Hyper-operations By Unconventional Means

The author makes use of infinite compositions and a limiting function to construct a ${\mathcal{C}}^{\mathrm{\infty }}$ tetration function $$\mathcal{F}$(t) = e \tet t$\mathrm{.}Asatetrationfunction,$$\mathcal{F}$$satisfies$e^{$\mathcal{F}$(t)} = $\mathcal{F}$(t+1)$\mathrm{.}Ofit,$$\mathcal{F}$$takes$(-2,\infty) \to $\mathbb{R}$$ bijectively with strictly monotone growth, and is continuously differentiable here. We then iterate this construction to derive arbitrary hyper-operations $e{\text{up}}^{k}t$. These hyper-operations are ${\mathcal{C}}^{\mathrm{\infty }}$ strictly monotone bijections of $({\alpha }_k,\mathrm{\infty })\to \mathbb{R}$ for $k$ even ($1-k > \alpha_k \ge -k$),and$$\mathcal{C}$^\infty$strictlymonotonebijectionsof$$\mathbb{R}$ \to (\alpha_k, \infty)$for$k$ odd. These hyper-operations satisfy the functional equation $e{\text{up}}^{k-1}(e{\text{up}}^{k}t)=e{\text{up}}^k(t+1)$ with the initial conditions $e{\text{up}}^{1}t=e^t$ and $e{\text{up}}^{k}0=1$.