ramanujan and tetration

[bo198214]

Code:

by the time he gets to chapter 4
  he is ready to return to iterated exponentiation
and after defining

F (x) = x
0

F   (x) = exp{F (x)} - 1
r+1           r

he decomposes the iteration in two different ways

         oo                 oo
        ---                ---
        \             j    \           j
F (x) = /    phi (r) x   = /    f (x) r
r      ---     j          ---   j
        j=0                j=0

I was really curious how Ramanujan's continuous iteration of \( e^x \) would look like. But now I am a bit disappointed. What he considers is not iterated \( e^x \) but iterated \( e^x-1 \)!
Not that this would be entirely trivial, however this is a case where the function to iterate has a fixed point at 0 and there is only one way to obtain (continuous/fractional/real/complex/analytic) iterates of the formal powerseries and that is regular iteration.

So, though its amazing that he considered the topic of regular iteration at such an early time, he does not contribute towards analytic tetration, where the difficulty is exactly this \( f(0)\neq 0 \).

[Ivars]

in the formal setting
jumping to iterating exponentiation misses a whole lot of other growth orders

in fact
we can start with iterating the original functions
the numerator over x
and the denominator over factorial
and this is key to the generalisation needed
because there are many ways to ensure the correct asymptotic order conditions
using a variety of iterative techniques to build function orders

all of these lie between the realm of the exponential and ramanujan's beast
and there is an infinite hierarchy even beyond
each waiting for a theory to develop and interesting relations to find

Thanks galathaea for answer to my musings and further development.
I am interested in tetration ( and further) because it is the next obvisously integer order of infinity.

If rules and analogies for discrete enumerable by some integers orders of infinity can be established, than later it should be possible to cover all intermediate ranges by extensions to fractional and rational and real and complex change of orders of infinity as usually.

So I am looking to start with integers that enumerate these discrete orders of infinity and then look back into what is between exponentiation and tetration- if needed.

I was trying to link it to combinatorics of branching tree chains but can not find any basic text about this subject to even understand the conventions people working in them use.

One thought about trees is that divergent series most likely end up in different (almost?) continuous orders of infinity via tree type ( bifurcation, trifurcation, n furcation etc) structure. The correspondance between a type of series and order of infinity ?

Ivars