Does the gamma function have unique integer solutions?

Question

I just want to know if the gamma function is the only function that pass through the points (1,1),(2,1),(3,2),(4,6) ... Or are there other functions that pass through the same points as the gamma function at all integers but not necessarily real numbers such as $$ f(0.5) = \sqrt\pi $$

Answer 1

There are infinitely many other functions that do so. For instance,

$$ H(x) = \Gamma(x) + \sin (2 \pi x) $$

is one such function, and is just as smooth as $\Gamma$. You can add to $\Gamma$ any function that's zero at all integers to get a new function that agrees with $\Gamma$ on the integers.

To get uniqueness, you need to insist on some other properties. This question -- The uniqueness of the Gamma Function -- has a fairly standard set listed at the top (and then asks whether a weaker set of conditions would suffice).