= Panjer recursion
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{title2=$g_r=\sum_{j=1}^r(a+bj/r)f_jg_{r-j}$}
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For positive integer severities with <probability mass function> $f_j$ and independent count in the <Panjer claim-count class>, the aggregate <probability mass function> starts at $g_0=p_0$ and satisfies the displayed recurrence. If $P,F,G$ are the count, severity and aggregate <probability generating functions>, respectively, then $G=P\circ F$ and $(1-aF)G'=(a+b)F'G$. Equating the coefficient of $z^{r-1}$ gives $r g_r=\sum_{j=1}^r[ a(r-j)+(a+b)j]f_jg_{r-j}$. The positivity assumption excludes zero severities; otherwise the initial value and recursion denominator change.
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