Questions tagged [continued-fractions]

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0answers
122 views

What is the nearest Ford circle for any point in $\mathbb R^2$

I want to draw Ford circles within a "distance Estimated system" (ray marching). Therefore, given a point $(x,y)$ from $\mathbb R^2$, I need the shortest distance to any circle with center $(p/q,1/2q^...
3
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90 views

Proof of continued fraction identity of subfactorial

This question is part of a wider conjecture I have formed with someone which has its roots in Raayoni et al. (2019). The subfactorial function can be written as $$!n=\frac{n!}e+\frac{(-1)^n}{n+2-\...
5
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98 views

Possible Birkhoff spectra for irrational rotations

Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the unit circle (think of it as of the interval $[0,1)$ with endpoints identified). Assume that $\alpha$ is irrational and consider the rotation by $\alpha$, ...
3
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1answer
166 views

Distribution of partial quotients of $\sqrt{d}$ with $X < d \leq 2X$

Recall that for a real number $\alpha$, it has a continued fraction expansion usually written as $$\displaystyle \alpha = [a_0; a_1, a_2, \cdots].$$ Moreover, $\alpha$ is rational if and only if its ...
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27 views

Maximal orders in Clifford algebras

Let $$ \mathcal{C}_n(R)=R\langle e_1,\ldots,e_n\rangle/(\{e_i^2+1\}, \{e_ie_j+e_je_i:i\neq j\}) $$ be the Clifford algebra for the negative definite quadratic form $-\sum_ix_i^2$ obtained by adjoining ...
6
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1answer
481 views

Are the coefficients of certain product of Rogers-Ramanujan Continued Fraction non-negative?

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$ The following equality is famous: $$\cfrac{q^{1/5}}{R(q)} = \prod_{k>0} \cfrac{(1-q^{5k-2})(1-q^{...
7
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2answers
305 views

Average number of iterations for the Euclidean algorithm to terminate

Let $N$ be a positive integer and $0 \leq s < N$. We try to divide $s$ into $N$ using the Euclidean algorithm: $N = q_1 s + r_1 $ $r = q_2 r_1 + r_2 $ $\vdots$ $r_{K-1} = q_{K-1} r_K$ If we ...
7
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2answers
160 views

Evaluation of hypergeometric type continued fraction

Is there a (possibly hypergeometric-type) explicit evaluation of the continued fraction $$a-\dfrac{1.(c+d)}{2a-\dfrac{2.(2c+d)}{3a-\dfrac{3.(3c+d)}{4a-\ddots}}}$$ Even the special case $d=0$, $a=1$ ...
2
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1answer
461 views

Does using continued fractions work to give a homeomorphism $\mathbb{Q}^+ \rightarrow (\mathbb{Q}^+)^2$?

Let $\mathbb{Q}$ be the topological space of rational numbers (with topology induced by inclusion in the real line) and let $\mathbb{Q}^+$ be the set of positive ($x>0$) rationals. I'm looking ...
3
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116 views

Why does division parallelize but not continued fractions and is there an analog of multiplication to continued fractions?

All the basic arithmetic operations $\times,+,/,-$ can be parallelized. However continued fraction representation of a rational number is not parallelized. The process of Euclid's algorithm looks ...
3
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2answers
637 views

Have new conjectures generated by the Ramanujan machine been proven?

Recently the following preprint was published with new automatically generated conjectures on generalizes continuous fractions, e.g., for the Euler constant: Raayoni, Pisha, Manor, Mendlovic, Haviv, ...
17
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1answer
1k views

Divergent Series & Continued Fraction (from Gauss' Mathematical Diary)

I've asked that question before on History of Science and Mathematics but haven't received an answer Does someone have a reference or further explanation on Gauß' entry from May 24, 1796 in his ...
7
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1answer
184 views

Riemann-Hilbert and Orthogonal polynomials

Sorry for perhaps naive questions, I am not at all a specialist in the subject but I need it for my research. I know that there are close relations between Riemann-Hilbert problems and orthogonal ...
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446 views

Representations of $\zeta(3)$ as continued fractions involving cubic polynomials

$\zeta(3)$ has at least two well-known representations of the form $$\zeta(3)=\cfrac{k}{p(1) - \cfrac{1^6}{p(2)- \cfrac{2^6}{ p(3)- \cfrac{3^6}{p(4)-\ddots } }}},$$ where $k\in\mathbb Q$ and $p$ is a ...
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82 views

“middle” partial denominator in continued fraction expansion of square roots

Suppose $d$ is a positive integer that is not a perfect square such that the negative Pell equation, $x^{2}-dy^{2}=-1$ has no solution. Then we know the minimal period of the continued fraction ...
8
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1answer
158 views

