ramanujan and tetration

[Gottfried]

Code:

bruce berndt's release of ramanujan's notebooks
  is one of the best resources for aspiring combinalgebraicists

(...)

Hi galathaea -

I have seen similar formulae for the fractional indexed infinite sum sometimes, but never got into it.
I try to translate this to the current case; correct me if I'm completely false yet.
Assume, I have a function f(n) which gives

f(n)= b^^n + b^^(n+1) + b^^(n+2) +...

where thus

f(n)-f(n+1) = b^^n

is it then possible to get from this the half-iterate

f(n+1/2) - f(n+3/2) = b^^(n+1/2) ?

Is that the idea?

If this would be right, then could I use the same idea with the function

g(n) = (-1)^n*( b^^n - b^^(n+1) + b^^(n+2) -... )

which I actually have (seem to have...)?

My problem is the understanding of the general concept and technique of such an idea. I think, it is somehow similar to what Euler did, but he uses also an integral as remainder, where I had no idea how to apply this in our context.

Gottfried