A famous artifact in the history of mathematics is the relation between the factorial $$n \to n! = \cases{n\cdot (n-1)!, n>0\\1, n = 0}$$
and the Gamma function;
$$x \to \Gamma(x) = \int_0^\infty t^{x-1}e^{-t}dt$$
We can prove that $\Gamma(n) = (n-1)!$
One nice property of this particular solution is that the recursive property of the factorial is extended to the reals $\Gamma(x+1) = x\cdot \Gamma(x)$.
We can find excuses to add extra conditions (for example logarithmic convexity) to narrow down the set of possible solutions so that the Gamma function is the only suitable candidate.
However, it is clear that if we don't add any further conditions, we can find infinitely many other functions which will interpolate between all the integer points smoothly.
Which other continuous functions can we find which fit the integer points of the factorial function? Any resource is welcome.
Own work : My own work is limited to a few google searches which for example found the case presented on Wikipedia: $$\cases{\Gamma(t+1)\cdot g(t)\\g(t) = g(t+1)\\g(0) = 1}$$
In other words we can multiply the gamma function with any function periodic on $[0,1]$ as long as it is $1$ at the end points.
This is appealing to me for example because it allows us to modulate the gamma function with any function having a Fourier Series.