Why is the gamma function so as it is

This is a historical note on the gamma function . The questio n is, why is ( n) for naturals n equal to (n 1)! and not equal to n! (the factorial function n! = 1·2 ··· n) ? Was A. M. Legendre responsible for this transformation, or was it L. Euler? And, who was the first who gave a representation of the so called Euler gamma function?