I've seen a couple of questions where some users provide some help on how to calculate the convolution of two independent variables $X\sim NB(r,p)$ and $Y\sim NB(s,p)$ link 1, link 2. However they always stop the demonstration when they arrive to:
$$\displaystyle \sum_{j=0}^k {j+r-1 \choose j}\cdot {k-j +s-1 \choose k-j}={k+r+s-1 \choose k}$$
How can this be proved?
Thank you very much
