If $X$ and $Y$ are random variables following the binomial law of parameters $n,p$ and $n,q$ then does $X+Y$ follow any known law?

Question

If $X$ and $Y$ are random independent variables following the binomial law of parameters $n,p$ and $n,q$ then what law does $X+Y$ follow?

$P(X+Y=z)=\sum\limits_{x=0}^nP(X=x)P(Y=z-x)=\sum\limits_{x=0}^n{n\choose x}p^x(1-p)^{n-x}{n\choose z-x}q^{z-x}(1-q)^{n-z+x}$

Is this some known probability mass function?

Answer 1

In general, the sum of two independent random variables is the convolution of their mass functions or density functions. This isn't a named mass function, but it is also easily recognizable.