Questions tagged [continued-fractions]

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Are these continued fractions of integrals known?

Simplified repost of Are these continued fractions of integrals known? on MSE EDIT: The period of the oscillations of $$f(s)=\dfrac1{1+\dfrac s{1+\dfrac{s^2/2!}{1+\dfrac{s^3/3!}{1+\cdots}}}}$$ ...
TheSimpliFire's user avatar
17 votes
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702 views

Picture of Lambert's proof that $\pi$ is irrational?

With a suitably generous notion of "picture proof" or "proof without words" or "geometric proof," there do exist such proofs of the irrationality of square roots and even ...
Timothy Chow's user avatar
14 votes
0 answers
344 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 ...
F Flicker's user avatar
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11 votes
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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 ...
Johann Cigler's user avatar
10 votes
1 answer
267 views

Bibliography request: Entropy for continued fractions

Given a strictly positive real number $x$ we set $e(x)=\log(1+x)$ if $x$ is an integer and $$e(x)=\log(1+x)+\frac{1+\lbrace x\rbrace}{1+x}\left(e(1/\lbrace x\rbrace)-\log(1+\lbrace x\rbrace)\right)$$ ...
Roland Bacher's user avatar
9 votes
0 answers
279 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}...
john mangual's user avatar
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8 votes
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465 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-...
Tito Piezas III's user avatar
7 votes
0 answers
242 views

Simple/Elementary derivation of Ramanujan's continued fraction for Hurwitz $\zeta(3,x+1)$

I came across this MSE post discussing a certain continued fraction for $\zeta(3)$ (more specifically, the Hurwitz zeta function $\zeta(s,z)$ at $s=3$) due to Ramanujan. I asked the original poster ...
D.R.'s user avatar
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On H. Cohen's four continued fractions for $\zeta(3), \zeta(5), \zeta(7)$?

After 6 years from this old MO post, I finally find in the literature polynomials of deg-$5$ for the continued fraction of $\zeta(5)$. I. Recurrences involving $\zeta(5)$ In Cohen's 2022 paper, ...
Tito Piezas III's user avatar
7 votes
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254 views

Can you identify this irrational number?

There is a certain number, say $v$. I can prove it is irrational. That would be more interesting if it is expressible in terms of known values ... zeta functions, Catalan's number, L-functions, etc. ...
Gerald Edgar's user avatar
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7 votes
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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 $\alpha$, ...
Dominik Kwietniak's user avatar
7 votes
0 answers
166 views

The Heine $q$-continued fraction

Let $q=e^{2\pi i \tau}$. The Heine continued fraction is $$H_2(\tau)=\frac1{q^{1/24}}\frac{\eta(2\tau)}{\eta(\tau)} =1+\cfrac{q}{1-q+\cfrac{q^3-q^2}{1+\cfrac{q^5-q^3}{1+\cfrac{q^7-q^4}{1+\ddots}}}}$$ ...
Tito Piezas III's user avatar
7 votes
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138 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-...
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Complexity of continued fraction arithmetic operations

Let $A = [a_0; a_1, \dots]$ and $B = [b_0; b_1, \dots]$ be continued fractions. Let's say that we want to compute $A+B$ or $A \cdot B$ while staying in the continued fraction representation. So, for ...
Oleksandr  Kulkov's user avatar
6 votes
0 answers
233 views

Continued fractions and class groups

Let $d$ be a positive integer. It is well-known (due to Lagrange) that the continued fraction of $\sqrt{d}$ is eventually periodic. Moreover, it is known that the equation $$\displaystyle x^2 - dy^2 = ...
Stanley Yao Xiao's user avatar
6 votes
0 answers
503 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]...
Nicolas Schmidt's user avatar
6 votes
0 answers
406 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 ...
john mangual's user avatar
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Lower bounds for period length of continued fraction of square root

