The Bounded Analytic Hyper-operators

In a previous paper \cite{ref1} we produced a sequence of analytic functions $\{\alpha \uparrow^n z\}_{n=0}^\infty$ when $1 \le \alpha \le e^{1/e}$ and $z$ was in the right half of the complex plane, the \emph{bounded analytic hyper-operators}. This was a sequence of functions where each function was the \emph{complex iteration} centered about $1$ of the previous function. We show $\alpha \uparrow^n z$ is holomorphic and bounded for $\Re(z) > 1-n$. We give a closed form expression for an analytic function in all variables $\alpha \uparrow^s z$, with $1 < \alpha < e^{1/e}$, $\Re(s)>1$ and $\Re(z) > 0$. This three variable function, when restricted to the real line; $\alpha \uparrow^t x$ when $x, t \in \mathbb{R}^+$ and $t \ge 1$; has initial conditions $\alpha \uparrow^t 1 = \alpha$ with $\alpha \uparrow^1 x = \alpha^x$ and satisfies the functional equation $\alpha \uparrow^{t} (\alpha \uparrow^{t+1} x) = \alpha \uparrow^{t+1} (x+1)$.