I could not find a tag for incomplete gamma function, so I've just used the gamma function one.
A rather curious fact about the normalised incomplete gamma function$${1\over n!}\operatorname{\Gamma}(n,\mu)=\int_\mu^\infty x^{n-1}e^{-x}dx$$ (or is that the complementary normalised incomplete gamma function?) is that it is equal to the hypervolume cut out by the hypersurface (a hyperhyperbola, maybe!) $$\prod_{k=1}^n x_k=e^{-\mu}$$ (the $x_k$ being coordinates in $n$-dimensional space) of the positive unit hyperbox with a corner at the origin, bounden by all the planes $x_j=0$ & $x_j=1$. (So it doesn't matter very much whether it's the IGF or the complementary IGF that is under consideration, as the other function is just the hypervolume in the other part of the box.
I know a rather gross proof of this that consists in integrating the whole of the present function's representation as such a hypervolume along the new axis to get the next function up in the series of increasing $n$: the general case of this can without tremendous difficulty be 'abstracted' from the first two or three by broaching yet another curious theorem of binomial coefficients; but I am not much enamoured of the idea of setting it out, and I suppose few of you are strongly desirous of seeing it.
But the question is - does anyone know a slick proof of the theorem that is the chief subject of this post? - by which I mean a proof that proceeds along somekind of shortcut or 'wormhole' through the space of general function-theory, without recourse to just taking the thing apart & reconstructing it cog-by-cog, as in my method of proof ... but I think the meaning of slick proof is well-enough known amongst us! I feel that there ought to be and probably is such a proof - my intuition just cannot rest in the idea that there isn't one.