33
votes
Accepted
What is the smallest set of real continuous functions generating all rational numbers by iteration?
It is enough with one continuous function. First, I'll give a simple example with one function which is discontinuous at one point. To do it, consider the function $$f:(0,\pi+1)\to(0,\pi+1)$$ with
$$
...
- 10.6k
26
votes
Accepted
Divergent series & continued fraction (from Gauss' mathematical diary)
The entry from May 24, 1796 is worked out in a more general form on February 16, 1797 [reproduced below from this scan]
$$1-a+a^3-a^6+a^{10}+\cdots=\frac{1}{\displaystyle 1+\frac{\strut a}{\...
- 188k
24
votes
Accepted
Representations of $\zeta(3)$ as continued fractions involving cubic polynomials
See NOTE below.
This MO inquiry is over 3 yrs old now.
By the date the question about the $\zeta(3)$ CF with $k=8/7$ was made (Feb, 2019), it can be answered in the negative nowadays, since it was '(...
- 2,836
22
votes
What is the smallest set of real continuous functions generating all rational numbers by iteration?
You only need one continuous function.
There exists a continuous function $f: \mathbb{R} \to \mathbb{R}$ with a dense orbit, according to this MathOverflow answer. As in Saúl's construction, you can ...
- 6,571
19
votes
Accepted
Is $\mathbb{Q}$ the orbit of a rational function under iteration?
As was mentioned in the comments by pregunton, it is possible to do using two rational functions. I claim it is not possible using just one. As Fedor Petrov suggests in another comment, this is ...
- 28.2k
18
votes
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Extending Apéry's proof to Catalan's constant?
Summary:
The continued fraction, the recurrence and the explicit form of the sequence are interchangeable and for the Apéry numbers, we don't know what come first. This extend to other constructions ...
15
votes
Ramanujan's $\tau(n)$ and continued fractions
Since the OP asked for other examples of this kind of numerology,I will give another one to support his observation
The function $\cos(\theta_{11})$ has the following closed form
$\cos(\theta_{11})=\...
- 256
14
votes
Accepted
$\text{SL}_2(\mathbb{Z})$ and continued fractions?
See Remark 2.2 here.
If you expand $p/q$ into a continued fraction then the successive convergents, as columns of a $2 \times 2$ matrix, have determinant $\pm 1$. Provided $p/q$ is in reduced form ...
- 50.6k
13
votes
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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"
We have $$C_2(-17,-6,-72)=-(5/8)L(\chi_{-3},2)$$ and
$$C_2(10,3,-9)=(1/2)L(\chi_{-3},2)$$ so both are proportional to what you
call Gieseking's constant but which is simply the value at 2 of the
L ...
- 13.1k
12
votes
Accepted
Are the coefficients of certain product of Rogers-Ramanujan Continued Fraction non-negative?
Notice that we can write
$$f(q)=\prod_{n\geq 1} (1-q^n)^{-\left(\frac{n}{5}\right)}$$
therefore
$$g(q)=\prod_{k\geq 1} f(q^k)=\prod_{n\geq 1} (1-q^n)^{-a(n)}$$
where $a(n)=\sum_{d|n}\left(\frac{d}{5}\...
- 85.6k
11
votes
Accepted
Average number of iterations for the Euclidean algorithm to terminate
This algorithm correspons to nearest integer continued fractions or centered continued fraction. The length of such fraction $l(a/b)$ can be expressed in terms of Gauss - Kuz'min statistics for ...
- 12.3k
11
votes
Accepted
On Zagier's missing continued fraction with multiple limits?
Set $Q=(1/2)L(\chi_{-3},2)$ (related to your Gieseking constant) and
$P=2\pi^2/81$. The limits are almost certainly (not proved),
\begin{align}
\lim_{m\to\infty}C_2(6m+0) &= -Q\\
\lim_{m\to\infty}...
- 13.1k
10
votes
Have new conjectures generated by the Ramanujan machine been proven?
The two formulas in the abstract were proven by relatively simple methods in a couple of days after the paper appeared on arxiv. See https://arxiv.org/abs/1907.05563
The rest of the formulas inside ...
- 209
10
votes
Accepted
Algebraic and rational parts of a real number
Let $\alpha$ be an irrational. We shall consider its continued fraction $[a_0;a_1,a_2,\dots]$. Recall some basic results about convergents of continued fractions (see e.g. here): letting $p_n,q_n$ be ...
- 28.2k
9
votes
When does $2$ arise when using the Euclidean algorithm to compute greatest common divisors?
Let me present a criterion that may or may not be more transparent than the definition.
