A is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number.

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Asymptotic solutions of a sparsely perturbed recurrence relation

Recurrence relation I am trying to find approximate solutions $T(n)$ of the recurrence relation $$ p\ T(n-1) - \left[p+q+\overline{S} + \varepsilon \tilde{S}(n)\right]T(n) + q\ T(n+1) = 0,\\ \text{...
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28 views

Textbooks on transcendence theory

Is there a nice, modern textbook (some lecture notes or survey would do, too) that covers the main results and methods from transcendence theory? Ideally, it should also have some good exercises. So ...
0
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1answer
45 views

How to find cases where $m^2$ is near to $2^A$?

In another problem here in MSE I ran into the question how I can (practically, in a program) find (positive) integer $m$ such that they are "near" to perfect powers of $2$, so $$ (0 \lt ) \qquad d_m ...
4
votes
2answers
137 views

Upward continued fractions

Has anybody seen "upward continued fractions", such as $$ \frac{1+\large{\frac{1+\large{\frac{1+...}{a_2}}}{a_1}}}{a_0} \quad? $$ These can be formed, for any real number $x$ with $0<x\le 1$, by ...
3
votes
1answer
135 views

Continued Fraction: Please prove $\frac{1}{e \gamma (x+1,1)}=x+\frac{1}{x+1+\frac{2}{x+2+\frac{3}{x+3+\frac{4}{\dots}}}}$

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 ...
5
votes
1answer
131 views

Rogers-Ramanujan continued fraction in terms of theta functions?

The Rogers-Ramanujan cfrac is, $$r = r(\tau)= \cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\ddots}}}$$ If $q = \exp(2\pi i \tau)$, then it is known that, $$\frac{1}{r}-r =\frac{\eta(\tau/5)}{\eta(5\...
5
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1answer
72 views

Two similar integration about continued fractions

Prove that \begin{align*} \int_0^{+\infty} \cfrac{\sin nx}{x + \cfrac{1}{x + \cfrac{2}{x + \cfrac{3}{x + \cdots}}}} \, dx &= \cfrac{\sqrt {\cfrac{\pi }{2}} }{n + \cfrac{1}{n + \cfrac{2}{n + \cfrac{...
1
vote
2answers
52 views

Are the numerator and the denominator of a convergent of a continued fraction always coprime?

Is it true that for a convergent $A_k = p_k/q_k$ of the continued fraction expansion $[a_0;a_1,a_2,\dots]$ the numerator $p_k$ and the denominator $q_k$ are always coprime? If yes, how would one show ...
20
votes
2answers
550 views

Evaluation of a continued fraction

Puzzle question... I know how to solve it, and will post my solution if needed; but those who wish may participate in the spirit of coming up with elegant solutions rather than trying to teach me how ...
0
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43 views

Prove that $p_nq_{n-1}$ - $p_{n-1}q_n=(-1)^{n-1}$ for $p_{-2}=0$ $p_{-1}=1$ $q_{-2}=1$ $q_{-1}=0$

Let $p_n$/$q_n$ for $n=0,1,2,..$ be the convergents of $a∈ R$ $p_{-2}=0$ $p_{-1}=1$ $q_{-2}=1$ $q_{-1}=0$ $p_n= a_np_{n-1}+p_{n-2}$ $q_n= a_nq_{n-1}+q_{n-2}$ I need to prove that $p_nq_{n-1} - p_{...
2
votes
1answer
48 views

Square root of continued fraction

Assuming I've been given a number in the form of a continued fraction. Is there a general way to write the square root of that number as continued question? For example, consider $$1+\sqrt{2} = 2+\...
5
votes
4answers
168 views

Find the continued fraction of the square root of a given integer [duplicate]

How to find the continued fraction of $\sqrt{n}$, for an integer $n$? I saw a site where they explained it, but it required a calculator. Is it possible to do it without a calculator?
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0answers
17 views

Q-D scheme, continued fractions

What is a Q-D scheme for a continued fraction? I am reading this text on numerical evaluation of the H-function and the author suggests using continued fractions as done by many other special ...
16
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1answer
370 views

Nested solutions of a quadratic equation.