EDIT: Dror answered this to my complete satisfaction. The simple continued fraction for $$ \sqrt{n^2 + 1}$$ has length exactly the same, no dependence on $n$!!!!! No lower bound of the type I wanted ...
Will Jagy's user avatar
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5 votes
0 answers
202 views

Very slow continued fraction convergence

Let $a(0)+b(0)/(a(1)+b(1)/(a(2)+b(2)/(a(3)+\dots)))$ be a continued fraction, and $p(n)/q(n)$ its $n$-th convergent. If it converges (i.e., $p(n)/q(n)$ tends to some limit $S$ as $n\to\infty$), then $...
Henri Cohen's user avatar
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107 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 ...
user132145's user avatar
5 votes
0 answers
322 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|\...
Elie Ben-Shlomo's user avatar
5 votes
0 answers
130 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 ...
Turbo's user avatar
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4 votes
0 answers
156 views

Simple continued fraction of Freiman's constant

The quadratic irrational $\frac{2221564096+283748\sqrt{462}}{491993569}$ is known as Freiman's constant and arises in the theory of continued fractions. I'm curious as to its simple continued ...
Jesse Elliott's user avatar
4 votes
0 answers
300 views

Is the set of approximating sequences for irrationals dominating?

Let $r\in [0,1]\setminus\mathbb{Q}$, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{N}_+=\mathbb{N}\setminus\{0\}$. For $n\in\mathbb{N}_+$ let $$\alpha_r(n)=\min\{|r-\frac{...
Dominic van der Zypen's user avatar
4 votes
0 answers
252 views

Why is Haven's discovery important?

Today my attention was caught by one of those little stories that appear when you open a certain browser: an inmate achieved a number theoretic breakthrough It is about continued fractions and I would ...
Manfred Weis's user avatar
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4 votes
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292 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 ...
Wolfgang's user avatar
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4 votes
0 answers
339 views

A question on M. Mignotte's Paper: "Petho's Cubics"

I have been reading the paper "Petho's Cubics" by M. Mignotte (appeared in Publ. Math. Debrecen, 56/3-4 (2000)) unfortunately I don't understand section 4 of the paper. I am hoping someone could help ...
RandomVisitor's user avatar
3 votes
0 answers
93 views

Infinite families of continued fractions for the Catalan constant

The recent answer to an old question of mine made me aware of The Ramanujan Machine. So it seems like so far, the number of continued fraction representations for $\zeta(3)$ of this polynomial kind is ...
Wolfgang's user avatar
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3 votes
0 answers
162 views

Reconstructing elements of $\mathbb Q$ in $\mathbb Z_p$

Can a rational number $a/b$ (with $b$ coprime to a prime number $p$) be recovered efficiently from a $p$-adic expansion of the form $$\frac{a}{b}=\sum_{j=0}^\infty x_jp^j,\ x_j\in\{0,\ldots,p-1\}\ ?$$ ...
Roland Bacher's user avatar
3 votes
1 answer
247 views

What is the sequence of badly approximable numbers to omit in Hurwitz' second Theorem for Diophantine Approximation to obtain better constants?

The well known result of Hurwitz on Diophantine approximation says that for any irrational $\alpha$ there are infinitely many integer numbers $p$ and $q$ such that $$ |\alpha -\frac{p}{q}|<\frac{1}{...
Thomas Lachmann's user avatar
3 votes
0 answers
501 views

Fundamental Units in $\mathbb{Z}[\sqrt{d}]$ with $d \equiv 1 \mod 4$

It is well known and often cited how the fundamental units for the number ring of $\mathbb{Q}[\sqrt{d}]$ look like. In the case of $d\equiv 2,3 \mod 4$ the number ring is $\mathbb{Z}[\sqrt{d}]$ and in ...
Thomas Lachmann's user avatar
3 votes
0 answers
228 views

