Thanks !
Dear Henri - How your answer relates to the specific original
question, arising from the fact that the continued fractions, which
are under discussion, are formed from the coefficients of the Engel's
expansion of certain number ?
Very limited in scope experimentation showed, by the way, that for 3
irrational numbers (out of 4, which I tried: 1) Pi-3; 2) Champernowne
constant; 3) Thue-Morse Constant) the above procedure (number->Engel's
expansion ->continued fraction) did not not converge, but yielded two
limit points.
However for the "e-1" - above procedure yielded complete convergence
to the SINGLE limit point ...
I also tried (one case only) to *simulate* the generation of the Engel
expansion coefficients using the (sequence) formula, which the for
first few terms produces values equal to the coefficients of Engel
expansion of Pi-3:
b_n = (n+2)*(n+1)/2 + 2*(-1)^n
(see
http://www.research.att.com/~njas/sequences/A135405)
In this case the continued fraction formed from such b_n also
yielded complete convergence to the SINGLE limit point ...
Thanks again,
Best Regards,
Alexander R. Povolotsky
===================================================
- Show quoted text -
On Fri, Jun 13, 2008 at 1:33 PM, Jean-Paul Allouche
<
Jean-Paul.Allouche@...> wrote:
> Dear Alexander
>
> Here is an answer of my posting concerning your question.
> best
> jp
>
>>>>>>>>>>>
>
> In response to JP Allouche's message, it is easy to check
experimentally that
> there are many examples of continued fractions of the type referred
to
> in his posting, in particular the reference to
>
http://www.austms.org.au/Publ/Bulletin/V72P1/pdf/721-5038-AnHi.pdf> For instance (but this is far from general), the cf in that paper is
a
> special case of the evaluation (q, a positive integers)
>
> (q-1)a+(q^0+qa^2)/((q-1)a+(q^1+qa^2)/((q-1)a+(q^2+qa^2)/....
>
> converge alternatively (even/odd) to qa and to qa+1/qa.
>
> This should be easy, and is reminiscent of Ramanujan's cf for theta
functions,
> although in that case |q|<1, while here q is a positive integer.
>
> Any references or ideas of proof ?
>
> H. Cohen
Date: Mon, 9 Jun 2008 22:52:52 -0400
Reply-To: Jean-Paul Allouche <[log in to unmask]>
Sender: Number Theory List <[log in to unmask]>
From: Jean-Paul Allouche <[log in to unmask]>
Subject: Engel and cont. fractions
Dear all I was asked the following question by Alexander R. 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. --- Has anyone partial answers
or hints for A. Povolotsky's question?
best jean-paul allouche >
--
Henri.Cohen@... wrote:
In response to JP Allouche's message, it is easy to check
experimentally that
there are many examples of continued fractions of the type referred to
in his posting, in particular the reference to
http://www.austms.org.au/Publ/Bulletin/V72P1/pdf/721-5038-AnHi.pdfFor instance (but this is far from general), the cf in that paper is a
special case of the evaluation (q, a positive integers)
(q-1)a+(q^0+qa^2)/((q-1)a+(q^1+qa^2)/((q-1)a+(q^2+qa^2)/....
converge alternatively (even/odd) to qa and to qa+1/qa.
This should be easy, and is reminiscent of Ramanujan's cf for theta
functions,
although in that case |q|<1, while here q is a positive integer.
Any references or ideas of proof ?
H. Cohen
