= Panjer recursion with zero severities
{c}
{title2=$g_0=G_N(f_0),\quad g_r=\frac1{1-af_0}\sum_{j=1}^r(a+bj/r)f_jg_{r-j}$}
For a <claim count distribution> in the <Panjer claim-count class> and independent nonnegative integer severities with mass $f_j$, the <probability generating function> composition is $G_S=G_N\circ G_X$. Combining it with $(1-az)G_N'=(a+b)G_N$ gives $(1-aG_X)G_S'=(a+b)G_SG_X'$. Equating coefficients yields the displayed recursion. Zero severities change both the initial mass and the denominator. For nondegenerate proper Panjer count laws $a<1$, so the denominator is positive. An identically zero count is handled directly if its redundant parameter choice makes the denominator vanish.
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