To further exemplify the relation to what I called Levenstein's equation, I'll start from scratch here.
Let's write:
\[
g_\lambda(e^{\lambda}w) = \frac{wf(g_\lambda(w))}{w+e^{\lambda}}
\]
Now, let's take the iteration:
\[
f^{\circ -n} \left(g_\lambda(e^{\lambda n}w)\right) = \varphi_\lambda(w)\\
\]
It became apparent very early on to sheldon, that there was a restriction on \(\lambda\) which allowed this object to converge, versus diverge. And additionally, there was no modification to the steps which could salvage it. I proved that Sheldon's guess was 100% correct--though I framed it as a result of "where \(\beta\) does converge, and where it can't".
Where by the function:
\[
\varphi_\lambda(e^{\lambda} w) = f(\varphi_\lambda(w))\\
\]
And this function will be holomorphic in a half plane in \(\lambda\), and holomorphic in \(w\) near the fixed point--but never in a neighborhood of the fixed point (there will be singularities near the fixed point).
GREAT WORK TOMMY! Your sum can be analytically continued to mine and Sheldon's approach, so this is fucking gold! Thanks a lot, Tommy!
EDIT:
I also called this the Mock Schroder Equation, because it was Schroder's equation with different multipliers.
Okay, this is even weirder! I think this is like the Complementary Levenstein, I'm too tired right now. Need to look at this tomorrow, lmao. But Great job again, Tommy!
I think you've just analytically continued \(\varphi_\lambda\)!
Let's write:
\[
g_\lambda(e^{\lambda}w) = \frac{wf(g_\lambda(w))}{w+e^{\lambda}}
\]
Now, let's take the iteration:
\[
f^{\circ -n} \left(g_\lambda(e^{\lambda n}w)\right) = \varphi_\lambda(w)\\
\]
It became apparent very early on to sheldon, that there was a restriction on \(\lambda\) which allowed this object to converge, versus diverge. And additionally, there was no modification to the steps which could salvage it. I proved that Sheldon's guess was 100% correct--though I framed it as a result of "where \(\beta\) does converge, and where it can't".
Where by the function:
\[
\varphi_\lambda(e^{\lambda} w) = f(\varphi_\lambda(w))\\
\]
And this function will be holomorphic in a half plane in \(\lambda\), and holomorphic in \(w\) near the fixed point--but never in a neighborhood of the fixed point (there will be singularities near the fixed point).
GREAT WORK TOMMY! Your sum can be analytically continued to mine and Sheldon's approach, so this is fucking gold! Thanks a lot, Tommy!
EDIT:
I also called this the Mock Schroder Equation, because it was Schroder's equation with different multipliers.
Okay, this is even weirder! I think this is like the Complementary Levenstein, I'm too tired right now. Need to look at this tomorrow, lmao. But Great job again, Tommy!
I think you've just analytically continued \(\varphi_\lambda\)!