Talk:Generalized continued fraction

message: The LaTeX in the series section is all messed up, starting with Euler's formula.  I don't know this stuff well so I would prefer to leave it to somebody else to fix.  I think frac needs to be changed to cfrac, and parentheses need to be moved around.  Adking80 (talk) 14:29, 29 June 2009 (UTC)
thread_title: Incorrect formatting in series section

message: Today I added a (partial) history section to this article. More information about developments in the 19th and 20th centuries is needed to bring the account up to the present day. If anyone who reads this is knowledgeable in that area, please contact me. Or add to the history section that's already started, if you like.
I'm particularly curious about one thing. I'm fairly certain that the uniqueness of the canonical continued fraction representation of an irrational real number was not a subject of discussion until sometime during the 19th century. It just makes sense that this result would have been roughly contemporaneous with Kronecker's research into Cauchy sequences and Dedekind's new idea of a cut in the rationals. But I'm not sure who came up with the canonical representation in the first place. Lagrange used canonical fractions for quadratic irrationals, but he didn't know about transcendental numbers, which didn't really become a hot issue until algebraic number theory was fairly far along. Does anybody have a clue, or an idea of how I should search for this factoid?  DavidCBryant 01:06, 16 December 2006 (UTC)
thread_title:  History section 

message: Can someone state the necessary and sufficient conditions for a convergent generalized continued fraction to converge to an irrational number? 
For instance if 
:
x = b_0 + \underset{i=1}{\overset{\infty}{K}} \frac{a_i}{b_i}\,
is known to converge and
: a_i \, and  b_i \,  are both functions of  i \,
and 
: a_i \, and  b_i \,  have positive integer values that increase as  i \, increases to infinity
  
does this imply that  x \, is irrational?
Frank M Jackson (talk) 17:44, 6 November 2008 (UTC)
replies: text: Have now found references to Lambert's Irrationality proofs for generalized continued fractions and have added a section with appropriate references. Frank M Jackson (talk) 11:22, 8 November 2008 (UTC)
thread_title: Conditions for irrationality

message: Since, to quote the section on Notation:
"The long continued fraction expression ... takes up a lot of space in a book (and it's not easy for the typesetter, either)"
would it not be better to use the short expression
: x = b_0 + \frac{a_1}{b_1+} \frac{a_2}{b_2+} \frac{a_3}{b_3+} \frac{a_4}{b_4+} \cdots\, 
in place of the long one
: x = b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + {\ddots\,}}}}} 
after the introduction?
Here's how the Examples section would appear after the suggested change:
========================================================================
Here are two continued fractions that can be built via Euler's identity.
:
\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots=
\frac{x}{1+}\frac{1^2x}{2-x+}\frac{2^2x}{3-2x}+\frac{3^2x}{4-3x}+\frac{4^2x}{5-4x}\cdots\,
:
\ e^x=1+x+\frac{x^2}{2!}+\cdots=
1+\frac{x}{1-}\frac{x}{x+2-}\frac{2x}{x+3-}\frac{3x}{x+4-}\frac{4x}{x+5-}\cfrac{5x}{x+6-}\cdots
More advanced techniques are necessary to construct the following examples.
:
\ e^{2m/n}=1+\frac{2m}{n-m+}\frac{m^2}{3n+}\frac{m^2}{5n+}\frac{m^2}{7n+}\frac{m^2}{9n+}\frac{m^2}{11n+}\cdots
Setting m = x and n = 2 yields
:
\ e^x=1+\frac{2x}{2-x+}\frac{x^2}{6+}\frac{x^2}{10+}\frac{x^2}{14+}\frac{x^2}{18+}\frac{x^2}{22+}\cdots
:
\pi=3+\frac{1}{6+}\frac{9}{6+}\frac{25}{6+}\frac{49}{6+}\frac{81}{6+}\frac{121}{6+}\cdots
:
\pi = \frac{4}{1 +} \frac{1}{3 +} \frac{4}{5 +} \frac{9}{7 +} \frac{16}{9 +} \frac{25}{11 +} \frac{36}{13 +} \frac{49}{15 +} \cdots
Glenn L (talk) 03:06, 5 March 2009 (UTC)
replies: text: Sounds good to me. Robinh (talk) 08:04, 5 March 2009 (UTC)
thread_title:  Notation: suggest changing to short expression in place of long 

