An integral formula for the reciprocal gamma function

Update: The solution to my original question, which was asking if given an OGF $F(z)$ for some sequence $\{f_n\}_{n \geq 0}$, whether there is an integral transform that generalizes the known OGF-to-EGF transform given by $$\widehat{F}(z) = \frac{1}{2\pi} \int_{-\pi}^{\pi} F\left(z e^{-\imath t}\right) e^{e^{\imath t}} dt,$$ can be answered with Fourier series and integral representations for the Hadamard product of two generating functions. I'm actually writing up a short note one this now, but the integral transform in the previous question is given by $$\sum_{n \geq 0} \frac{f_n z^n}{(2n+1)!!} = \frac{1}{2\sqrt{2\pi}} \int_{-\pi}^{\pi} F\left(z e^{-\imath t}\right) e^{\frac{1}{2}\left(e^{\imath t} -\imath t\right)} \operatorname{erf}\left(\frac{e^{\imath t/2}}{\sqrt{2}}\right) dt.$$