Let $y(z) = \sum_k y_k z^k$ be a holomorphic function in a vicinity of the point $z=0$. Define its Borel transform $By$ as a function $By(z) = \sum_k \frac {y_k}{k!} z^k$.

The well-know formula allows to reconstruct $y(z)$ from $By(z)$. It is given by an application of a simple integral operator. Namely $$y(z) = \int_{0}^\infty e^{-t} By(tz) dt.$$

I was wondering, if it is possible to construct a similar operator realizing the Borel transform ? A functional on the space of holomorphic functions, sending $z^k \mapsto \frac 1{k!}$ does the trick. So basically, I am interested in a realization of such a functional as an integral.