I got a function that is very close to tetration:
Let us take the function:
\( \vartheta(w) = \sum_{n=0}^\infty \frac{w^n}{n!(^ne)} \)
Then \( \vartheta \) is exponential order zero.
If \( \int_0^\infty |\vartheta(-w)|w^{\sigma-1}\,dw<\infty \) for \( 0 < \sigma < 1 \)
Then
\( \int_0^\infty \vartheta(-w)w^{s-1} \,dw = \frac{\Gamma(s)}{(^{-s} e)} \)
Which IS TETRATION and satisfies the recursion. However, I haven't been able to prove absolute convergence of that integral.
BUT! We can take:
\( \phi(s) = \frac{1}{\Gamma(s)}\int_0^\infty e^{-\lambda w}\vartheta(-w)w^{s-1}\,dw \)
for \( 0<\lambda<\epsilon \) and \( \phi(s) \approx \frac{1}{(^{-s}e)} \) for small \( \epsilon \) hopefully not too small to blow up.
Furthermore this can be made even better for \( \forall s \in \mathbb{C} \)
\( \frac{1}{(^s e)} \approx \psi(s) = \frac{1}{\Gamma(-s)} (\sum_{n=0}^\infty \frac{(-1)^n}{n!(^n e)(n-s)} + \int_1^\infty e^{-\lambda w}\vartheta(-w)w^{-s-1}\,dw) \)
These functions are so close to tetration it makes me want to punch a hole in the wall that I can' prove absolute convergence. -_- lol. It's Real to real and interpolates tetration too. The limit as \( \lambda \to 0 \) is tetration in both cases if it converges. but both functions are not the same \( \psi \neq \phi \).
Let us take the function:
\( \vartheta(w) = \sum_{n=0}^\infty \frac{w^n}{n!(^ne)} \)
Then \( \vartheta \) is exponential order zero.
If \( \int_0^\infty |\vartheta(-w)|w^{\sigma-1}\,dw<\infty \) for \( 0 < \sigma < 1 \)
Then
\( \int_0^\infty \vartheta(-w)w^{s-1} \,dw = \frac{\Gamma(s)}{(^{-s} e)} \)
Which IS TETRATION and satisfies the recursion. However, I haven't been able to prove absolute convergence of that integral.
BUT! We can take:
\( \phi(s) = \frac{1}{\Gamma(s)}\int_0^\infty e^{-\lambda w}\vartheta(-w)w^{s-1}\,dw \)
for \( 0<\lambda<\epsilon \) and \( \phi(s) \approx \frac{1}{(^{-s}e)} \) for small \( \epsilon \) hopefully not too small to blow up.
Furthermore this can be made even better for \( \forall s \in \mathbb{C} \)
\( \frac{1}{(^s e)} \approx \psi(s) = \frac{1}{\Gamma(-s)} (\sum_{n=0}^\infty \frac{(-1)^n}{n!(^n e)(n-s)} + \int_1^\infty e^{-\lambda w}\vartheta(-w)w^{-s-1}\,dw) \)
These functions are so close to tetration it makes me want to punch a hole in the wall that I can' prove absolute convergence. -_- lol. It's Real to real and interpolates tetration too. The limit as \( \lambda \to 0 \) is tetration in both cases if it converges. but both functions are not the same \( \psi \neq \phi \).
