I have a conjecture I haven't been able to prove. I think the Fractional Calculus approach I took for hyper operators in between 1 and eta will work for bases inbetween \( e^{-e} \le b < 1 \)
To be explicit:
\( (^z b) = \frac{1}{\G(1-z)} (\sum_{n=0}^\infty (^{n+1} b) \frac{(-1)^n}{n!} + \int_1^\infty (\sum_{n=0}^\infty (^{n+1} b) \frac{(-w)^n}{n!}) w^{-z}\,dw) \)
This is of course if there exists an exponentially bounded solution. Since these bases do not induce a periodic tetration there requires more thought into how to solve this case.
Constructing the Koenigs function \( \psi \) of \( b^z \) produces a function \( \psi(b^z) = \lambda \psi(z) \) where \( -1 < \lambda < 0 \). Using regular Koenig iteration we can construct a tiny area about the fixed point \( z_0 \) which \( \psi^{-1} (z) \) exists. If we can induce the Koenigs function such that \( \exp_b^{\circ s+n} (z) = \psi^{-1}(\lambda^{s+n}\psi(z)) \) is a well defined function for \( |z - z_0 |<\delta \) and \( n \) large enough so that this function is defined for all \( \Re(s) > 0 \) we can extend this to all \( z \) in the immediate basin (including 1) using Fractional Calculus and similar techniques that I used here. This would give \( \exp_b^{\circ s+n}(1) = (^{s+n} b) \), and then using more tricks \( ^s b \) can be recovered.
The trick is of course ensuring that \( \psi^{-1}(\lambda^{s+n}\psi(z)) \) defined for \( s \) in some \( \Omega \) can be analytically continued to the right half plane, and then further is appropriately bounded. This would require analytically continuing the koenig function and as well its inverse.
Unless someone knows if there exists a solution such that \( |(^z b)| < M e^{\rho |\Re(z)| + \alpha |\Im(z)|} \) for \( M,\rho, \alpha \in \mathbb{R}^+ \) with \( \alpha < \pi/2 \), I see no other way of making this method work.
However there is excitement, because certain holomorphic functions with a multiplier that is not real positive and between 0 and 1 can be complex iterated using fractional calculus.
To give further motivation of this technique, consider the function \( \phi_\lambda(\xi) = \lambda \xi + \sum_{n=2}^\infty a_n\xi^n \) for \( 0 < \lambda < 1 \)
We can iterate this using fractional calculus and for \( \xi \) in the immediate basin of attraction, z in the right half plane, \( \phi^{\circ z}_\lambda(\xi) \) is analytic in \( \lambda \) and consequently the iterate must converge in the Fractional calculus transform for \( \lambda \) in an epsilon neighbourhood of (0,1). The suspicion is that it will converge for all \( 0<|\lambda|<1 \) on suitable \( \phi \). I'm betting the exponential function is one of these suitable functions.
To be explicit:
\( (^z b) = \frac{1}{\G(1-z)} (\sum_{n=0}^\infty (^{n+1} b) \frac{(-1)^n}{n!} + \int_1^\infty (\sum_{n=0}^\infty (^{n+1} b) \frac{(-w)^n}{n!}) w^{-z}\,dw) \)
This is of course if there exists an exponentially bounded solution. Since these bases do not induce a periodic tetration there requires more thought into how to solve this case.
Constructing the Koenigs function \( \psi \) of \( b^z \) produces a function \( \psi(b^z) = \lambda \psi(z) \) where \( -1 < \lambda < 0 \). Using regular Koenig iteration we can construct a tiny area about the fixed point \( z_0 \) which \( \psi^{-1} (z) \) exists. If we can induce the Koenigs function such that \( \exp_b^{\circ s+n} (z) = \psi^{-1}(\lambda^{s+n}\psi(z)) \) is a well defined function for \( |z - z_0 |<\delta \) and \( n \) large enough so that this function is defined for all \( \Re(s) > 0 \) we can extend this to all \( z \) in the immediate basin (including 1) using Fractional Calculus and similar techniques that I used here. This would give \( \exp_b^{\circ s+n}(1) = (^{s+n} b) \), and then using more tricks \( ^s b \) can be recovered.
The trick is of course ensuring that \( \psi^{-1}(\lambda^{s+n}\psi(z)) \) defined for \( s \) in some \( \Omega \) can be analytically continued to the right half plane, and then further is appropriately bounded. This would require analytically continuing the koenig function and as well its inverse.
Unless someone knows if there exists a solution such that \( |(^z b)| < M e^{\rho |\Re(z)| + \alpha |\Im(z)|} \) for \( M,\rho, \alpha \in \mathbb{R}^+ \) with \( \alpha < \pi/2 \), I see no other way of making this method work.
However there is excitement, because certain holomorphic functions with a multiplier that is not real positive and between 0 and 1 can be complex iterated using fractional calculus.
To give further motivation of this technique, consider the function \( \phi_\lambda(\xi) = \lambda \xi + \sum_{n=2}^\infty a_n\xi^n \) for \( 0 < \lambda < 1 \)
We can iterate this using fractional calculus and for \( \xi \) in the immediate basin of attraction, z in the right half plane, \( \phi^{\circ z}_\lambda(\xi) \) is analytic in \( \lambda \) and consequently the iterate must converge in the Fractional calculus transform for \( \lambda \) in an epsilon neighbourhood of (0,1). The suspicion is that it will converge for all \( 0<|\lambda|<1 \) on suitable \( \phi \). I'm betting the exponential function is one of these suitable functions.