Continuous Hyper Bouncing Factorial

[JmsNxn]

You may know about my Hyper Bouncing Factorial function denoted , it is like the Bouncing Factorial, but with a starting number of two and you use exponentiation.
How could the Hyper Bouncing Factorial function be defined for non integers, like how the exponential factorial could possibly be extended to non integers in this way?

Lmao, good luck Catullus. This is way too hard to be solved.

Where as the Bouncing factorial looks something like:

\[
\prod_{c=1}^x \Gamma(c+1)\,
\]

Which is analytic in \(x\), and you'd just be adding an additional step to get the analytic bouncing factorial. The hyper bouncing factorial function is just impossible, because we can't even do this kind of hyper factorial.

I spent a good amount of time on the Hyper factorial and the closest I came to was very far away from the hyper factorial. I used it to justify that these things "could" be solvable.

I managed to solve:

\[
\Upsilon(s+1) - \Upsilon(s) = e^{s\Upsilon(s)}\\
\]

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.