A function with certain conditions on derivatives at 0

[phoenixthoth]

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.

 

[fresh_42]

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.

$f^{(n)}(x)$

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

$A=C^\infty(I)\, , \,0\in I\subseteq \mathbb{R}$ open

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}.

$S_c := \{\,f\in C^\infty(I)\,|\,f^{(n+1)}(0)=c\cdot f^{(n)}(0)\text{ for all }n\in \mathbb{N}_0\,\}$

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}?

$T_c := \{ \, f \in C^\infty (I) \,|\, f^{(n+1)} (x) =c \cdot f^{(n)} (x) - c\cdot (f^{(n)}(x))^2 \text{ for all }n\in \mathbb{N}_0\, , \,x\in I\,\}$

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?

$f(x)=\sum_{k=0}^\infty a_k\dfrac{x^k}{k!}\, , \,a_{k+1}=ca_k -ca_k^2 \stackrel{?}{\Longrightarrow} f$

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.

It looks as if the ansatz with a Taylor series is the best approach. For $S_c$ there are only local conditions, so it makes sense to look at the local Taylor series. For $T_c$ there could be a possibility of an exponential function $A(x)e^{B(x)}$ if at all. But at first sight it could well be that $T_c= \{\,0\,\}$.