Firstly, I needed to define three functions to make it as compact as possible:
$$\left \{ x \right \}=x-\left \lfloor x \right \rfloor$$
$$f(x)=\left \lfloor 1-\left \{ x \right \} \right \rfloor$$
$$f_{1}(x)=f\left (\frac{x-1}{c_{1}} \right )\cdot f\left (\frac{x-1}{c_{2}} \right )$$
such that$${c_{1}}\cdot {c_{2}}\neq \frac{p}{q} (irrational) $$ and the equation itself is:
$$\Gamma (x)=\frac{\Gamma (\left \{ x \right \}+1)}{\left \{ x \right \}^{\frac{x-1 +\left | x-1 \right |}{2(x-1)+f_{1}(x)}}+f(x)}\cdot\prod_{n=f(x)}^{\left \lfloor \left | x-1 \right | \right \rfloor+f_{1}(x)}\left ( \left \{ x \right \}+\frac{\left | x \right |}{x}n \right )^{\frac{\left | x-1 \right |}{x-1+f_{1}(x)}}+0.5f_{1}(x)$$
Now, its idea came to me as constructing the whole gamma function from the interval [1,2] in which it is defined using its recurrence relation. It was originally defined on the non-integer values only, but with some additional terms it was defined on all values just like the original one, but all the additions were intuitive and conceivable except the 0.5f1(x) at the right, and I need someone to explain why we need it.
And because I'm just a math hobbyist high schooler, I was so excited when I completed its derivation and I needed to know if it would have any uses or if we could get something new out of it.