[FOM] 605: Integer and Real Functions

[Mitchell Spector]

Harvey Friedman wrote:
 > ...
> So now I close this food for thought with the n!. in N, we have
>
> f(n+1) = (n+1)f(n), f(0) = 1
>
> with the unique solution n!.
>
> In R|>=0 this equation has lots and lots of even analytic solutions.
> So what we would like to have is a TWO VARIABLE functional equation
> with a initial condition that uniquely determines factorial on R|>=0.
> And then see what that two variable functional equation means on N.
> And even if it does NOT mean anything on N, then try to argue that on
> R|>=0, it still serves a fundamentally important purpose.
 > ...


The Bohr-Mollerup theorem is perhaps in the spirit of what you're looking for in connection with 
factorials.


BOHR-MOLLERUP THEOREM:

   The Gamma function on the positive reals is the unique function f mapping the positive reals to 
the positive reals such that:

   (1) f(x+1) = x f(x);

   (2) f is logarithmically convex (meaning that log f(x) is concave upward);

   and (3) f(1) = 1.


Of course, n! = Gamma(n+1) for non-negative integers n.  It is perhaps regrettable that Gamma(x) was 
adopted into the canon of standard functions instead of what we now call Gamma(x+1), which extends 
the factorial function itself to the complex plane.



However, I think that the naturalness of the Gamma function really comes from complex analysis.  The 
Gamma function is closely related to, and easily defined from, the simplest entire function whose 
zeros are precisely the positive integers.  This is in the same vein as the fact that the sine 
function is easily defined from the simplest entire function whose zeros are the precisely the integers.

Of course, you also need an initial condition in order to pin down a specific function; otherwise 
there's a constant factor that can be taken to be whatever you want.


"Simplest" here refers to the (very natural) canonical infinite-product representation of a function 
with a specified set of zeros.

I would conjecture that "simplest" could be interpreted in some formal sense, based on formulas of 
logic, or length of computations, but I don't know if any results of that sort are known.


Mitchell Spector