For two ordinary generating functions $F(x) = \sum_i f_i x^i$ and $G(x) = \sum_i g_i x^i$, one can compute
$$(F \odot G)(x) = \sum_i f_i g_i x^i = \frac{1}{2\pi} \int_0^{2\pi} F(\sqrt{x} e^{it}) G(\sqrt{x} e^{-it}) dt$$
Is there something similar for exponential generating functions of the form $F(x) = \sum_i f_i \frac{x^i}{i!}$?