Limit of quotients of elements of special Fibonacci matrices

Let $F_n$ be the $n$-th Fibonacci number, started with $F_0=0,F_1=1$, and consider the matrices $$M_n=\pmatrix{F_{n+3} & F_{n+1} \\ F_{n+2} & F_{n}}.$$ Let $$\pmatrix{\alpha_n & \beta_n \\...
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48 views

Bound for truncation error of continued fraction for $E_1(z)$

Let $z \in \mathbb C \setminus(-\infty,0)$. It is known that $$E_1(z) = \cfrac{e^{-z}}{z+\cfrac{1}{1+\cfrac{1}{z+\cfrac{2}{1+\cfrac{2}{z+\cfrac{3}{1+\cdots}}}}}}.$$ For example, see http://functions....
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1answer
94 views

Distinctness of quadratic surd continued fraction convergent ratio limit

In this question on math.stackexchange.com I have made two conjectures the first of which I have proved. The second has not been settled. I post it here to seek a proof. Given a quadratic surd $\sqrt ...
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43 views

Reduction of a Jacobi-type continued fraction

I am trying to reduce the following Jacobi-like continued fraction(or J-fraction): $$f(z)=z+K_{n=1}^{\infty}\frac{R_{n}k^2}{z+Q_{n}}$$ where, $$R_{n}=n\left(n-\frac{1}{2}\right),\; Q_n=n\left(n+m+\...
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252 views

Theory of Irrational Tangles?

According to one possible definition, an $n$-tangle $T$ is a subset $T \subseteq \Bbb{R}^2\times [0,1] =: X$ that is homeomorphic to a disjoint union $[0,1] \times n := [0,1] \amalg \ldots \amalg [0,1]...
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2answers
234 views

Periods of the continued fraction expansions of Galois-conjugate quadratic-irrationals

Question: Given a quadratic irrational $x = a + b\sqrt{D}$ ($a,b \in \Bbb{Q}$, $D \in \Bbb{N}_{> 0}$ square-free) and its Galois conjugate $x' = a - b\sqrt{D}$, is it true that the continued ...
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215 views

Transcendental Continued Fractions

Liouville famously showed the existence of a transcendental number by considering $\alpha =\sum\limits_{n=0}^\infty10^{-n!}$ and showing that it did not satisfy 'Liouville's criterion' $|\alpha-p/q|\...
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75 views

Different characterizations of Liouville numbers

Usually, Liouville numbers are defined as follows: $x$ is Liouville if for ever $i\in\mathbb N$ there exist $n,m\in\mathbb Z$ such that \begin{equation} \left|x-\frac nm\right|<\frac1{m^i}. \end{...
3
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1answer
178 views

Matrix continued fractions

I am aware of the classical continued fraction in the field of real numbers, but recently I have come across the term matrix continued fraction and when I checked on the internet there are varieties ...
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2answers
165 views

Does the set of Diophantine $m$-tuples has full measure?

We say that an $m$-tuple $\omega=(\omega_1,\ldots,\omega_m)$ satisfies the Diophantine condition of order $\nu \geq 0$ if there is a constant $C>0$ such that for all natural $q$ and integer $p_1,\...
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345 views

Relation between a continued fraction and partitions

I am interested in the continued fraction $$\sum\limits_k {{z^{{2^k} - 1}}} = \frac{1}{{1 - \frac{{{T_0}z}}{{1 - \frac{{{T_1}z}}{{1 - \frac{{{T_2}z}}{{1 -{ \ddots }}}}}}}}}.$$ OEIS A104977 states ...
1
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1answer
119 views

Simultaneous Diophantine Condition and Growth Rate of Convergents Denominators

Let $\omega=(\omega_1,\ldots,\omega_{m})$ be an $m$-tuple of real numbers. Let $|\omega|_{m}:=\sup\limits_{1 \leq j \leq m}|\omega_j|_{1}$ be a metric on flat torus $\mathbb{T}^{m}=\mathbb{R}^{m}/\...
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151 views

whether the quotient of continued fraction of algebraic irrational number is bounded or not is similar or equivalent to Collatz conjecture?

