Questions tagged [continued-fractions]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
7 votes
0 answers
243 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
  • 741
3 votes
1 answer
75 views

Evaluating the generalized continued fraction obtained from the factorization of a bivariate polynomial equation

Happy New Year, MO community! We need someone expert in Generalized Continued Fractions (GCFs), with a deep knowledge of the GCFs’ convergence properties, to solve the following problem. PROBLEM ...
Monk's user avatar
  • 125
1 vote
1 answer
196 views

Calculating the value of periodic continued fractions with $a_i\in\lbrace 0,1\rbrace$

Question: How can the value of continued fractions of the form $$y:=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\begin{align}\ddots& \\ &a_{n-1}+\cfrac{1}{a_n+y}\end{align}}}}}$$ $$...
Manfred Weis's user avatar
  • 12.6k
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
0 votes
1 answer
256 views

Factorization trees and (continued) fractions?

This question is inspired by trying to understand the lexicographic sorting of the natural numbers in the fractal at this question: Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , ...
mathoverflowUser's user avatar
4 votes
2 answers
275 views

Approximating a fraction with a given denominator

Let $M$, $N$ be large natural numbers (say ~200 bits). Let $L$ be a smaller number, (say ~100 bits). I want to approximate the fraction: $$\frac{M}{N} \sim \frac{k}{L+r}$$ where $r$ is at most $L$. In ...
mtheorylord's user avatar
2 votes
1 answer
189 views

What can we say about the reciprocal of a reduced regular continued fraction?

For positive integers $a>b>0$, we can represent $a/b$ uniquely as $$\frac{a}{b}=a_1-\cfrac{1}{a_2-\cfrac{1}{\cdots-\cfrac{1}{a_n}}}=:[a_1,\dots,a_n]^{-}$$ with $a_i\geq 2$, and this is called ...
blancket's user avatar
  • 161
4 votes
0 answers
159 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
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
6 votes
1 answer
113 views

Finding the continued fraction for the "tails" of $\eta(3)$

I am interested in the continued fractions for the "tails" or "correction term" of the series sum of specific constants. For example, the Madhava's correction term for $\pi/4$: $$ \...
Nanhui Lee's user avatar
0 votes
0 answers
79 views

Numerical detection of Cantori

It is known that as parameters vary in Hamiltonian system, KAM tori can break [1,2]. How to construct numerically the breaking tori? The most relevant paper that I could find is [3,4]. But it uses ...
0x11111's user avatar
  • 493
2 votes
1 answer
142 views

A bijection between odd natural integers and compositions

Given an odd natural integer $2a-1$ with $a\geq 1$, associate to it recursively the composition $\psi(1)=\emptyset$ and $\psi(2^{-n}a)+(n+\delta_{>1}(m))$ if $a=2^n m$ with $m$ odd where $\delta_{&...
Roland Bacher's user avatar
6 votes
0 answers
100 views

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
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
0 votes
1 answer
128 views

Series reversion using something like continued fraction

Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$. Let $$ F(x)=\sum\limits_{m\geqslant 0}f(m)x^m $$ Define the operator $\operatorname{SR}$, which is associated with the series ...
Notamathematician's user avatar
18 votes
0 answers
697 views

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
5 votes
1 answer
174 views

On four Ramanujan-type "Legendrian" sequences with a 3-term recurrence?

I. Recurrences In a previous post, it was mentioned how Almkvist-Zudilin did a computer search for solutions to the recurrence relation, $$(n+1)^3s_{n+1}=(2n+1)(an^2+an+b)s_n+c\,n^3s_{n-1}$$ within a ...
Tito Piezas III's user avatar
7 votes
0 answers
419 views

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
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
7 votes
0 answers
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
  • 40.2k
8 votes
2 answers
576 views

On Zagier's missing continued fraction with multiple limits?

I. Zagier's continued fraction As pointed out by Gorodetsky in his answer, Zagier evaluated the continued fractions associated with his six sporadic sequences excepting the one for $(-9,-3,-27)$. Let $...
Tito Piezas III's user avatar
6 votes
1 answer
250 views

On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence

In a previous MO post, H. Cohen suggested Gorodetsky's 2021 paper which discussed $6+6+3=15$ "sporadic sequences". The first 6 are Zagier's sporadic sequences, the second 6 are by Almkvist-...
Tito Piezas III's user avatar
10 votes
2 answers
648 views

On 12 cfracs: for Catalan's $K$, Gieseking's $\kappa$, and $\pi^2$, $\pi^3$, plus three for $\zeta(3)$ using Zagier's "six sporadic sequences"

I. Some functions As these will be used in the continued fraction evaluations below, recall the Riemann zeta function $\zeta(s),$ and Dirichlet beta function $\beta(s),$ $$\beta(s) = \sum_{n=1}^\infty\...
Tito Piezas III's user avatar
-1 votes
2 answers
105 views

How to solve the following infinite ladder fraction? ( Through pen & paper ) [closed]

The fraction continues till infinity as shown in the image :
Chandra Prakash Bairagi's user avatar
0 votes
0 answers
108 views

Improving Diophantine approximation by rescaling

Let $\lambda\in(0,1)$ be an irrational number such that its continued fraction expansion is bounded (for example, an irrational quadratic number, whose continued fraction is periodic). It is known ...
RaffaeleScandone's user avatar
3 votes
1 answer
164 views

