Holomorphic semi operators, using the beta method

[tommy1729]

...

This is because:

\[
\left(\alpha \uparrow^s z+1\right) - \left(\alpha \uparrow^{s-1} \left( \alpha \uparrow^s z\right)\right)\Big{|}_{s \in \mathbb{N}} = 0\\
\]

I snipped the part I completely understood.

This statement seems intuitively correct to me.
But Im still a bit troubled.
why positive integer s is sufficient and implies it for positive real s.

Maybe im tired.
I can see both are in the same vector space so that is good.

And we have countable parameters what seems good too ( implying countable s will be sufficient )

Maybe im lazy now.

But this matters alot

regards

tommy1729