It’s of course a coincidence, but it’s nice to learn that the first reference concerning the well-beloved Gamma function is in a paper from 1729. It’s by Euler, of course, and a copy of the original latin paper can be found here, as well as a link to an English translation.
Whittaker and Watson stated that the first definition is the following infinite product
$latex \frac{1}{z\Gamma(z)}=\prod_{n\geq 1}{\Bigl(1+\frac{z}{n}\Bigr)\Bigl(1+\frac{1}{n}\Bigr)^{-z}}$
for all complex numbers z, except the poles of the function, which are
$latex z=0,\ -1,\ -2,\ldots, -n,\ \text{ for } n\geq 1.$
In fact, Euler writes it slightly differently, namely (in a form equivalent to)
$latex \Gamma(z+1)=\prod_{k\geq 1}{\frac{k^{1-z}(k+1)^z}{k+z}}.$
I did not remember having seen this formula, though I guess I probably had it in front of me at least once (I will check if it appears in Titchmarsh’s book, which would not be surprising). It looks a bit like, but is different, from the Weierstrass product
$latex \frac{1}{z\Gamma(z)}=e^{\gamma z}\prod_{n\geq 1}{\Bigl(1+\frac{z}{n}\Bigr)e^{-z/n}}\quad ,$
where γ is the Euler constant. (This definition has the problem of requiring first to say what this constant is).
It turns out that this original definition was exactly suited to some purpose for which the Weierstrass product, frustratingly, just barely didn’t quite work. So I was happy when I found it in the big tables of Gradhstein and Ryzhik. However, those tables are sometimes unfortunately wobbly when it comes to the range of validity of their formulas, and in that case stated that the product was valid for
$latex \text{Re}(z)>0,$
(which did not suit me), so it was rather more satisfactory to see it in Whittaker and Watson. In fact, it is easy to derive it from the Weierstrass product and the definition of the Euler constant γ.
By the way, there is a recurrent conflict about the Gamma function: is the “right” function really Γ(z), or Γ(z+1), which interpolates the factorial rather better? In analytic number theory, various reasons suggest the standard definition is indeed the right one, but I’ve recently starting vacillating, and my current conclusion is that there are two canonical (Gamma) functions who just happen to be related by translation. Or maybe three, where the third is 1/Γ(z). Or four, with also 1/Γ(z+1).
