[a_wild_dandan]

There ISN'T one correct extension of a factorial function.

The properties of the standard gamma function are great, but some might prefer the properties of Hadamard's or Luschny's alternative gamma function.

Like many arbitrary extensions in mathematics -- be it the factorial function or the division function which deals with 1 / 0 differently -- it's a matter of taste and convenience!

Interestingly the arbitrariness of some mathematical choices seem to unnerve folks on an almost existential level. My guess is that it conflicts with their expectation of capital T truth from mathematics.