Take the 2-minute tour ×  If I have a holomorphic function
$f(z)$, then I can write it as $$f(z)=\sum_{n=0}^\infty a_n
z^n.$$ So these functions can be viewed as a generating function
of the coefficients $\{a_n\}$ which have a certain structure.
There are nice finite differential problems, solved exactly by
some sequences $\{a_n\}$ encoded in a generating function. From
what I see, they are like a set of numbers with incredibly low
Kolmogorov complexity (see comments). The property of being
holomorphic is tied to the derivation $f'(z)$ being
defined/existent/computable for the function. And this
derivation is also used in the "algorithm" for generating the
$a_n$'s, namely the series expansion. Is this a valuable
perspective?  In this same vain, I wonder if there is a class of
functions, let's denote one by $F$, for which there are unique
$A$, such that  $$F(z)=\int A(t)z^t\ \mathbb{d}t.$$  Thoughts:
This reminds me of the Laplace transform, which is essentiall
the same thing, only with $z\equiv\mathbb{e}^{-s}