The first person who gave a representation of the so called gamma function was Daniel Bernoulli in a letter to Goldbach from 1729-10-06. The letter can be seen here.

The formula reads in modern notation as given by Gronau in the article cited in the answer:

$ x! = \lim_{n\rightarrow \infty}\left(n+1+\frac{x}{2}\right)^{x-1} \prod_{i=1}^n\frac{i+1}{i+x} $

Gronau also observes that "Numerical experiments show that the formula of Bernoulli converges much faster to its limit than that of Euler ...", "that of Euler" refers here to a formula Euler has given in a letter to Goldbach dated 1729-10-13.

Gronau writes: "Euler who, at that time, stayed together with D. Bernoulli in St. Petersburg gave a similar representation of this interpolating function. But then, Euler did much more. He gave further representations by integrals, and formulated interesting theorems on the properties of this function."

Though this justifies the name 'Euler gamma function' Euler's representation was historically only second to Daniel Bernoulli's.

The correspondence between Goldbach, Daniel Bernoulli and Euler which undoubtedly gave birth to the gamma function is well documented in Paul Heinrich Fuss's „Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIeme siècle ..“, St. Pétersbourg, 1843.