Zero-truncated claim-count distribution ($G_{N\mid N>0}(z)=\frac{G_N(z)-G_N(0)}{1-G_N(0)}$)

For a nonnegative integer claim count $N$ with $p=\mathbb{P}(N=0)<1$, the conditional law given $N>0$ has masses $\mathbb{P}(N=n)/(1-p)$ for $n\ge 1$. Its probability generating function is $(G_N(z)-p)/(1-p)$. Truncation at zero is meaningful only when the positive-count event has positive probability.