I vaguely recall that whether the quotient of continued fraction of algebraic irrational number is bounded or not is similar or equivalent to Collatz conjecture, could any one give the reference? or ...
10
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1answer
201 views

Distribution of good diophantine approximations

Let $\langle x \rangle: \mathbb{R} \to (-1/2,1/2]$ be the periodic function with period $1$ which is $x$ for $x \in (-1/2,1/2]$. Is there some function $D(a,b)$ of real numbers $a<b$ such that, for ...
3
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1answer
235 views

Continued Fraction of Random Variables

So this is my first post in mathoverflow. I posted this problem in Mathstack, an I've also put a bounty on it, but did not get any response. If anyone can at least point out a reference on this ...
6
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2answers
2k views

Distribution of distances of successive zeros of $f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$

Let $\alpha$ and $\beta$ be incommensurate real numbers. Consider the function $f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$ and its positive zeros $x_k(\alpha,\beta)$. Fix $\alpha$ and ...
11
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0answers
214 views

quasiperiodic continued fractions

Is anything known about continued fractions in which the sequence of integers is quasiperiodic? Quasiperiodic is meant here in the sense of 1D quasicrystals. For example, draw an irrationally-sloped ...
5
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3answers
325 views

Irrationality of generalized continued fractions

An infinite simple continued fraction $$\frac{1}{b_1 + \frac{1}{b_2 + \frac{1}{b_3+\dots}}} (b_i\in\mathbb Z)$$ is irrational. Now for a generalized continued fraction: $$\frac{a_1}{b_1 + \frac{a_2}...
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0answers
141 views

Has anybody studied continued fractions in function spaces?

For the text below, define $f^\infty(x) = \lim_{n\to\infty} f^n(x)$ where $f^n = \underbrace{f \circ \ldots \circ f}_{n}$. Usually 'continued fraction' means continued fraction in $\mathbb{R}$. For ...
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1answer
275 views

Some nice functional equations for $q$-continued fractions

Given $\large q=e^{2\pi i \tau}$. Define, $$\alpha(\tau) = \sqrt2\,q^{1/8}\prod_{n=1}^\infty\frac{ (1-q^{4n-1})(1-q^{4n-3})}{(1-q^{4n-2})(1-q^{4n-2})}$$ $$\beta(\tau) = q^{1/5}\prod_{n=1}^\infty\frac{ ...
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141 views

The Heine $q$-continued fraction

Let $q=e^{2\pi i \tau}$. The Heine continued fraction is $$H_2(\tau)=\frac1{q^{1/24}}\frac{\eta(2\tau)}{\eta(\tau)} =1+\cfrac{q}{1-q+\cfrac{q^3-q^2}{1+\cfrac{q^5-q^3}{1+\cfrac{q^7-q^4}{1+\ddots}}}}$$ ...
0
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1answer
175 views

limit of denominator in continued fraction expansion algebraic?

I asked this on MathStackExchange but didn't get an answer, so I'm trying it here: Let $\alpha$ be an algebraic number and denote with $\frac{p_n}{q_n}$ the $n$-th convergent of $\alpha$ that we get ...
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1answer
1k views

Can we use the Rogers-Ramanujan cfrac to parameterize the Fermat quintic $x^5+y^5=1$?

Define $\color{blue}{q=e^{2\pi i \tau}}$ and Dedekind eta function $\eta(\tau)$. Note: I found these relations empirically, but their consistent forms suggest they can be rigorously proven. I. $p=2$...
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0answers
77 views

Stern-Stolz in $p$-adic case

I'm trying to figure out if the following statement is trivial or not: For $b_i \in \mathbb{C}_p$ (the complete $p$-adic field), if $\sum |b_i|_p < \infty$, then the continued fraction $b_0+\...
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245 views

when is the Brun continued fraction periodic?

I was hoping to figure this one out on my own. There's this nice paper by Avila on various "subtractive" Euclidean algorithms. Here is one he attributes to Viggo Brun: $$ (x,y,z) \mapsto \text{sort}...
2
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2answers
237 views

Growth rate for the average of the entries in the fundamental period of the continued fraction expansion of $\sqrt{n}$

(Cross-posted from stackexchange: https://math.stackexchange.com/questions/1976296/what-is-known-about-the-average-of-the-partial-quotients-in-the-fundamental-peri) I'd like references concerning ...
5
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2answers
378 views

Constant related to continued fraction of quadratic irrationals

Let $d$ be a positive, non-square integer, and define $c_d$ to be the smallest positive number with the following property: for all pairs of co-prime integers $(p,q)$ with $q > 0$, the inequality $...
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0answers
77 views

Criterion for irrational numbers of constant type 2

From Kuiper's and Niederreiter's book Uniform distribution of sequences, Ch.2, § 3, I learn that an irrational number $\alpha\in \mathbf{R}\smallsetminus \mathbf{Q}$ is of constant type $\eta$ if ...
8
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2answers
630 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{...
8
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3answers
443 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
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0answers
258 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
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1answer
134 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 $d$?
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75 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+...
18
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2answers
652 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
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1answer
264 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 ...