Please, forgive my ignorance and my silly points and observations.... but really, I have not time to put in it the required time and effort.
First, I'd bet 100$ that those equations do not hold for non integer ranks... but they put in some way a constraint on the rank infinitesimally approaching integers ranks. So I hoped there could be a way to use those identities to force something on the ranks in \((0,\epsilon) \cup (1-\epsilon,1+\epsilon)\cup (2-\epsilon,2)\) - or regard this as a sum of open balls in \(\mathbb C\).
Secondly, the \(2<s+1>2=2<s>2=4\) identity is just one special case of the identities I showed above. That's why I believe that that is no not going to hold for non-integer ranks. In other words, since the rule breaks for zero-th rank, I'd expect an oscillating behavior of that function hitting the value \(4\) only at integers.
If I'm not misunderstanding completely what you are doing, that would rule out your choice of fixing to zero one coordinate of your surface... that would pump the complex dimension to three probably making impossible to parametrize it with just two complex parameters.
I'm sure you have good reason to claim that those identities are not going to help with your pasting-solution works, so I trust you.
Last thing: I cited infinite composition because those recurrence relations I showed you have as solutions finite iterated compositions, as I made evident... just as they were Gamma-function-like recurrence relations, but in the rank variable!! So, maybe, composition calculus could jump in, find the limit of those compositions, and then obtain non-integer ranks.
ps: just ignore my remark about the surface... I went back re-reading again what you wrote about it... and I'm not sure I know what I'm talking about... srry.
First, I'd bet 100$ that those equations do not hold for non integer ranks... but they put in some way a constraint on the rank infinitesimally approaching integers ranks. So I hoped there could be a way to use those identities to force something on the ranks in \((0,\epsilon) \cup (1-\epsilon,1+\epsilon)\cup (2-\epsilon,2)\) - or regard this as a sum of open balls in \(\mathbb C\).
Secondly, the \(2<s+1>2=2<s>2=4\) identity is just one special case of the identities I showed above. That's why I believe that that is no not going to hold for non-integer ranks. In other words, since the rule breaks for zero-th rank, I'd expect an oscillating behavior of that function hitting the value \(4\) only at integers.
If I'm not misunderstanding completely what you are doing, that would rule out your choice of fixing to zero one coordinate of your surface... that would pump the complex dimension to three probably making impossible to parametrize it with just two complex parameters.
I'm sure you have good reason to claim that those identities are not going to help with your pasting-solution works, so I trust you.
Last thing: I cited infinite composition because those recurrence relations I showed you have as solutions finite iterated compositions, as I made evident... just as they were Gamma-function-like recurrence relations, but in the rank variable!! So, maybe, composition calculus could jump in, find the limit of those compositions, and then obtain non-integer ranks.
ps: just ignore my remark about the surface... I went back re-reading again what you wrote about it... and I'm not sure I know what I'm talking about... srry.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)