The factorial function: convex functions, the Bohr-Mollerup-Artin theorem, and some formulae

Abstract

How might we define x!? This is the next in my series of articles [1-3]; by now we know quite a bit about the factorial function, except that we have no real definition of it, nothing that would allow us to evaluate, say, ½!. So how do we move on to the next stage? There is a range of strategies. We could find or manufacture a suitable function f that satisfies the basic relations and adopt it as our definition.