message: Here are two continued fractions that can be built via Euler's identity.
:
\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots=
\frac{x}{1+}\frac{1^2x}{2-x+}\frac{2^2x}{3-2x}+\frac{3^2x}{4-3x}+\frac{4^2x}{5-4x}\cdots\,
:
\ e^x=1+x+\frac{x^2}{2!}+\cdots=
1+\frac{x}{1-}\frac{x}{x+2-}\frac{2x}{x+3-}\frac{3x}{x+4-}\frac{4x}{x+5-}\cfrac{5x}{x+6-}\cdots
More advanced techniques are necessary to construct the following examples.
:
\ e^{2m/n}=1+\frac{2m}{n-m+}\frac{m^2}{3n+}\frac{m^2}{5n+}\frac{m^2}{7n+}\frac{m^2}{9n+}\frac{m^2}{11n+}\cdots
Setting m = x and n = 2 yields
:
\ e^x=1+\frac{2x}{2-x+}\frac{x^2}{6+}\frac{x^2}{10+}\frac{x^2}{14+}\frac{x^2}{18+}\frac{x^2}{22+}\cdots
:
\pi=3+\frac{1}{6+}\frac{9}{6+}\frac{25}{6+}\frac{49}{6+}\frac{81}{6+}\frac{121}{6+}\cdots
:
\pi = \frac{4}{1 +} \frac{1}{3 +} \frac{4}{5 +} \frac{9}{7 +} \frac{16}{9 +} \frac{25}{11 +} \frac{36}{13 +} \frac{49}{15 +} \cdots
Glenn L (talk) 03:06, 5 March 2009 (UTC)
replies: text: Sounds good to me. Robinh (talk) 08:04, 5 March 2009 (UTC)
thread_title: ============================================================

message: ¿How many external links does wiki accept?
[User:Mirificium|Mirificium]] (talk) 22:32, 20 March 2010 (UTC)
thread_title: EXTERNAL LINKS

message: The article on Fundamental recurrence formulas discusses a term that may not be widely used and has content that would be better presented in context here. I propose merging it into this article. RockMagnetist (talk) 17:20, 14 January 2013 (UTC)
thread_title:  Merge discussion 

message: === Partial numerators ===
Is it worth making it explicit that a transformation that makes the partial numerators equal to 1 does not make the continued fraction regular or in canonical form, since the partial denominators will not in general, perhaps ever, be integers? Ian S (talk) 14:09, 12 January 2014 (UTC)
thread_title:  Particular equivalence transformations 

message: Is it worth making it explicit that a transformation that makes the partial numerators equal to 1 does not make the continued fraction regular or in canonical form, since the partial denominators will not in general, perhaps ever, be integers? Ian S (talk) 14:09, 12 January 2014 (UTC)
thread_title:  Partial numerators 

message: No references are cited and it is far from obvious that the sequence {di} where d1 = 1/b1 and otherwise dn+1 = 1/(bnbn+1) makes the partial denominators equal to 1. Ian S (talk) 14:09, 12 January 2014 (UTC)
thread_title:  Partial denominators 

message: Does anyone have a reference for it? I'd like to know who invented it, and if there are any faster converging series Muu9 (talk) 14:23, 13 September 2022 (UTC)
thread_title:  Question about the 3rd CF for Pi 

message: :
The golden number is a funny choice of example for this; it has been called "the most irrational number" because its convergence is the slowest possible! —Tamfang (talk) 03:08, 7 June 2023 (UTC)
thread_title:  the golden ratio