On the convolution of the negative binomial random variables

In this note we are concerned with the sums S=Y1+Y2+⋯+YnS=Y1+Y2+⋯+Yn, where every constituent follows the negative binomial distribution with arbitrary parameters. We derive the exact probability mass function and the cumulative probability function of S. We also show that one can relate to the distribution of S as a mixture negative binomial distribution.