Partial Values for Knuth's Up-Arrows











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$3 uparrow 4 $ is $3^4$, and $3uparrow uparrow 3$ 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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    $3 uparrow 4 $ is $3^4$, and $3uparrow uparrow 3$ 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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      $3 uparrow 4 $ is $3^4$, and $3uparrow uparrow 3$ 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?






      power-towers







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      $3 uparrow 4 $ is $3^4$, and $3uparrow uparrow 3$ 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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      William Grannis

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