Does anyone think that tetration by a non-integer will ever be defined … really properly?
Great mathematicians struggled with finding an implementation of the non-integer factorial for a long time to no avail … and then eventually Leonhardt Euler devised a means of doing it, & by a sleight-of-mind that was just so slick & so simple in its essence … and yet so radical! Is there any scope for another such sleight-of-mind as that, whereby someone might do similarly for tetration, or have they all been used-up? It’s impossible to conceive of how there might be any truly new conceptual resource left of that kind. But it’s actually tautological that that is so, because the kind I am talking about is precisely the kind that is essentially radically new, & of even how it might be thought hitherto unconceived!
But the attempts I have seen so far at defining tetration by general real number look to me for all the world like mere interpolation – some of them indeed very thorough & cunning & ingenious (insofar as I can follow them atall) – but lacking that spark of essential innovation that is evinced in Euler’s definition of the gamma function.
Just incase it seems I have gotten lost in philosophy, I’ll repeat the question: will there ever be a definition of tetration by general real number that resolves the matter as thoroughly as Euler’s definition of the gamma-function resolved the matter of factorial of general real number?
