08/09/2022, 07:22 AM
(08/09/2022, 05:34 AM)Catullus Wrote:(08/09/2022, 05:01 AM)JmsNxn Wrote: I managed to solve:How did you solve it, and what solution did you obtain?
\[
\Upsilon(s+1) - \Upsilon(s) = e^{s\Upsilon(s)}\\
\]
Which turns out to be an entire function. Trying to even get close to anything near hyper factorial from here is leagues beyond me though. I have no idea. My goal was to solve "some kind of" difference equation involving exponentials.
Take the function:
\[
q(s,z) = z + e^{sz}\\
\]
If you take the sequence of functions:
\[
\begin{align}
&q(s-1,z)\\
&q(s-1,q(s-2,z))\\
&q(s-1,q(s-2,q(s-3,z)))\\
&\vdots\\
&\Omega_{j=1}^\infty q(s-j,z)\bullet z\\
\end{align}
\]
Then the limit was \(\Upsilon\), where \(z\) acts as an open parameter (the (spectrum) parameters exist for \(\Re(z) > 0\)). This function satisfies the equation, by construction:
\[
q(s,\Upsilon(s,z)) = \Upsilon(s+1,z)\\
\]
Which by linear substitution satisfies:
\[
\Upsilon(s+1,z) - \Upsilon(s,z) = e^{s\Upsilon(s,z)}\\
\]
These questions get really complicated though. And Catullus, I can't answer them. I can prove general normality theorems, and convergence theorems. But everything to do with the hyper factorial is anomalous and nulls every element of my proofs.
