Is there a functional equation for extending this to a smooth real function?
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1$\begingroup$ It sure seems like a rather silly question. However, a priori one could say the same thing about the sequence of factorials. So I will upvote it, just for kicks... $\endgroup$– Lev BorisovOct 8, 2014 at 1:33
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2$\begingroup$ Silly or not it has a Wikipedia page: en.wikipedia.org/wiki/Tetration $\endgroup$– Noam D. ElkiesOct 8, 2014 at 1:57
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$\begingroup$ Noam D. Elkies : interesting page, thanks. $\endgroup$– David LampertOct 8, 2014 at 10:57
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1 Answer
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Let $$ f(x,y)=x^{f(x,y-1)}, $$ $$ f(x,1)=x. $$ Then your function is given by $g(n)=f(n,n)$.
answered Oct 8, 2014 at 1:26