The conclusion of the following link
Does Gamma function a solution for known Ordinary differential equation?
is
"$\Gamma(x)$ function is impossible to characterize by algebraic differential equation"
So 'Differential equation' is not good weapon for characterizing gamma.
But I found some articles in stackexchange math, asserting that "time scale calculus is a part of mathematics which unifies recurrence relation and differential equation".
But I know nothing of time scale calculus. Can I hope "time scale calculus" is a good weapon for characterizing Gamma from the factorial recurrence relation $x_n = n x_{n-1}$ ?
Bohr–Mollerup theorem is about extending factorial recurrence relation to Gamma function. But one can hope similar logic for other similar recurrence relation to some function. Does time scale calculus deal with such problems ?