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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.

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Ilk
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    Define $H_n$, please! –  Oct 01 '20 at 07:13
  • Does the question make sense now? – Ilk Oct 01 '20 at 07:38
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    The definition is still deficient. $H_1(x,y)$ crashes the recursion. Please provide a reference of where you picked this definition from. –  Oct 01 '20 at 17:07
  • That's simply $H_1(x,y):=:if:y:=:0:then:x:else:H_0(x,H_1(x,y-1))$ which further reduces to $H_1(x,y):=:if:y:=:0:then:x:else:H_1(x,y-1)+1$, which I think is well well-defined as y is decreasing and there is a base case y = 0. – Ilk Oct 02 '20 at 04:47
  • Sorry, I didn't assume that you are having your arguments $x,y\in\mathbb{N}$. For such numbers, yes of course, the recurrence terminates. Meanwhile, see: https://math.stackexchange.com/questions/2314123/are-there-nontrivial-equations-for-hyperoperations-above-exponentiation?rq=1. If nobody else comments, I'll write a couple of things about this, a little later. –  Oct 03 '20 at 17:47
  • Ah sorry, should be fixed now. The first link in that questions mentions a result by Gurevich about no non-trivial equations for n > 2, for their definition, which would be n > 3 in here. This might be what I was looking for, but I have to go through the technical details. – Ilk Oct 03 '20 at 21:03
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    @Nift: Yes. Also make sure this agrees with your results for BOTH $x,y\in\mathbb{N}$. If $x$ is real or complex and $y\in\mathbb{N}$, then it is conceivable that there are such non-trivial equations with solutions in terms of these functions you mention. But such equations would look like $H_m(x,k)=H_n(x,k)$, so they are not like what you specified ($H_m(x,x)=H_n(x,x)$), so that's a different matter, altogether. –  Oct 04 '20 at 00:57

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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 $m<2$.

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