Gottfried Wrote:Just some speculation:
Then we may take another series in the iterator; I don't remember at the moment which, but I think, there are some divergent series of rational or even integer summands, for whose value the replacement by I makes sense/is consistent. Then we had the same way
... f°c(f°b(f°a(x))) = f°(a+b+c+...)(x) = f°I(x)
But don't know, whether this makes sense ...
I have seen mentionings of so called Motzkin (first) and Shroder (second) divergent series having value \( -I \):
\( 1+2+4+9+21+51+...= 1+2+6+22+90+...= -I \)
Also, Catalan numbers based divergent series turn out interesting complex value:
\( 1+1+2+5+14+42+..=e^{-I*\pi/3} \)
It is all explained here: Divergent Series III
However, I have not seen such series giving \( I \)? Perhaps there are. \( -I \) is significant enough, but \( I \) would be a clear case to look at. These 3 sums are all located in lower half plane. Are there any in upper?
What seems clear, that complex counting and combinatorics are related, although no one seems to know how.
Ivars