Oh, Tommy
I was not advocating to use this as a method of interpolating. I'm just saying, IT IS A METHOD OF INTERPOLATING. It's absolutely garbage, if you ask me. It contains next to zero functional data. But there is always a solution for this. For example, let's use \(\beta(z)\), base \(2\) arbitrary multiplier \(\lambda\). You can even make a faster growing \(\beta\) I don't really care.
\[
F(z) = \frac{\beta(2z)}{\Gamma(-z)} \sum_{n=0}^\infty \frac{2 \uparrow \uparrow n}{\beta'(2n)A(2n)(z-n)}
\]
for \(A(2n)\) the derivative of the recipricol gamma.
And voila, it interpolates tetration. It's hot garbage though. Probably not monotone, probably extra chaotic. Would be a bitch to even attempt at showing that it satisfies tetration equation; where spoiler alert, it doesn't.
I was just explaining that you can interpolate infinitely many different ways. And it's always possible. That's all I meant. Wasn't advocating for actually using it, lol.
I was not advocating to use this as a method of interpolating. I'm just saying, IT IS A METHOD OF INTERPOLATING. It's absolutely garbage, if you ask me. It contains next to zero functional data. But there is always a solution for this. For example, let's use \(\beta(z)\), base \(2\) arbitrary multiplier \(\lambda\). You can even make a faster growing \(\beta\) I don't really care.
\[
F(z) = \frac{\beta(2z)}{\Gamma(-z)} \sum_{n=0}^\infty \frac{2 \uparrow \uparrow n}{\beta'(2n)A(2n)(z-n)}
\]
for \(A(2n)\) the derivative of the recipricol gamma.
And voila, it interpolates tetration. It's hot garbage though. Probably not monotone, probably extra chaotic. Would be a bitch to even attempt at showing that it satisfies tetration equation; where spoiler alert, it doesn't.
I was just explaining that you can interpolate infinitely many different ways. And it's always possible. That's all I meant. Wasn't advocating for actually using it, lol.