 Solution

If $a+b=0$, then $p_1=(a+b)p_0=0$. Applying the count recurrence successively gives $p_n=0$ for every $n\ge 1$. Normalization forcesThus the claim count distribution is a point mass at zero, and the aggregate claims model also has $S=0$ almost surely. In particular the recursion starts at $g_0=1$ and gives no positive-loss probabilities.