# Expressions in “continued” monotone functions

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?

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.