My previous answer, in much simplified form: take $g(z):=\frac{1-\sqrt{1-z}}{z}$ and $h(z):=f(z)g(z)=\frac{1-\sqrt{1-z}}{z}\,\frac{1+z}{2}$. Then $g$ and $h$ are pgf's.
Iosif Pinelis
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