Generalized arithmetic operator

[hixidom]

\( (x[s]y)[s-1]x=x[s](y+1) \)

I can't seem to prove or disprove that statement in my system, except by the following discrepancy: Wikipedia has \( a[4]b=a^{a^{b-1}} \), which doesn't agree with what I have.
For example:
According to Wiki, \( a[4]4=a^{a^3} \)

According to what I have, \( a[4]4=[3]^2 a=[3][3]a=[3](a^a)=(a^a)^{(a^a)} \).

So I don't think that that recursion relation results in the same set of hyperoperators. Do you think that the equation with the summation still applies? What exactly should I take from that equation? Is it to be evaluated numerically?

Thanks for the response.
hixidom