
17answers
7k views

Applications of finite continued fractions

I know some applications of finite continued fractions. Probably you know more. Can you add anything? (For for Applications of periodic continued fractions I have made a special topic.) 1) (Trivial) ...
3
votes
1answer
216 views

When two Dedekind sums are equal

The (classical) Dedekind sum $s(h,k)$ is defined as $$s(h,k)=\sum_{r=1}^{k-1}\frac{r}{k}\bigg(\frac{hr}{k}-\Big[\frac{hr}{k}\Big]-\frac{1}{2}\bigg)$$ for $\gcd(h,k)=1$. A natural question is, when ...
1
vote
1answer
107 views

Division methods for divergent continued fractions

I hadn't even noticed before entering the subject above the parellel with the title of an earlier question posted here titled Summation methods for divergent series. And it's just mutatis mutandis as ...
6
votes
0answers
359 views

Why does the Rogers-Ramanujan continued fraction $R(q)$ appear in Emma Lehmer's quintic?

Define the Ramanujan theta function $f(a,b)$ as, $$f(a,b) = \sum_{n=-\infty}^{\infty} a^{n(n+1)/2}\,b^{n(n-1)/2}$$ and the Dedekind eta function, $$\eta(\tau) = q^{1/24}\prod_{n=1}^{\infty} \left(1-...
0
votes
0answers
88 views

Partial quotients restricted to a thin set

Zaremba's conjecture asserts that if there exists some integer $A > 1$ such that for the set of numbers $x \in [0,1]$ whose partial quotients in the continued fraction representation are all ...
11
votes
1answer
603 views

Poles from the Continued Fraction Expansion of the Tangent Function?

Consider the well known continued fraction expansion $$ z \tan z = \frac{z^2}{1 - \cfrac{z^2}{3- \cfrac{z^2}{5 - \ldots}}} $$ of the tangent function going back to Euler and Lambert (Lambert used it ...
2
votes
1answer
124 views

Fact similar to Ostrowski numeration for reals

I have to prove this fact (found in an article without proof). Let $\alpha \in \mathbb{R}$ be an irrational number. Let $\alpha = [a_0;a_1,a_2,\ldots]$ be the continued fraction expansion. We call $\...
2
votes
0answers
72 views

Name of a difference of continuants

I am getting ready to publish the manuscript http://arxiv.org/pdf/1408.4631v2.pdf and I am trying to do due diligence on a quantity I study before it gets published. (This is cross-posted from http:...
10
votes
1answer
311 views

Repetend digit graphs for $1/n$ in base $b$

Here is a decimal expansion of $\frac{1}{34}$: $$(1/34)_{10}=0.02941176470588235\overline{2941176470588235}\ldots$$ And here is a graphical representation of the 16-digit "repetend," as a directed ...
18
votes
3answers
2k views

Is any particular algebraic number known to have unbounded continued fraction coefficients?

The continued fraction $$[1;1,2,3,4,5,\dots]=1+\cfrac{1}{1+\cfrac{1}{2+\cdots}}, $$ for instance, is known explicitly as a ratio of Bessel function values and is (I believe - SS) known to be ...
4
votes
2answers
448 views

Are there any patterns in simple continued fraction expansions of algebraic real numbers?

As we know there are patterns in simple continued fraction expansion of quadratic algebraic numbers,are there any patterns in simple continued fraction expansions of other algebraic real numbers?...
6
votes
4answers
685 views

Orbits of the projective special linear group on $\mathbf{Q} \cup \{\infty\}$

The group $\mathrm{PSL}_2(\mathbf{Q})$ of fractional linear transformations $x \mapsto (ax+b)/(cx+d)$ such that $a,b,c,d \in \mathbf{Q}$ and $ad-bc = 1$ acts on $\mathbf{Q} \cup \lbrace \infty\rbrace$....
28
votes
5answers
2k views

What is the effect of adding 1/2 to a continued fraction?

Is there a simple answer to the question "what happens to the continued fraction expansion of an irrational number when you add 1/2?" A closely related question is "what happens to such an expansion ...
15
votes
2answers
2k views

Is there an elementary proof of a result about the parity of the period of the repeating block in the continued fraction expansion of square roots

It is a known fact that for a Prime $P$, $P\equiv 1$ mod $4$ iff the length of the period in the repeating block for the continued fraction expansion of $\sqrt{P}$ is odd. I have an elementary proof ...
7
votes
1answer
615 views

Continued fraction representation of Zeta

A question at math.SE is asking for references. The fraction is quite nice! Check it out and post some references if you know of any. I found this at arxiv, but it doesn't apply to Zeta.
18
votes
1answer
640 views

Is anything known about which numbers appear in the continued fraction expansion of $\pi$?

This question is mostly idle curiosity, and certainly is not related to any research activities of my own. The motivation and background are as follows. I am currently teaching a Freshman Seminar in ...
10
votes
5answers
1k views

Euler's divergent series sum n!*(-1)^n: what is known about the resulting constant?

Much of the theory of continued fractions has been developped by Euler in the 18th century. The little survey "Euler: continued fractions and divergent series (and Nicholas Bernoulli)", mentions ...
2
votes
1answer
308 views

How to prove that the odd continued fraction approximants of ln(1+X) are upper bounds?

The odd order continued fraction approximants for $\ln(1+X)$ are $$X,\quad \frac{X^2+6X}{4X+6,}\quad \frac{X^3+21X^2+30X}{9X^2+36X+30,}\quad \dots.$$ In "Some bounds for the logarithmic function", ...
6
votes
2answers
1k views

Gauss-Kuzmin Theorem (continued fractions) - why is important?

As we know, Gauss wrote that \begin{equation} \lim_{n \rightarrow \infty} \lambda \left(\tau^n \leq x\right) = \frac{\log(1+x)}{\log2}, \quad 0 \leq x < 1, \end{equation} with $\lambda$ is Lebesgue ...
2
votes
1answer
72 views

Expressions in “continued” monotone functions

Recall continued fractions: http://en.wikipedia.org/wiki/Continued_fraction Now take a look at this question: http://math.stackexchange.com/questions/601846/the-limit-of-displaystyle-lim-n-to-infty-...
7
votes
2answers
397 views

How Symmetric is Diophantine Approximation using Fractions with Square Denominators?

Let $S$ be an infinite set of positive integers. Let us say that a "best S-approximation" to a real irrational $r$ is a rational number $p/q$, with $p$ and $q$ integers and $q \in S$, such that for ...
2
votes
0answers
690 views

Applications of the length of the continued fraction

The fractions $n/q$ and $n/(n-q)$ can be representated by Hirzebruch continued fractions (also called Jung-Hirzebruch CF), the length of each we denote by $s$ and $t$. Is there any bound for $s+t$?