(1)

\begin{align*} H_{4}\left(a,n\right) & =a\uparrow^{2}n\\ & =\log_{a}\left\{ a\uparrow\left(a\uparrow^{2}n\right)\right\} \\ & =\log_{a}\left\{ a\uparrow^{2}\left(n+1\right)\right\} \\ & =\log_{a}H_{4}\left(a,n+1\right)\\ & =\log_{a}^{m\circ}H_{4}\left(a,n+m\right)+\sum_{k=1}^{m}\left\{ \log_{a}^{\left(k-1\right)\circ}H_{4}\left(a,n+k-1\right)-\log_{a}^{k\circ}H_{4}\left(a,n+k\right)\right\} \\ & =\log_{a}^{m\circ}H_{4}\left(a,n+m\right) \end{align*}

(2)

(1)より、
\begin{align*} H_{4}\left(a,-1\right) & =\log_{a}H_{4}\left(a,0\right)\\ & =\log_{a}1\\ & =0 \end{align*}