The growth of certain continued fractions

I was recently looking into an old problem of Hardy which studies the distribution of integers of the form $2^a 3^b \leq x$, where $a,b\geq 0$. Letting $N(x)$ denote the number of pairs $(a,b)$ ...
Joshua Stucky's user avatar
0 votes
2 answers
207 views

Natural extension of the Gauss map

Let $G:(0,1)\to(0,1)$ be the Gauss map, i.e., $G(x)=\left\{\frac1{x}\right\}$, which is known to act as the shift on the space of continued fraction expansions. Question. Is there an explicit ...
Nikita Sidorov's user avatar
10 votes
1 answer
269 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
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
  • 11.5k
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
6 votes
1 answer
615 views

Are these continued fractions for the "tails" of $\zeta(3)$ and of the Catalan constant known?

For polynomials $a=a(x)$ and $b=b(x)$, define the continued fraction $$f(a,b):=a(1)+ \lower 2pt\overset{\infty }{\underset{n=1}{\mathbb{\LARGE K}}}~\dfrac{b(n)}{a(n+1)}=a(1)+\cfrac{b(1)}{a(2) + \cfrac{...
Wolfgang's user avatar
  • 13.2k
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
  • 13.2k
2 votes
1 answer
409 views

Period of continued fraction expansion and class number

Let $d>1$ be not a square. Then the continued fraction expansion of $\sqrt d$ is $[a_0; \overline{a_1,\dots,a_\ell}]$, where $a_0=\lfloor \sqrt d\rfloor$ and $a_\ell=2a_0$. Thus, $\ell=\ell(d)$. ...
Nikita Sidorov's user avatar
1 vote
0 answers
84 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
9 votes
2 answers
622 views

Is $\mathbb{Q}$ the orbit of a rational function under 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. In the ...
Ivan Meir's user avatar
  • 4,782
2 votes
0 answers
72 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
  • 4,782
33 votes
2 answers
2k views

What is the smallest set of real continuous functions generating all rational numbers by iteration?

I recently came across this problem from USAMO 2005: "A calculator is broken so that the only keys that still work are the $\sin$, $\cos$, $\tan$, $\arcsin$, $\arccos$ and $\arctan$ buttons. The ...
Ivan Meir's user avatar
  • 4,782
9 votes
1 answer
691 views

$\text{SL}_2(\mathbb{Z})$ and continued fractions?

I know the following facts: $\text{SL}_2(\mathbb{Z})$ is generated by everyone's favorite matrices \begin{equation*} S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \end{equation*} and \...
Sprotte's user avatar
  • 1,065
1 vote
1 answer
179 views

A continued J fraction for $a_n = \frac{1}{(n+1)^2}$?

The following is called a J continued fraction: $$\cfrac{\alpha_0}{1+a_0x-\cfrac{b_1x^2}{1+a_1x-\cfrac{b_2x^2}{1+a_2x-\cdots}}}$$ where the constants are real numbers. Let $\alpha_n= \frac{1}{(n+1)^2}$...
VSP's user avatar
  • 233
0 votes
0 answers
133 views

A question on continued J-fraction

Consider the following two continued fractions $A$ and $B$: $$\frac{\alpha_0}{1+a_0x-\frac{b_1x^2}{1+a_1x-\frac{b_2x^2}{1+a_2x-\cdots}}}$$ $$\frac{\beta_0}{1+c_0x-\frac{d_1x}{1+c_1x-\frac{d_2x}{1+c_2x-...
VSP's user avatar
  • 233
4 votes
2 answers
256 views

A special class of weighted Motzkin paths

Consider Motzkin paths with the following weight: All up-steps and the horizontal steps on height $0$ have weight $1$, all down-steps have weight $t$ and the horizontal steps on even heights have ...
Johann Cigler's user avatar
3 votes
1 answer
246 views

Distance formula for continued fractions

In the book Neverending fractions from Borwein, van der Poorten, Shallit and Zudilin, there is the so called distance formula (Theorem 2.45, p. 43) stated: $$\alpha_1\alpha_2\cdot...\cdot\alpha_n=\...
Marcus's user avatar
  • 396
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
16 votes
1 answer
2k views

Extending Apéry's proof to Catalan's constant?

I've been looking into Apéry's irrationality proof of $\zeta (3)$, and one of the first questions I instantly had, was how did he derive the following continued fraction? $$\begin{equation*} \zeta (3)=...
KStarGamer's user avatar
3 votes
1 answer
249 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
5 votes
1 answer
272 views

Continued fraction associated to KdV solitons

Background (may be skipped by those interested only in the basic question and not important associations): “An essay on continued fractions” by Euler (translated by Myra and Bostwick Wyman) contains ...
Tom Copeland's user avatar
  • 9,937
3 votes
0 answers
505 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
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
2 votes
1 answer
194 views

Fibonacci-Motzkin paths and J-type continued fractions

Recall that a Motzkin path is a piece-wise linear planar path connecting points in the integer lattice quadrant $\Bbb{Z}_{\geq 0} \times \Bbb{Z}_{\geq 0}$ beginning at the origin $(0,0)$ and ending at ...
Jeanne Scott's user avatar
  • 1,847
2 votes
1 answer
292 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

1
2 3 4 5