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This question is inspired by the concept of "Differential Inclusion".

The ring of entire holomorphic functions is denoted by $Hol(\mathbb{C})$.

Is there a complete classification of all $f\in Hol(\mathbb{C})$ such that the $Hol(\mathbb{C})$-module generated by $\{f,f',f'',\ldots,f^{(n)},\ldots\}$ would be equal to the whole ring $Hol(\mathbb{C})$?

Is there a complete classification of all $f\in Hol (\mathbb{C})$ such that the $\mathbb{C}$_ module (or $Hol (\mathbb{C})$_ module generated by $\{f^{(n)}, f^{(n+1)}, \ldots\}$ is independent of $n$?

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    $\begingroup$ For the second question, every solution to linear ordinary differential equations of the form $$ f = \sum_{k = 1}^K \alpha_k f^{(k)} $$ where the $\alpha_k$ are holomorphic functions satisfies your condition, so there are a lot of functions of this type. (How would you even classify all ODEs of the above type?) $\endgroup$
    – Willie Wong
    Commented Nov 15, 2017 at 20:27
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    $\begingroup$ An obvious necessary condition is that $f$ can't have zeros of arbitrarily high order. Maybe I'm too naïve, but to me it feels as if this condition might well be sufficient also. $\endgroup$
    – Christian Remling
    Commented Nov 15, 2017 at 23:16

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