Asymptotic expansions for the continued fraction $[1,x,x^2,x^3,\cdots]$

The $n$-th convergent is defined as $$R_n(x) = \frac{P_n(x)}{Q_n(x)}=[1;x,x^2,\cdots,x^n]=1+\frac{1}{x+}\frac{1}{x^2+}\frac{1}{x^3+\cdots}\frac{1}{x^n}$$ where $P_n(x), Q(x)$ are polynomials ...
Vincent Granville's user avatar
3 votes
0 answers
152 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 ...
Turbo's user avatar
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3 votes
0 answers
244 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^...
Nicco's user avatar
  • 246
3 votes
0 answers
128 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 ...
Cassius Manuel's user avatar
3 votes
0 answers
510 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 ...
Wolfgang's user avatar
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3 votes
0 answers
269 views

Cassels' algorithm vs. “divided cells” algorithm

Cassels' algorithm mentioned in link text looks similar to Delone's “divided cells” algorithm. Are there any differences in these algorithms?
Alexey Ustinov's user avatar
3 votes
1 answer
332 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-\...
TheSimpliFire's user avatar
2 votes
0 answers
61 views

Elementary recursion for the A258173

Let $a(n)$ be A258173 i.e. sum over all Dyck paths of semilength $n$ of products over all peaks $p$ of $y_p$, where $y_p$ is the $y$-coordinate of peak $p$. A Dyck path of semilength $n$ is a $(x,y)$-...
Notamathematician's user avatar
2 votes
0 answers
72 views

Is the continued fraction of a constructible number special in some way?

Rationals have finite CF and quadratic have periodic CF. CF in turn can be represented in terms of the modular group SL2(Z), e.g. using the standard generators S(z)=-1/z and T(z)=z+1. On the other ...
Lucian Ionescu's user avatar
2 votes
0 answers
148 views

Finding similar Zudilin-Cohen recurrence relations and cfracs for $\frac{\zeta(4)}{13}$?

I. Two recurrence relations The first one was also discussed in this MO post. We have the similar, \begin{align} (n+1)^5 u_{n+1} &= (2n + 1)(9n^2 + 9n + 3)(15n^2 + 15n + 4)u_n +3n^3(9n^2-1)u_{n-1}\...
Tito Piezas III's user avatar
2 votes
0 answers
84 views

Curious sequences associated to continuous fractions

Given a strictly positive initial rational number $x_0$ in $\mathbb Q_>$ we define a sequence $x_0,x_1,\ldots$ recursively by setting $x_{n+1}=x_n+1/S(x_n)$ for $S(x)=a_0+a_1+\ldots+a_k$ where $[...
Roland Bacher's user avatar
2 votes
0 answers
71 views

Is there a finite set of polynomials generating all rational numbers by iteration?

In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb{Q}$ by iteration starting from 0. Surprisingly one continuous function suffices. The ...
Ivan Meir's user avatar
  • 4,762
2 votes
1 answer
290 views

(Exponential) Mixing property for Gauss map - going from cylinders to intervals

I'm trying to understand the proof of a mixing property of the Gauss map from the paper - 'Some metrical theorems in number theory' and I'm getting confused by the logic in a step. The Gauss map $T$, ...
Bim Binsella's user avatar
2 votes
0 answers
150 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 ...
edom's user avatar
  • 29
2 votes
0 answers
276 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 ...
Adam Gyenge's user avatar
2 votes
0 answers
108 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 Name/...
Barry's user avatar
  • 1,501
2 votes
0 answers
729 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 $s+t$?
wanyuan Xu's user avatar
1 vote
0 answers
200 views

Proving that quotient of orthogonal polynomials is a Padé approximant of Stieltjes transform

This question is reposted from Math Stack Exchange (you can see the original post here). The motivation for reposting is that I feel like the question isn't getting much attention in MSE - if there is ...
Rodrigo's user avatar
  • 51
1 vote
0 answers
83 views

Quasiperiodic sequence, finite differences, recursion

Consider a sequence $\{an\}$ consisting of fractional parts of the numbers $an$ for natural numbers $n$ where $a$ is an irrational number. Its $n$th value is in the interval $(x;y)$ for numbers $n = ...
Станислав Крымский's user avatar