To be precise:
Given a function $f(x)$ I want a function $g(t, x)$, $x,t \in \mathbb{R}$ or $\mathbb{C}$, such that $g(t, x)$ equals the $t$-time application of $f$ to the value $x$ for $t \in \mathbb{N}$,
This means, $g(0, x) = x, g(1, x) = f(x), g(2, x) = f(f(x))$, and g should be continuous in $t$.
It feels like I want something like an "analytic continuation", like the gamma function extending the discrete factorial function into a smooth transformation. Is there some algorithm that can be applied to get some kind of taylor expansion or even a closed form systematically?
For example,
if $f(x) = x+c$, then I want $g(t, x) = x+ct$
if $f(x) = ax$, then $g(t, x) = a^t x$
if $f(x) = ax+b$, then $g(t, x) = a^t x + \frac{b(a^t-1)}{a-1}$
the last one I managed to figure out the closed form for, but e.g. for $f(x) = x^2 + c$ the iterated function explodes into increasingly higher degree polynomials that I don't know how to handle.
My formal understanding of calculus is on the level of standard university courses, so if this goes way way deeper, please be gentle!