A quadratic equation of the form $x^2+bx+c=0$ can be solved with the classical formula that gives all solutions. Here I want discuss some other methods to find one solution. The best known is by ...
7
votes
2answers
256 views

a conjecture of certain q-continued fractions

Given the squared nome $q=e^{2i\pi\tau}$ with $|q|\lt1$, define, $$\begin{aligned}F(q)=\cfrac{1-q^2}{1-q^3+\cfrac{q^3(1-q)(1-q^5)}{1-q^9+\cfrac{q^6(1-q^4)(1-q^8)}{1-q^{15}+\cfrac{q^9(1-q^7)(1-q^{11})}...
1
vote
1answer
41 views

Convert partial fraction to continued fraction?

Lets say you have a partial fraction of the form: $$ f(x) = a_0 + \sum_{n=0}^{\infty} \frac{a_n}{\lambda_n + x} $$ Can anyone explain to me, in mildly plain English, how to convert this partial ...
0
votes
1answer
21 views

Show simple continued fraction with Euclid's Algorithm

I have this math problem, I have to show a simple continued fraction from equations. Here's the question. Use these equations: $$397 = 204(1) + 193$$$$204 = 193(1) + 11$$$$193 = 11(17) + 6$$$$11 =...
5
votes
1answer
77 views

Where to learn about continued fractions?

I sought a convergent subsequence of $\sin n$, and I met a proof that utilizes continued fractions. I always stumble upon things related to them as well. So, I would like to learn about them. What is ...
8
votes
2answers
161 views

the ratio of jacobi theta functions and a new conjectured q-continued fraction

Given the squared nome $q=e^{2i\pi\tau}$ with $|q|\lt1$, define $$\begin{aligned}H(q)=\cfrac{2(1+q^2)}{1-q+\cfrac{(1+q)(1+q^3)}{1-q^3+\cfrac{2q^2(1+q^4)}{1-q^5+\cfrac{q^3(1+q)(1+q^5)}{1-q^7+\cfrac{q^...
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0answers
16 views

Rapidly converging regular infinite continued fractions - anything special about them?

Say we have a simple continued fraction with rapidly increasing terms. Then it obviously converges very quickly and has a very good rational approximation. But is there anything special or ...
5
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2answers
247 views

The ratio of jacobi theta functions

Let $q=e^{2\pi i\tau}$. If $\theta_2$ and $\theta_3$ are jacobi theta functions , is it true that the ratio of the two functions can be expressed as a continued fraction of the form $$ \frac{\theta_2(...
0
votes
3answers
65 views

How Can I calculate this expression?

I have this repeating expression $5+\dfrac {6} {5+\dfrac {6} {5+..}}$ I saw a solution on a book. which is: $5+\dfrac {6} {5+\dfrac {6} {5+..}}=x$ $5+\dfrac {6} {x}=x$ $x^2-5x-6=0$ $x=6 $ or $x=...
0
votes
1answer
92 views

Determine convergents for square root

All square roots can be represented as a continued fraction. The fraction can be calculated to $n$ terms (e.g. $\sqrt{2}$ is $[1; 2, 2, 2, 2...]$) So the continued fraction for $\sqrt{2}$ to $3$ ...
5
votes
2answers
43 views

Can we define sum and product of two irrational numbers using Cauchy sequences of their simple continued fraction convergents?

There is a lot of questions about sum and product of irrationals here, so I hope you'll bear with me. Simple continued fraction is a very convenient way to represent any number since every real ...
4
votes
1answer
60 views

While resolving a continued fraction by simplifying it into a quadratic equation, does the extraneous root have any significance?

Suppose we want to find the value of the the following expression, with infinite terms: $$ y = {1\over 1+ {1\over 1+ {1\over 1+ {1\over \ddots}}}} $$ To solve this, we follow the following procedure, ...
5
votes
1answer
197 views

conjectured new generating function of fibonacci numbers

I conjecture a new generating function for the fibonacci numbers $F_{n}$. Given,the following conjectured q-continued fraction $$\chi(q)=\cfrac{1}{1+q-\cfrac{(1+q^2)}{1+q^3+\cfrac{q^2(1-q)(1-q^3)}{1+...
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0answers
38 views

What kinds of functions can be expanded into a form of an infinite continued fraction?

