Is there a family of probability distributions for ...?

Is there a family of probability distributions for ...?

Is there a family of probability distributions for
$P(x|y,n)=\frac{\Gamma(n+y)}{\Gamma(n)\Gamma(y)}(x+1)^{-n-y}x^{y-1}$ ?
$n>0$ and $y>0$
Has it an explicit expression for the CDF?