The continued-fractions tag has no usage guidance.

**4**

votes

**1**answer

112 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

**0**answers

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

**2**answers

399 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

**1**answer

203 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

**3**answers

974 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

**0**answers

60 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

**0**answers

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

**2**answers

275 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

**1**answer

188 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

**0**answers

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

**0**answers

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

**3**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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 $|ab|<1$ ...

**3**

votes

**0**answers

78 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

**1**answer

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

**7**answers

961 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

**1**answer

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

**1**answer

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

**1**answer

107 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

**0**answers

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

**1**answer

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

**3**answers

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

**2**answers

441 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

**2**answers

289 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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

276 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

**1**answer

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

**0**answers

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

**2**answers

606 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

**1**answer

380 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

**1**answer

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

**2**answers

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

**5**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

833 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

**2**answers

821 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

**0**answers

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

**2**answers

436 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

**2**answers

388 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 ...