In this 1943 paper by Hudson on the theory of elastic waves in beams the author offers an interesting way to calculate a function defined as $$ \theta_n (z)=\frac{z J'_n (z)}{J_n (z)} $$ with $z$ ...
3
votes
1answer
68 views

Coloring rational numbers

Here is my problem. Fix a color for the number $1$, for example yellow. Choose another color, for example green. Now, for a positive rational denoted $x$, there are two rules : $x$ and $1/x$ have ...
3
votes
0answers
99 views

conjectured identity of the product of two theta functions

Looking into the discussion in this post,I was naturally led to consider the following general identity Given the two jacobi theta functions,$$\theta_2(q)=\sum_{n=-\infty}^\infty q^{(n+1/2)^2}$$ and $...
5
votes
2answers
193 views

a certain simple continued fraction

Given the golden ratio: $$\phi=\frac{1+\sqrt{5}}{2}$$ and the following simple continued fraction: $$G(q,k)=\cfrac{1}{1-q+\cfrac{1}{1-{q^3}^k+\cfrac{1}{1-{q^5}^k+\cfrac{1}{1-{q^7}^k+\ddots}}}}$$ For ...
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0answers
69 views

Two irrational numbers are congruent iff the tails of their infinite continued fractions eventually coincide

We say that a real number $\alpha$ is $congruent$ to real number $\beta$ if there exist integers a, b, c and d with ad-bc=+1 or -1 and such that $$\alpha=\frac{a\beta +b}{c\beta+d}$$ I need to prove ...
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1answer
43 views

Why is this continued fraction expansion what it is?

We have to find the continued fraction expansion of the roots of $1553 t^2 + 6014 t + 5820 = 0$, that is, $(\sqrt{14356} - 6014) / 3106$ Simplifying, $(\sqrt{3589}- 3007) / 1553$ The continued ...
8
votes
1answer
122 views

are these two continued fractions equivalent?

I would like to pose the following conjecture.Given $$\phi(q) =\cfrac{1}{1-q+\cfrac{q(1-q)^2}{1-q^3+\cfrac{q^3(1-q^2)^2}{1-q^5+\cfrac{q^5(1-q^3)^2}{1-q^7+\ddots}}}}$$ and $$\psi(q)=\cfrac{-q}{1-q+\...
3
votes
1answer
91 views

an interesting q-series and a certain continued-fraction

My aim is to find a rigorous proof of the following conjectured identity.Given $$1+q+q^2-q^4-q^5+q^7+q^8-q^{10}-q^{11}+\ddots=\cfrac{1}{1-q+\cfrac{(q^3)}{1-q^3+\cfrac{q^2(1+q)(1+q^3)}{1-q^5+\cfrac{q^...
1
vote
2answers
167 views

a conjectured new generating function of narayana's sequence

In the 14th century ,an Indian mathematician T.V Narayana came up with a sequence now named after him.The sequence satisfies the recurrence $$a_{n}=a_{n-1}+a_{n-3}$$ Starting with $a_{0}=a_{1}=1$, ...
12
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0answers
146 views

Bi-linear relation between two continued fractions

We know that any positive real number $x$ can be represented as a simple continued fraction $$x = a_{0} + \dfrac{1}{a_{1} + \dfrac{1}{a_{2} + \dfrac{1}{a_{3} + \cdots}}} = [a_{0}, a_{1}, a_{2}, a_{3}, ...
6
votes
3answers
97 views

Find the value of this infinite term

goes on till infinity. I get two solutions by rewriting the term in the form of the equation $x = 3-(2/x)$, which are $1$ and $2$. But in my opinion this term should have only one possible value. ...
2
votes
0answers
41 views

Riemann Zeta continued fraction approximants

In the paper Continued-Fraction Expansions for the Riemann Zeta Function and Polylogarithms by Djurdje Cvijovic and Jacek Klinowski, there is a claim that I cannot reproduce. In the abstract they ...
4
votes
1answer
220 views

When $x=2$ in the infinitely continued fraction $x+\frac{1}{x^2+\frac{1}{x^3+\frac{1}{x^4+…}}}$, what algebraic value does it converge to?

