6b018f34_Columbia_UMS_Spring_2015__Title
[Date] Feb 18 [Speaker] Joshua Seaton [Abstract] Graph the points \((n, n!)\) in the plane for positive integers \(n\); there are a continuum of curves that go through these points. So why does the Gamma function -- among the infinitely-many other candidates -- deserve to be regarded as the 'factorial function'? The Bohr-Mollerup-Artin Theorem tells us why, which reveals that a function satisfying a couple of mild 'factorial'-like properties necessarily has to be the Gamma function. We first go through this brief, neat result. Then, in the second part of this talk, we look at the p-adic side of things (we will introduce p-adic numbers gently here, I promise.) In the ring of p-adic integers, the usual integers are dense. So -- in high contrast to the setting over the real numbers -- an integer-valued function on the integers (if p-adically continuous) will lift uniquely to a function on the p-adic integers! After some minor tweaking to the function that maps \(n\) to \(n!\), we will take a brief look at its (unique) p-adic extension, the p-adic Gamma function. [Title] 
Extrapolating factorials: the Gamma function, classical and p-adic
Columbia UMS Spring 2015
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