# Partial Values for Knuth's Up-Arrows

$$3 \uparrow 4$$ 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?

• 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 – SmileyCraft Nov 16 '18 at 22:13
• Paulsen and Cowgill already took care of that for us. myweb.astate.edu/wpaulsen/tetration2.pdf – William Grannis Nov 16 '18 at 22:29
• 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. – mathlove Nov 17 '18 at 6:23