Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*...

[Gottfried]

Hi Mr. Helms,

If we consider the solution to the infinite product exp(x) = M * x * f(x) * f^[2](x) *..., it has the same solution as the one we get for M = 1 because of the fraction appears in your calculation which makes M/M = 1. So, it's likely that a factor appears but it seems your calculation doesn't counts it. It seems more like the constant of integration but except that it isn't integration - not very useful, but just my 2 cents.

Balarka
.

Hi Balarka -

yes, that seems also to me the reason. But why is that factor just exp(1)? In the discussion in MSE I've tried to give an answer to this, but that what I got is merely an intuition yet, not yet a usable formalism.

I've one time also looked at the gamma-function in terms of functional iteration and arrived at the "incomplete gamma" where a seemingly similar effect appears. It might be interesting to compare this.(If you're interested, here's the link: http://go.helms-net.de/math/musings/Unco...gGamma.pdf , see specifically at pg 12)

What I'm after with this is to see, how iterates of exp(x) might look expressed with that function f(x). (or whether it might be more sensical to discuss iterates of exp(x-1) in this context... )

Gottfried