(adsbygoogle = window.adsbygoogle || []).push({}); a function with certain conditions on derivatives at 0-generating functions

i don't know how to do superscripts, so let's say [f,n](x) is the nth derivative of f at x.

let A be the set of infinitely differentiable functions from I to R where I is some possibly infinite open interval containing 0.

for any c in R,

let S_c={f in A such that [f,n+1](0) = c [f,n](0) (1 - [f,n](0)) for all n>=0}.

what are the properties of S_c?

ideally, i'd like a statement like f is in S_c iff f is of the following algebraic form...

is S_c a proper subset of

T_c:={f in A such that [f,n+1](x) = c [f,n](x) (1 - [f,n](x)) for all n>=0 and all x in I}?

a closely related way to ask this is if

f(x)=SUM[(a[k] x^k)/k!, {k,0,oo}] and for all k,

a[k+1]=c a[k](1-a[k]),

what is f?

i'm trying to find a generating function for the sequence of iterates for the discrete logistic sequence. the things i've seen about generating functions only refer to linear recurrence relations.

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# A function with certain conditions on derivatives at 0

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