Panjer recursion with zero severities ($g_0=G_N(f_0),g_r=\frac{1}{1-a{f}_0}{\sum }_{j=1}^r(a+bj/r)f_j{g}_{r-j}$)

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 $(1-az)G_N^{\prime }=(a+b)G_N$ gives $(1-a{G}_X)G_S^{\prime }=(a+b)G_S{G}_X^{\prime }$. 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.