Say you have the infinitely continued fraction: $$x+\frac{1}{x^2+\frac{1}{x^3+\frac{1}{x^4+...}}}$$ When $x=1$, you can see that it's $$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+...}}}$$ which converges ...
3
votes
2answers
96 views

Why am I getting two answers for 8th root of continued fraction

Find value of $x$: $x=\sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207-....and\,so\, on}}}$ On solving ,we have $x^8=2207-\frac{1}{x^8}$ $x^8+\frac{1}{x^8}=2207$ $x^4+\frac{1}{x^4}=47$ $x^2+\frac{1}{x^2}...
2
votes
2answers
78 views

Continued Fraction Counting Problem

The house of my friend is in a long street, numbered on this side one, two, three, and so on. All the numbers on one side of him added up exactly the same as all the numbers on the other side of him. ...
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0answers
83 views

Is there an advantage in using continued fractions for Catalan or Fibonacci-Lucas primality tests?

I am studying the basic theory about continued fractions and also reviewed here at MSE former questions to learn more. While reviewing the questions and answers, I found references to the Fibonacci ...
3
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0answers
152 views

a q-continued fraction related to the octahedral group

Let $q=e^{2\pi i\tau}$. If $u(\tau)$ is Ramanujan's octic continued fraction, $$u(\tau)=\cfrac{\sqrt{2}\,q^{1/8}}{1+\cfrac{q}{1+q+\cfrac{q^2}{1+q^2+\cfrac{q^3}{1+q^3+\ddots}}}}$$ is it true that ...
12
votes
1answer
131 views

Continued fraction estimation of error in Leibniz series for $\pi$.

The following arctan formula for $\pi$ is quite well known $$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\tag{1}$$ and bears the name of Madhava-Gregory-Leibniz series after ...
3
votes
0answers
117 views

Continued fraction for $\int_{0}^{\infty}(e^{-xt}/\cosh t)\,dt$

In one of the comments to a question I posted on MSE, I got this wonderful continued fraction $$\int_{0}^{\infty}\frac{e^{-xt}}{\cosh t}\,dt = \frac{1}{x +}\frac{1^{2}}{x +}\frac{2^{2}}{x +}\frac{3^{2}...
8
votes
1answer
133 views

Are the unit partial quotients of $\pi, \log(2), \zeta(3) $ and other constants $all$ governed by $H=0.415\dots$?

Khinchin showed that given the simple continued fraction of a real number, $$r = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1} {\ddots}}}$$ then it is almost always true that the partial quotients $a_i$ ...
5
votes
2answers
346 views

What is the geometric average of 2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10…?

It is known that the Khinchin constant is not the geometric mean of the first $n$ coeffecients, as $n$ approaches infinity, of the continued fraction of e, which is $$[2; 1, 2, 1, 1, 4, 1,1, 6, 1, 1, ...
0
votes
2answers
63 views

Visualisation of the reciprocal of an continued fraction?

If: $$a=\cfrac{l}{m+\cfrac{n}{o+\cfrac{p}{q+\cdots}}}$$ Then could you help me visualize $1/a$? I really don't understand it. Thank you so much!
3
votes
1answer
186 views

A new $q$-continued fraction of order $12$

I think I may have discovered a $q$-continued fraction of order $12$ with a form different from that established by Mahadeva Naika. Let $q=e^{2i \pi \tau}=\exp(2i \pi \tau)$, then, $$\begin{aligned} ...
2
votes
1answer
56 views

Continued fraction manipulation

I have the following continued fraction $$ \frac{1}{a_1x+}\;\;\frac{1}{b_1+}\;\;\frac{1}{a_2x+}\;\;\frac{1}{b_2} $$ The paper I am reading then converts this to the following continued z-fraction ...