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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 defining $x_0:=x$ and inductively $$ a_n:=\lfloor{x_n}\rfloor+1\qquad\text{and}\qquad x_{n+1}:=a_nx_n-1. $$ It is easy to check that $1\le a_1\le a_2\le a_3\le\dots$, and that the sequence of $a_i$'s is eventually constant if and only if $x$ is rational. This procedure yields expressions like the one displayed above, since $$ x = \frac{1+x_1}{a_0}=\frac{1+\large{\frac{1+x_2}{a_1}}}{a_0} =\frac{1+\large{\frac{1+\Large{\frac{1+x_3}{a_2}}}{a_1}}}{a_0} = \cdots $$ I looked for these in google, wikipedia, and standard texts like Perron's, but couldn't find them. Have they been studied? I ask because a high school student invented these before my eyes today, and I'd like to tell him what he's rediscovered (assuming that in fact these have been studied before).

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  • $\begingroup$ Yes. They exist, but are not widely used. $\endgroup$
    – Lucian
    Jul 20, 2014 at 19:47
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    $\begingroup$ @Lucian: could you give me a reference or keyword? $\endgroup$ Jul 20, 2014 at 19:49
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    $\begingroup$ I think one big reason these are rarer is because the fact that you can 'distribute' the division means that they can be equivalently written as $\frac1{a_0}+\frac1{a_0a_1}+\frac1{a_0a_1a_2}+\cdots$. I'd swear I've seen some discussion of the latter, particularly in conjunction with the series for the exponential (note that setting $a_i=i+1$ gives you a 'continued fraction' for $e$), but I'll have to go digging for refs. $\endgroup$ Jul 20, 2014 at 19:56
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    $\begingroup$ Ahh! Here it is. This is apparently known as the Engel expansion : en.wikipedia.org/wiki/Engel_expansion $\endgroup$ Jul 20, 2014 at 20:08
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    $\begingroup$ @Steven: thanks for the answer, and the kudos -- what the student did was even more impressive than it sounds, since we gave him zero background information and simply asked him to prove on day 1 that for any positive real numbers $a,b$ there exist nonzero integers $p,q$ with $|qa-p|<b$. He invented these Engel expansions on the spot, and used them to prove what we asked. $\endgroup$ Jul 20, 2014 at 20:17

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Expanding my comments into an answer: by distributing the divisions by $a_0$, $a_1$, $\ldots$ successively you can rewrite such an upward continued fraction in the equivalent form $\frac1{a_0}+\frac1{a_0a_1}+\frac1{a_0a_1a_2}+\cdots$. This is known as the Engel expansion of the number, and their coefficients have some interesting limiting properties (in particular, for almost all real numbers the coefficients grow exponentially); the Wikipedia article on them should offer up several good pointers for more information.

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  • $\begingroup$ The Engel expansion for $e$ is particularly nice. $\endgroup$ Jul 20, 2014 at 20:21
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Five years ago or so I worked on some "tree-type" continued fractions... $$ T = 1+\frac{\displaystyle 1+\frac{\displaystyle 1+\frac{\displaystyle 1+\frac{1+\cdots\;}{2+\cdots\;} }{\displaystyle 2+\frac{2+\cdots\;}{4+\cdots\;} } }{\displaystyle 2+\frac{\displaystyle 2+\frac{2+\cdots\;}{4+\cdots\;} }{\displaystyle 4+\frac{4+\cdots\;}{8+\cdots\;} } } }{\displaystyle 2+\frac{\displaystyle 2+\frac{\displaystyle 2+\frac{2+\cdots\;}{4+\cdots\;} }{\displaystyle 4+\frac{4+\cdots\;}{8+\cdots\;} } }{\displaystyle 4+\frac{\displaystyle 4+\frac{4+\cdots\;}{8+\cdots\;} }{\displaystyle 8+\frac{8+\cdots\;}{16+\cdots\;} } } } \;. $$

But it never went very far. I did submit one of these as a problem... Math. Mag. 79 (2006) p.151

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  • $\begingroup$ I've seen this in a quite long treatize with the name "the fifth operation" ("la quince operationi" or the like of someone fellow from Spain) and he even has made a book out of it. Don't have the link at the moment, perhaps this can be found using google or the webarchive. $\endgroup$ Nov 2, 2015 at 8:08
  • $\begingroup$ Here is a better reference, but not yet a link. Try webarchive with it: The Fifth Operation / La Quinta Operacion by Domingo Gómez Morín (2002-01-22) I don't know exactly why, but in the newsgroup sci.math he was handled as a slight crackpot because of the consequences he wanted to derive from his idea of such fractal continued fractions... $\endgroup$ Nov 2, 2015 at 8:13
  • $\begingroup$ Ah, here it is: web.archive.org/web/20081221211317/http://mipagina.cantv.net/… $\endgroup$ Nov 2, 2015 at 8:18
  • $\begingroup$ I had a recent question about this kind of fractions. They can be reworked into the generalized continued fraction form $\endgroup$
    – Yuriy S
    Mar 8, 2017 at 16:31
  • $\begingroup$ The author of the book mentioned in my earlier comment added a comment with the current link, but in an answer-box(now deleted because it is no answer to the question). Here is his comment : any thanks to Gottfried Helms for pointing out my work. The updated link that fully answer this question and other comments on this thread is: domingogomezmorin.wordpress.com The aforementioned bifurcated fractions are fully explained there as well as some other related stuff. $\endgroup$ Mar 9, 2017 at 1:51
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You can find plenty on information on those special continued fractions at the following link (American Mathematical Monthly):

https://www.facebook.com/AmerMathMonthly/photos/a.250425975006394.53155.241224542593204/1055084257873891/?type=3&theater

Also at: https://youtu.be/mjSHenvXsEs

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