Question

An infinite continued fraction of $x^n$

score 153 · Answer 1 · 2023-11-18 13:19:14

153

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An infinite continued fraction of $x^n$

Published on 18 Nov 2023 - 13:19

score 25 · Answer 2 · 2023-11-30 09:06:49

25

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Stuck at showing recursion property $q_n=c_nq_{n-1}+q_{n-2}$ of the continuants $q_n$ of $x$'s continued fraction expression $x=\frac{p_n}{q_n}$.

Published on 30 Nov 2023 - 9:06

score 59 · Answer 3 · 2023-12-01 10:58:40

59

Views

Trouble understanding Khinchin's proof of the recursion formulae $p_k=a_kp_{k-1}+p_{k-2};q_k=a_kq_{k-1}+q_{k-2}$ for continued fractions

Published on 01 Dec 2023 - 10:58

score 79 · Answer 4 · 2023-12-12 18:20:58

79

Views

Continued Fraction of the Square Root of the Ratio of Two Primes $\equiv 3 \mod 4$

Published on 12 Dec 2023 - 18:20

score 219 · Answer 5 · 2023-12-17 08:46:31

219

Views

A new Dottie number via Continued Fraction?

Published on 17 Dec 2023 - 8:46

score 61 · Answer 6 · 2023-12-20 00:23:21

61

Views

Proof $x^2-dy^2=-1$ is unsolvable in integers if $\sqrt{d}$ has continued fraction expansion of even period

Published on 20 Dec 2023 - 0:23

score 355 · Answer 7 · 2023-12-23 00:10:01

355

Views

Functional equation: $f(x) = x + 1 + \frac{f(x+1)}{f(x+2)}$ , $f(t) = 0$?

Published on 23 Dec 2023 - 0:10

score 120 · Answer 8 · 2023-12-24 20:22:48

120

Views

Natural continued fraction

Published on 24 Dec 2023 - 20:22

score 118 · Answer 9 · 2023-12-26 06:46:02

118

Views

Other "natural" simple continued fractions with closed-forms?

Published on 26 Dec 2023 - 6:46

score 25 · Answer 10 · 2023-12-26 16:01:36

25

Views

Given that x has the continued fraction expansion x = [1,$\bar{s}$], show that $ x = 1+ \frac{2}{s+\sqrt{4+s^2}}$

Published on 26 Dec 2023 - 16:01

score 42 · Answer 11 · 2023-12-29 10:44:37

42

Views

A ratio of Bessel functions with a similar continued fraction expansion to $e^{1/m}\,$?

Published on 29 Dec 2023 - 10:44

score 64 · Answer 12 · 2024-01-01 22:12:30

64

Views

Series involving continued fractions

Published on 01 Jan 2024 - 22:12

score 72 · Answer 13 · 2026-03-27 13:03:26

72

Views

Understanding the proof of a transcendental criterion regarding palindromic continued fraction.

Published on 27 Mar 2026 - 13:03

score 499 · Answer 14 · 2024-01-11 10:28:08

499

Views

Closed form of integral $\int_{0}^{\infty}\frac{e^{-x}}{1+x/\cdots}dx$ with continued fraction

Published on 11 Jan 2024 - 10:28

score 131 · Answer 15 · 2024-01-11 22:14:23

131

Views

Does the continued fraction $\frac{2}{3 - \frac{2}{3 - \ddots}}$ have two values?

Published on 11 Jan 2024 - 22:14

List Question

An infinite continued fraction of $x^n$

Stuck at showing recursion property $q_n=c_nq_{n-1}+q_{n-2}$ of the continuants $q_n$ of $x$'s continued fraction expression $x=\frac{p_n}{q_n}$.

Trouble understanding Khinchin's proof of the recursion formulae $p_k=a_kp_{k-1}+p_{k-2};q_k=a_kq_{k-1}+q_{k-2}$ for continued fractions

Continued Fraction of the Square Root of the Ratio of Two Primes $\equiv 3 \mod 4$

A new Dottie number via Continued Fraction?

Proof $x^2-dy^2=-1$ is unsolvable in integers if $\sqrt{d}$ has continued fraction expansion of even period

Functional equation: $f(x) = x + 1 + \frac{f(x+1)}{f(x+2)}$ , $f(t) = 0$?

Natural continued fraction

Other "natural" simple continued fractions with closed-forms?

Given that x has the continued fraction expansion x = [1,$\bar{s}$], show that $ x = 1+ \frac{2}{s+\sqrt{4+s^2}}$

A ratio of Bessel functions with a similar continued fraction expansion to $e^{1/m}\,$?

Series involving continued fractions

Understanding the proof of a transcendental criterion regarding palindromic continued fraction.

Closed form of integral $\int_{0}^{\infty}\frac{e^{-x}}{1+x/\cdots}dx$ with continued fraction

Does the continued fraction $\frac{2}{3 - \frac{2}{3 - \ddots}}$ have two values?

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