Re: derivative of x!

From: Zdislav V. Kovarik (kovarik_at_mcmaster.ca)
Date: 09/09/04 

Date: Thu, 9 Sep 2004 18:13:59 -0400

On Fri, 10 Sep 2004, Rob Johnson wrote:
[much deleted]
> Now why is the fact that the logarithm is convex a good reason to use
> Gamma(x+1)? I know that that is one of the defining features of Gamma,
> but why is it important when finding an analytic function which matches
> x! on the non-negative integers?
>
> Rob Johnson <rob@trash.whim.org>
> take out the trash before replying
>
 Perhaps because the statement

   "If a <= b <= c then

   (a!)^(c-b) * (c!)^(b-a) >= (b!)^(c-a)"

is a statement about triples of positive integers,
provable in elementary arithmetic (use induction on c)
(and you can see the connection with logarithmic convexity).

So, Bohr-Mollerup Theorem states that to extend this elementary
inequality (re-written to use the candidate for Gamma function)
to all positive reals forces us to use Gamma function itself.

Cheers, ZVK(Slavek).

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