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4
votes
1answer
113 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
56 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}{ ...
17
votes
2answers
402 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
204 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
977 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, ...
0
votes
0answers
61 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
131 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
2answers
276 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 ...
6
votes
1answer
189 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 ...
5
votes
0answers
95 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 ...
7
votes
0answers
58 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 ...
10
votes
3answers
462 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 ...
4
votes
2answers
120 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 $[a_0; a_1, ... ]$?
1
vote
0answers
133 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 $$ ...
4
votes
1answer
133 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 ...
4
votes
0answers
222 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 $|ab|<1$ ...
4
votes
0answers
79 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 ...
6
votes
1answer
196 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 $\ell$. ...
10
votes
7answers
964 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.
3
votes
1answer
209 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
93 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 ...
2
votes
1answer
108 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
70 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 ...
9
votes
1answer
303 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
442 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 ...
2
votes
2answers
290 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 ...
7
votes
1answer
314 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 $a_i = \lfloor \alpha_i \rfloor$, ...
0
votes
0answers
83 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 ...
18
votes
1answer
631 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 ...
7
votes
1answer
599 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.
2
votes
1answer
277 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", ...
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: ...
6
votes
0answers
346 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} ...
5
votes
2answers
609 views

About two 'negative' continued fractions whose sum equals $1$

Letting $a_1,a_2,\cdots,a_r$ be integers which are larger than or equal to $2$, let us define $$[a_1,a_2,\cdots,a_r]=\cfrac{1}{a_1-\cfrac{1}{a_2-\cfrac{1}{\ddots-\cfrac{1}{a_r}}}}$$ (Note that the ...
8
votes
1answer
382 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 $$ x = [a_1, a_2, a_3, \dots] = \cfrac{1}{a_1+\cfrac{1}{a_2+\dots}}, \qquad a_i \in \mathbb{N}. $$ ...
3
votes
1answer
277 views

Complex continued fractions with given digits

For which sets of Gaussian integers $A\subseteq\mathbb{Z}[i]$ is the set of continued fractions with digits in $A$ $$ C(A)=\{a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cdots}}\mid a_i\in A\} $$ totally ...
4
votes
2answers
449 views

Numerical coincidence?

(Nobody's answered this one on stackexchange after several days.) My brother built a garage whose horizontal cross-section is a rectangle that measures $45$ feet by $30$ feet. To make sure the right ...
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 ...
1
vote
2answers
218 views

Linkage between singularities of algebraic varieties and continued fractions

I have an impression that there is linkage or relation between singulariry of algebraic variety and continued fraction when I read some book on resolution of singularity or algebraic geometry.Could ...
4
votes
0answers
244 views

Expressions of $tanh$ type whose continued fractions have two shifts per period

This is a follow-up of another thread about quasi periodic continued fractions, a.k.a. Hurwitz fractions, with some linear shifts. I seem to have found the pattern of a subclass of them, as given ...
3
votes
0answers
278 views

quasi periodic continued fractions and powers of e, tanh, tan

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 ...
11
votes
0answers
474 views

Connection between Infinite continued fractions 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 + ...
6
votes
1answer
361 views

Previous work on this generalization of continued fractions?

The 2x2 matrix representation of a continued fraction makes it clear that we're multiplying together a bunch of group elements. Inversion is essentially freely adjoining a generator to a Coxeter ...
5
votes
1answer
322 views

3-D continued fractions

Which theorems from classical theory of continued fractions have 3-(or multi-) dimesional analogs? Of cause classical one is a periodicity of Klein polyhedra. Probably there are some more...
10
votes
2answers
834 views

Length of Hirzebruch continued fractions

Suppose $a,b$ are two natural numbers relatively prime to $n$ and to each other. Assume $n\geq ab+1$. Suppose further that $\frac{a}{b}\equiv k \pmod{n}$ for some $k\in \lbrace 1,2,\dots, n-1\rbrace$ ...
10
votes
2answers
823 views

The complete list of continued fractions like the Rogers-Ramanujan?

I have two questions about q-continued fractions, but a little intro first. Given Ramanujan's theta function, $$f(a,b) = \sum_{n=-\infty}^{\infty}a^{n(n+1)/2}b^{n(n-1)/2}$$ then the following, ...
6
votes
0answers
297 views

Birth-Death Process associated with Orthogonal Polynomials

I have read in various places the following objects are related: orthogonal polynomials birth-death processes Lattice paths continued fractions After a lot of searching online, I found sketches ...
2
votes
2answers
438 views

Palindromic continued fraction

Here's what I hope is a final question outside of my area that I need to understand a problem about stable vector bundles on $\mathbb{P}^2$. Thank you everybody for your help so far! Suppose I have ...
8
votes
2answers
389 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 ...