Recall continued fractions: <http://en.wikipedia.org/wiki/Continued_fraction>


Now take a look at this question: <https://math.stackexchange.com/questions/601846/the-limit-of-displaystyle-lim-n-to-infty-exp-1-exp-2-exp-3-ldots-exp/601890#601890>


You cannot help but notice that there is some resemblance between recursively applying something like 1/x:


$ x \to a\_n + 1 / x $


and recursively applying something like exp(-x):


$ x \to a\_n + \exp( -x ) $


It may be worth investigating the propertied of the map


$ x \to a\_n + f( x ) $


for functions $f$ that are monotone asymptotically decreasing to $0$ to ensure convergence.


Thus the question: were such things investigated before, apart from generalized continued fractions?



Expressions in "continued" monotone functions
In other words you want to write $f$ as a limit $g\_1\circ g\_2\circ g\_3\circ\ldots$, where
$g\_n(x)=a\_n+\exp(-x)$. Putting $h\_n(t)=\exp(- g\_n(-\log t))$, we obtain
$$h(t)=\exp f(\log t)=h\_1\circ h\_2\circ\ldots,$$
now $h\_n(t)=\lambda\_n e^{-t}, \lambda\_n=\exp(- a\_n)$. Infinite compositions of such functions and their limits were
studied since XVIII century (with no much success). Of the recent papers, you may look at
MR1400837 
Bender, C., Vinson, J.
Summation of power series by continued exponentials 
J. Math. Phys. 37 (1996), no. 8, 4103–4119, and references there.


