The uniqueness of the Gamma Function

Question

It is a theorem that any function $f$ defined for positive real numbers satisfying

$f(1)=1$
$f(x+1)=x\cdot f(x)$
$f$ is log convex

is identically equal to the gamma function. (Condition 2 means that this function interpolates a shifted factorial function.)

Now, a beginner (such as myself) might ask: What if we weaken condition 2 by instead requiring $f$ to be merely convex, not log convex?

I would imagine that such functions would look not too different, since intuitively, I can't wildly deviate the graph of the gamma function if I want to maintain condition 2 and stay convex.

Just a follow-up musing---What if instead of condition 3, we require convexity and infinite differentiability? Do we still uniquely determine the gamma function?

Artin's book The Gamma Function contains some other results on uniqueness that depend on continuity or continuous differentiability only but require Legendre functional equation. — lhf, Apr 12 '11 at 10:59
See also http://mathoverflow.net/questions/23229/importance-of-log-convexity-of-the-gamma-function . — lhf, Apr 12 '11 at 11:01

score 1 · Answer 1 · answered Mar 13 '11 at 09:47

1

(This should be either a comment or CW.)

Peter Luschny studied a number of gamma-like functions that do not have the log-convex imposition; you might want to look into them for inspiration.

answered Mar 13 '11 at 09:47

3

I do not think Luschny's examples satisfy condition 2. Any functions satisfying 1 and 2 must have a singularity at the origin. – Johan Mar 13 '11 at 10:20

The uniqueness of the Gamma Function

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