The functions considered are positive real for positive real x, and monotonically increasing to infinity. Given such a function f(x), define the (nonlinear) operator * by

f*(x) := 1 + x/f(1) + x^2/( f(1)f(2) ) + x^3/( f(1)f(2)f(3)) + ...

= 1 + Sum[ x^n/ Product[f(k) , k = 1 to n] , n = 1 to infinity].

The new function f* is again positive real for positive real x and monotonically increasing to infinity. (In fact it is entire, not that we need that.) So this process can be iterated. It seems that, no matter what function we start with, the result converges pointwise.

The functions considered are positive real for positive real $x$, and monotonically increasing to infinity. Given such a function $f(x)$, define the (nonlinear) operator $f\mapsto f^*$ by $$ f^*(x) := 1 + \frac {x}{f(1)} + \frac{x^2}{ f(1)f(2)} + \frac{x^3}{ f(1)f(2)f(3)} +\dots = 1 + \sum\limits_{n=1}^{\infty} \frac{x^n}{\prod\limits_{k=1}^n f(k)}. $$ The new function $f^*$ is again positive real for positive real $x$ and monotonically increasing to infinity. (In fact $f^*$ is an entire function, not that we need that.) So this process can be iterated. It seems that, no matter what function we start with, the result converges pointwise.