The Bounded Analytic Hyper-operators

Abstract

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