Factorials as Multiplicative Integrals
In which we try to figure out what what’s going on with double-factorials.
This was formerly part of the previous post about \(n\)-spheres, but I started adding things to it and decided to split them up. It is not necessary to read the original previous post first, but this does sort of function as a sequel to it since it’s the direction my investigation has gone. Both articles are essentially unwieldy dumps for notes and calculations that I’ve done and make a record of. But maybe they’ll be useful as a survey of various related ideas, if anyone else is curious about this and comes across them.
My main finding is that I now believe we should be thinking of factorials as multiplicative integrals, like this:
\[\frac{n!}{m!} = \prod_m^n d^{\times}(x!)\]And in particular, the factorials we’re used to have an implicit lower bound on that integral: the value \(n!\) is really \(\frac{n!}{0!} = \prod_0^n d^{\times}(x!)\). This means we never really “see the value” of \(0!\), because \(0!\) is equivalent to \(0!/0! = 1\). This interpretation seems to remove a bunch of ambiguity in the various definitions/analytic continuations of factorials on non-integer numbers (as well as explaining why those definitions don’t mess up the usual combinatoric sense of factorials).