galathaea Wrote: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