Hey, Mphlee
This very much reminds me of Andy's slog method for Tetration. Specifically the initial iteration. By which, you would have analycity for \(\sigma \in \mathbb{R}^+/\mathbb{N}\). The trick would be making it analytic at the natural numbers.
These constructions appear more commonly in logic theory/computation theory. They don't really appear in analysis, because, sadly, I think it would be very difficult to massage this construction into analycity. Even if we could massage, it would definitely be very hard to prove convergence of the Taylor expansion which results.
For example, what you could do to massage this is write:
\[
\zeta^1(\epsilon) = b^{\epsilon}\\
\]
Your right, this creates a continuous solution at the endpoints \(\sigma =0,1\). But it wouldn't be continuous in the first derivative. To make it continuous in the first derivative we need that:
\[
\frac{d}{d\epsilon}\Big{|}_{\epsilon = 0,1}\zeta_b^2(\epsilon) +_{k+1} n = \frac{d}{d\epsilon}\Big{|}_{\epsilon =0,1}b +_{k+\epsilon} n\\
\]
Now, you'd have to make an iterative procedure to successively get this to go to the second derivative:
\[
\frac{d^2}{d\epsilon^2}\Big{|}_{\epsilon=0,1} \zeta_b^3(\epsilon) +_{k+1} n = \frac{d^2}{d\epsilon^2}\Big{|}_{\epsilon =0,1}b +_{k+\epsilon} n
\]
Then, if you create a good enough sequence of functions \(\zeta^n\), such that they converge uniformly (for analycity you're going to want this on a non trivial domain in \(\mathbb{C}\), which would be very tricky), then \(\lim_n \zeta^n= \zeta\) would give you an analytic solution (theoretically). But there'd be a lot of steps. This is very similar to the main principle of Andy's slog. So you'd probably get a matrix formula for the various taylor coefficients about \(0\) and \(1\), and assuring they align (they satisfy the conjugate identity). Then your stuck with showing the taylor series converges, which will certainly be the hardest part.
The idea you are suggesting does exist. I can only point to Joel David Hamkins talking about it, which is very similar to what you are doing here. https://mathoverflow.net/questions/14678...-the-reals, read the first answer, which is a very similar breakdown of what you have here, but he does it more generally. This though, largely subscribes itself to logic/computation stuff like that, as it isn't concerned necessarily with much more than continuity (or in some cases continuously differentiable).
I don't quite understand what you are getting at with \(\Theta\), could you elaborate?
This very much reminds me of Andy's slog method for Tetration. Specifically the initial iteration. By which, you would have analycity for \(\sigma \in \mathbb{R}^+/\mathbb{N}\). The trick would be making it analytic at the natural numbers.
These constructions appear more commonly in logic theory/computation theory. They don't really appear in analysis, because, sadly, I think it would be very difficult to massage this construction into analycity. Even if we could massage, it would definitely be very hard to prove convergence of the Taylor expansion which results.
For example, what you could do to massage this is write:
\[
\zeta^1(\epsilon) = b^{\epsilon}\\
\]
Your right, this creates a continuous solution at the endpoints \(\sigma =0,1\). But it wouldn't be continuous in the first derivative. To make it continuous in the first derivative we need that:
\[
\frac{d}{d\epsilon}\Big{|}_{\epsilon = 0,1}\zeta_b^2(\epsilon) +_{k+1} n = \frac{d}{d\epsilon}\Big{|}_{\epsilon =0,1}b +_{k+\epsilon} n\\
\]
Now, you'd have to make an iterative procedure to successively get this to go to the second derivative:
\[
\frac{d^2}{d\epsilon^2}\Big{|}_{\epsilon=0,1} \zeta_b^3(\epsilon) +_{k+1} n = \frac{d^2}{d\epsilon^2}\Big{|}_{\epsilon =0,1}b +_{k+\epsilon} n
\]
Then, if you create a good enough sequence of functions \(\zeta^n\), such that they converge uniformly (for analycity you're going to want this on a non trivial domain in \(\mathbb{C}\), which would be very tricky), then \(\lim_n \zeta^n= \zeta\) would give you an analytic solution (theoretically). But there'd be a lot of steps. This is very similar to the main principle of Andy's slog. So you'd probably get a matrix formula for the various taylor coefficients about \(0\) and \(1\), and assuring they align (they satisfy the conjugate identity). Then your stuck with showing the taylor series converges, which will certainly be the hardest part.
The idea you are suggesting does exist. I can only point to Joel David Hamkins talking about it, which is very similar to what you are doing here. https://mathoverflow.net/questions/14678...-the-reals, read the first answer, which is a very similar breakdown of what you have here, but he does it more generally. This though, largely subscribes itself to logic/computation stuff like that, as it isn't concerned necessarily with much more than continuity (or in some cases continuously differentiable).
I don't quite understand what you are getting at with \(\Theta\), could you elaborate?