So I managed to come up with a limit that seems to be approaching the gamma function for all non integer positive numbers, here it is
$$ \lim{x\to \infty} \sum_{a=0}^x \frac {\prod_{n=0}^x (p-n)}{(p-a)(a!)^2(-1)^{a+x}(x-a)!} = \frac {1}{\Gamma (p+1)} $$
However my only method of checking it is with desmos which can't calculate factorials for 170 and up and I don't have a proof for it , it just seems to work
Here is a desmos link to show it and compare it with other know formulas :https://www.desmos.com/calculator/ls7zx6ungh