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Just a note

The negative binomial distribution function is often given as the expansion of [p-q]-k where p = m/k and q = [k+m]/k

This confuses many people because p and q are defined quite differently in the binomial distribution and, by some authors, even when used within the terms of the negative binomial expansion.

To avoid this confusion, we prefer to use the parameters m and k, for example as:
[p - q]-k =   [( m  ) - ( k+m  )] -k  = 1
kk
and rearrange these terms as need be.