Simplify ${\Gamma(\beta)\Gamma(u) \over \Gamma(\beta+u)}$ Is it possible that the expression below be simplified to a simpler form: $${\Gamma(\beta)\Gamma(u) \over \Gamma(\beta+u)}$$ whereby $\beta$ is a variable

This is the Beta function \- not because you have an argument denoted with beta i.e. $β$ \- but because it is called so. One form of the Beta function is the one you posted $$B(β,u):=\dfrac{\Gamma(β)\,\Gamma(u)}{\Gamma(β+u)} $$

whereas in the link that I gave you can see its general definition and its properties. If the arguments $β,u$ are for example integers then, yes it can be furthered simplified.