Timeline for Why is the Gamma function shifted from the factorial by 1?

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Dec 12, 2021 at 15:58	comment	added	LSpice		Oh, I see, you mean that $\Gamma$, $\zeta$, and $\eta$ are evaluated at the same argument. I misunderstood you to be saying that $\Gamma$ is evaluated at the same parameter as appears in the integral on the left. But perhaps this is an argument that $\zeta$ and $\eta$ should be shifted as well as $\Gamma$ ….
Dec 12, 2021 at 15:16	comment	added	Lucian		@LSpice: Without the aforementioned shift, they would obviously not possess the same argument; not unless one would have alternately shifted the $\zeta$ and $\eta$ functions by $-1.$
Dec 12, 2021 at 15:06	comment	added	LSpice		It seems a bit strange to describe these as "$\Gamma$ or $\zeta$ or $\eta$ functions of the same argument" when each of them is actually evaluated at $\Gamma(z + 1)$—exactly the shift that the question is discussing! (Of course, @S.Carnahan's point shows how one may consistently and sensibly get rid of that shift.)
Jan 30, 2016 at 14:51	history	edited	Lucian	CC BY-SA 3.0	Fixed Row Alignment.
Sep 16, 2014 at 10:26	comment	added	S. Carnahan♦		You can replace some of the $z+1$ terms with $z$ by replacing $dx$ with the natural scale-invariant measure $d^\times \! x = \frac{dx}{x}$.
Sep 16, 2014 at 4:43	history	answered	Lucian	CC BY-SA 3.0