Extended negative binomial distribution

For a natural number m ≥ 1 and real parameters p, r with 0 < p ≤ 1 and –m < r < –m + 1, the probability mass function of the ExtNegBin(m, r, p) distribution is given by

${\displaystyle f(k;m,r,p)=0\qquad {\text{ for }}k\in \{0,1,\ldots ,m-1\}}$

and

${\displaystyle f(k;m,r,p)={\frac {{k+r-1 \choose k}p^{k}}{(1-p)^{-r}-\sum _{j=0}^{m-1}{j+r-1 \choose j}p^{j}}}\quad {\text{for }}k\in {\mathbb {N} }{\text{ with }}k\geq m,}$

where

${\displaystyle {k+r-1 \choose k}={\frac {\Gamma (k+r)}{k!\,\Gamma (r)}}=(-1)^{k}\,{-r \choose k}\qquad \qquad (1)}$

is the (generalized) binomial coefficient and Γ denotes the gamma function.

Using that f ( . ; m, r, ps) for s ∈ (0, 1] is also a probability mass function, it follows that the probability generating function is given by

${\displaystyle {\begin{aligned}\varphi (s)&=\sum _{k=m}^{\infty }f(k;m,r,p)s^{k}\\&={\frac {(1-ps)^{-r}-\sum _{j=0}^{m-1}{\binom {j+r-1}{j}}(ps)^{j}}{(1-p)^{-r}-\sum _{j=0}^{m-1}{\binom {j+r-1}{j}}p^{j}}}\qquad {\text{for }}|s|\leq {\frac {1}{p}}.\end{aligned}}}$

For the important case m = 1, hence r ∈ (–1, 0), this simplifies to

${\displaystyle \varphi (s)={\frac {1-(1-ps)^{-r}}{1-(1-p)^{-r}}}\qquad {\text{for }}|s|\leq {\frac {1}{p}}.}$