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I recently found out about Knuth's arrowup notation. Wikipedia, among other websites, only shows a definition for $a \uparrow^n b$ where $n \in \Bbb{N}_0, a \in \Bbb{R}, b \in \Bbb{N}$ as following:

$ a \uparrow^n b = \begin{cases} a^b, & \text{if $n = 1$} \\ 1, & \text{if $b = 0$} \\ a \uparrow^{n-1} (a \uparrow^n (b-1)) , & \text{otherwise} \\ \end{cases} $ (From wikipedia)

I wonder whether Knuth (or maybe someone else) also defined this notation for $n, a, b \in \Bbb{R}$. I would then expect $a \uparrow^{1/n} b$ to be the inverse $a \uparrow^{n} b$. And maybe $a \uparrow^{-n} b$ would mean ${1\over a \uparrow^{n} b}$.

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  • $\begingroup$ What property would you want this to have? You could define these, I suppose, but what would it give you? It's not like you can compose them usefully - what is would $a \uparrow^{q} b$ be now when $q$ is rational? There isn't a nice way to "compose" binary functions to get a new binary function - I suppose you could define $a \uparrow^{x/y} b = (a \uparrow^{1/y} b) \uparrow^{x} b$. But I don't see that as useful, and it wouldn't be true that $a \uparrow^{p} b \uparrow^{q} b = a \uparrow^{pq} b$ in general - indeed, that isn't true for natural $p,q$. So what is the goal? $\endgroup$ – Thomas Andrews Apr 30 '13 at 19:40
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We attempt to extend to zero by using the inductive case backwards:

$$a\uparrow^1b=a^b=a\uparrow^0 (a\uparrow^1 (b-1))$$

$$a^b=a\uparrow^0 a^{b-1}$$

So a sensible definition of $a\uparrow^0b$ is just $ab$.

Can we extend to negative numbers?

$$ab=a\uparrow^{-1}(a(b-1))$$

$$a\uparrow^{-1}b=a+b$$

Here we'll hit a bit of a snag:

$$a+b=a\uparrow^{-2}(a+b-1)$$

Really the sensible definition here is that $a\uparrow^{-2}b=b+1$. We only have one argument now, really. $a$ is irrelevant. Plugging this in gives

$$b+1=a\uparrow^{-3}b$$

But of course, this is the same as $a\uparrow^{-2}b$. So we don't actually get anything new here. You can't keep breaking it down into smaller things: adding one isn't a "repeated operation" in any real sense.

I'd imagine an extension to $\mathbb{Q}$, forget $\mathbb{R}$, to be anywhere between arbitrary and impossible. The major issue here is that this definition fundamentally relies on induction. This needs a base case, and a sense of a next element, both of which pin down the natural numbers (or some such thing that terminates at the left end) as the natural setting.

This is where I'd like to say something about other induction-based ideas with continuous extensions (e.g. factorials and the gamma function), but I can't really see a way of linking the ideas used in that extension (or similar ones) to this case.

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