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$3 \uparrow 4 $ is $3^4$, and $3\uparrow \uparrow 3$ is $3^{3^{3^3}}$, etc. For those of you unfamiliar, here is a wiki page on the notation. Clearly, up-arrow expressions, as they are usually defined, only have meaning when the number of arrows is in the naturals. Is there an extension, similar to $\Gamma (x)$ and the factorial function, that extends this to all reals, or even complex?

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  • $\begingroup$ Maybe first look for a definition of $a\uparrow\uparrow b$ for non-integer $b$ before we start looking at a non-integer number of arrows :P $\endgroup$ – SmileyCraft Nov 16 '18 at 22:13
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    $\begingroup$ Paulsen and Cowgill already took care of that for us. myweb.astate.edu/wpaulsen/tetration2.pdf $\endgroup$ – William Grannis Nov 16 '18 at 22:29
  • $\begingroup$ Here says that there are many different notation styles that can be used to express tetration. In the same page, you can see some explanations about the extensions. $\endgroup$ – mathlove Nov 17 '18 at 6:23

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