Are there any texts on solutions to equations involving hyperoperations? Let's define $H_n(x,y) : \mathbb{N}_0 \rightarrow \mathbb{N}_0 $ to be the n-th hyperoperation, in particular:
$H_n(x,y)=\begin{cases} y+1 & n=0 \\ x & n=1,y=0 \\ 0 & n=2,y=0 \\ 1 & n\geq3,y=0 \\ H_{n-1}(x,H_{n}(x,y-1)) & \text{otherwise} \\ \end{cases}$
This comes from Hyperopration article on wikipedia. The particular problem I am looking at is for fixed $m, n.$ What inverses or other operations are required to give a closed form solution of $H_m(x,x)=H_n(x,x)$ for $x?$ I would expect these to be some generalizations of Lambert W function, but I cant find what are they called.
The special case of $H_m(x,x)=H_n(x,x)$ is remarkably simple to solve. Assuming $m<n$, one can easily verify that $x\ge3$ has no solutions since $H_m(x,x)$ will be significantly smaller than $H_n(x,x)$. One can then verify that all of $x=0,1,2$ are always solutions except for a small handful of cases when $m<2$.