Let $x$ be real and let $n$ be a natural number.
Consider the following functional equation :
$$f(0) = 0$$ $$f(1)=1$$ $$f(x+1) = 1 + f(x) + f(x/2)^2 + f(x/3)^3 + f(x/4)^4+...$$
What are the solutions and what are the asymptotics ? How many free parameters do we have ? Are all solutions analytic ?
obviously related
$$g(0) = 0$$ $$g(1)=1$$ $$g'(x) = 1 + g(x/2)^2 + g(x/3)^3 + g(x/4)^4+...$$
And
$$h(0) = 0$$ $$h(1)=1$$ $$h(n+1) = 1 + h(n) + h(n/2)^2 + h(n/3)^3 + h(n/4)^4+...$$
Where the division is rounded below to the integer part.
Notice the connection to number theory.
If $n$ is close to an interval with many primes, it will relatively grow slower as usual. ( A similar thing probably happens for $f$ and $g$ )
My brutal estimate for $f,g,h$ and $x>0$ is
$$f(x) = O(g(x)) = O(h(x)) = O(\frac{\exp(x)}{x+1})$$
Using big-O notation.
Maybe asymptotics relate to number theory functions ? Or to differential equation ideas ?
I noticed
$$f(x) = O(\frac{\exp(x)}{x+1}) = O(\pi(\exp(x)))$$
maybe ??
Notice the higher derivatives of $f(x)$ can be given by the defining equation and taylors theorem. Not sure if that helps.
I wonder if and how $f(-x)$ and $f(x)$ relate and similar with $g$.
The focus here is on the real imput $x$ or natural integer $n$, although comments about complex $z$ ($f(z),g(z)$) are welcome ofcourse.