Panjer claim-count class ($p_n=(a+b/n)p_{n-1}$)

The nonnegative integer laws whose probabilities satisfy this recurrence for all $n\ge 1$. Their probability generating function solves $(1-az)G^{\prime }(z)=(a+b)G(z)$. This is the $(a,b,0)$ count class used in the Panjer aggregate algorithm; the count recurrence itself should be distinguished from that aggregate recursion. It includes the Poisson distribution, binomial distribution, and negative binomial distribution with suitable parameters.