Can a Periodic Function and Gamma Function Solve a Functional Equation?

[shmoe]

Well, if interpolation of the factorial is your only goal, then the functional equation f(n+1)=nf(n),\mbox{ for }n=1,2,3,\ldots being satisfied would suffice; yet a solution to such is not unique, both the gamma function and the Barnes G-function are solutions to the above functional equation.

Doesn't the Barnes function grow much faster and satisfy G(n+1)=Gamma(n)*G(n)? How can this possibly interpolate factorial? Barnes is the one with zeros of order of order |n| at negative integers n isn't it?

Of course there's lots of ways to extend factorial, you can always connect the dots. My point was if you were introducing gamma with this goal, then the limit definition would involve less "where on Earth did that come from". To me it requires the least motivation via hindsight.

 

[benorin]

My bad, bad memory that is: If u(x) is any function such that \forall x,u(x+1)=u(x) (i.e. u is has a period of unity,) then u(x)\Gamma (x) is a solution to said functional equation.