Dear all
I was asked the following question by Alexander Povolotsky
(
apovolot@...).
>>>>
Start with an Engel expansion $X := \{x_1, x_2, ...\} =
\frac{1}{x_1} + \frac{1}{x_1 x_2} + ...$ (the $x_i$'s are positive integers).
Now look at the generalized continued fraction
$1/(1+x_1/(1+x_2/(1+x_3...)))$ noted $1 / 1 + x_1 / 1 + x_2 /...$.
This continued fraction does not converge in general, but has two limit
points.
To what extent do the arithmetical nature of these limit points reveal
the nature of the real number $X$ we started from (or vice versa)?
>>>>
What I found myself in the literature is the following.
Take the generalized continued fraction $b_1 / a_1 + b_2 / a_2 ...$
where the $a$'s and the $b$'s are positive integers.
It has two limit points given as the limits of $p_{2k}/q_{2k}$
and $p_{2k+1}/q_{2k+1}$. See in particular
D. Angell, M. D. Hirschhorn, A remarkable continued fraction
Bull. Austr. Math. Soc. 72 (2005) 45--52
(
http://www.austms.org.au/Publ/Bulletin/V72P1/pdf/721-5038-AnHi.pdf ).
As indicated in that paper it seems difficult to evaluate the two limit
points for "simple" sequences of $a$'s and $b$'s.
But I don't have any clue for the initial question.
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Has anyone partial answers or hints for A. Povolotsky's question?
best
jean-paul allouche
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