The pmf of Negative Binomial-Inverse Gaussian is
$Pr(X=x)=\displaystyle\binom{r+x-1}{x}\left[\sum_{j=0}^{x}(-1)^{j}\binom{x}{j}\exp\left\{\frac{\psi}{\mu}\left[1-\sqrt{1+\frac{2(r+j)\mu^{2}}{\psi}}\right]\right\}\right]$
how to prove it's pmf always >0 and it sum add up to 1?