10/17/2009, 09:20 PM
Let me rewrite this so I understand it better.
(1) \( ({+}b)^h(x) = x + bh = \mathcal{S}[{+}b]^{-1}(b^h\mathcal{S}[{+}b](x)) \)
(2) \( ({\times}b)^h(x) = xb^h = \mathcal{S}[{\times}b]^{-1}(b^h\mathcal{S}[{\times}b](x)) \)
(3) \( (\!{\uparrow}b)^h(x) = x^{b^h} = \mathcal{S}[\!{\uparrow}b]^{-1}(b^h\mathcal{S}[\!{\uparrow}b](x)) \)
and
(4) \( \mathcal{S}[{+}b](x) = a^{x/b} \) for almost any a.
(5) \( \mathcal{S}[{\times}b](x) = x \)
(6) \( \mathcal{S}[\!{\uparrow}b](x) = \log_a(x) \) for almost any a.
so what I think you mean is that you can set \( a=b^b \) in (4), and \( a=b \) in (6) so that there is a common base? That is quite interesting. Honestly, I had never seen that before. This effectively reduces the problem of "nice" noninteger lower hyperoperations to the problem of "nice" noninteger tetration.
I am well aware of another similar situation with the commutative hyperoperations, where the general form can be written:
\( H_{n+2}(a, b) = \exp^{n}\left(\log^{n}(a)\log^{n}(b)\right) \)
this can also be considered a way of reducing noninteger hyperoperations to noninteger tetration.
(1) \( ({+}b)^h(x) = x + bh = \mathcal{S}[{+}b]^{-1}(b^h\mathcal{S}[{+}b](x)) \)
(2) \( ({\times}b)^h(x) = xb^h = \mathcal{S}[{\times}b]^{-1}(b^h\mathcal{S}[{\times}b](x)) \)
(3) \( (\!{\uparrow}b)^h(x) = x^{b^h} = \mathcal{S}[\!{\uparrow}b]^{-1}(b^h\mathcal{S}[\!{\uparrow}b](x)) \)
and
(4) \( \mathcal{S}[{+}b](x) = a^{x/b} \) for almost any a.
(5) \( \mathcal{S}[{\times}b](x) = x \)
(6) \( \mathcal{S}[\!{\uparrow}b](x) = \log_a(x) \) for almost any a.
so what I think you mean is that you can set \( a=b^b \) in (4), and \( a=b \) in (6) so that there is a common base? That is quite interesting. Honestly, I had never seen that before. This effectively reduces the problem of "nice" noninteger lower hyperoperations to the problem of "nice" noninteger tetration.
I am well aware of another similar situation with the commutative hyperoperations, where the general form can be written:
\( H_{n+2}(a, b) = \exp^{n}\left(\log^{n}(a)\log^{n}(b)\right) \)
this can also be considered a way of reducing noninteger hyperoperations to noninteger tetration.