If $a,b\ge0$ are coprime, let $b^{-1}\bmod a$ denote the least $u\ge0$ such that $ub\equiv1\pmod a$. (I.e., $b^{-...
- 47.1k
8
votes
Accepted
Can we use the Rogers-Ramanujan cfrac to parameterize the Fermat quintic $x^5+y^5=1$?
(Courtesy of a comment by Nemo who suggested Huber's paper.)
Part I. $x^5+y^5 = 1$
In "A Theory of Theta Functions to the Quintic base",
Tim Huber defines four theta functions which can be ...
- 12.6k
8
votes
Irrationality of generalized continued fractions
There are such examples in Ramanujan's Notebooks, Part 2, page 116
- 5,624
8
votes
Accepted
Irrationality of $e^{x/y}$
I think this might be a solution.
The Continued Fraction Expansion of the hyperbolic tanh function discovered by Gauss is
$$\tanh z = \frac{z}{1 + \frac{z^2}{3 + \frac{z^2}{5 + \frac{z^2}{...}}}} \\\\$...
- 177
7
votes
Limit of quotients of elements of special Fibonacci matrices
One can prove by induction that some of given ratios have nice continued fraction expansions:
\begin{gather*}
\frac{\alpha_n}{\beta_n }=[2;1^n,2,1^{n-1},\ldots,2,1,1,2,1]\to 1+\varphi=\varphi^2;\\
\...
- 12.3k
7
votes
Accepted
Riemann-Hilbert and orthogonal polynomials
Let $P_{n}(z)=\gamma_{n}z^{n}+\cdots$ be a sequence of orthonormal polynomials with respect to some weight $w$ on $\mathbb{R}$. Given an $n\geq0$, consider the following Riemann-Hilbert problem for ...
- 4,034
7
votes
Have new conjectures generated by the Ramanujan machine been proven?
Already many people pointed, but the proof of the first identity of the abstract is the following in short.
$3+\displaystyle \frac{-1}{
\displaystyle 4+ \frac{-2}{
\displaystyle 5+ \frac{-3}{
\...
- 171
7
votes
Is $\mathbb{Q}$ the orbit of a rational function under iteration?
A rational function is as a self-map of $\mathbb P^1$. With that understanding, as was noted earlier, it is possible to generate all of the points $\mathbb P^1(\mathbb Q)$ by starting with the point $...
- 47.4k
7
votes
Accepted
On four Ramanujan-type "Legendrian" sequences with a 3-term recurrence?
Stupid of me. As O. Gorodetsky mentions, these are classical:
$$F_1=(91\zeta(3)-2\pi^3\sqrt{3})/432$$
$$F_2=(28\zeta(3)-\pi^3)/64$$
$$F_3=(117\zeta(3)-2\pi^3\sqrt{3})/243$$
In addition, note that ...
- 13.1k
6
votes
Accepted
Are these two $q$-continued fractions equivalent?
In this answer we shall follow a path Ramanujan must have likely taken at some point, though we have no solid evidence that he actually did. We prove the claim in two different ways part I and Part II....
- 256
6
votes
Periods of the continued fraction expansions of Galois-conjugate quadratic-irrationals
The periodic part of the continued fraction for the Galois conjugate is always the mirror image of the periodic part for the original quadratic irrational. Here we are viewing the periodic part as a ...
- 20k
6
votes
Continued Fraction of Random Variables
A general approach to problems of this type, worked out for a slightly different continued fraction,
$$Y_n=X_n+1/X_{n-1},$$
is described in Random Continued Fractions: A Markov Chain Approach (2004). ...
- 188k
5
votes
Irrationality of generalized continued fractions
There is a family of generalized continued fractions which give irrational or rational values depending on a very simple but nontrivial parametrization. I'd like to use the notation
$$c = a_0 + {b_0\...
- 5,342
5
votes
Connection between Infinite continued fractions, elliptic integrals and AGM
This is not an answer.
As was noted in the comments, the function $C(x)$ can be expressed in terms of the Gamma function, see Corollary 1 on page 145 in Ramanujan's Notebook II which attributes this ...
- 8,597
5
votes
Have new conjectures generated by the Ramanujan machine been proven?
In the year or so since this question first appeared on MO, there has been further progress on proving some of these identities, for example by Kadyrov and Mashurov and Dougherty-Bliss and Zeilberger.
...
- 82.6k
5
votes
Accepted
Evaluation of hypergeometric type continued fraction
This is found in [1] $\S 82$, Satz 5. It covers the case where the numerator $a_n$ is polynomial of degree $2$ in $n$ and the denominator $b_n$ is degree $1$.
If I plugged in correctly, we get ...
- 41.1k
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