What are the properties of this exotic function transform, which divides Taylor series coefficients by $(n!)$?

Question

While investigating a model in particle physics, I encountered an exotic function transform. The form that appears in our application acts on analytic functions. In its original form, it can be written in terms of a contour integral, but it looks the most transparent in terms of Taylor series, $$f(z) = \sum_n \frac{f^{(n)} z^n}{n!} \, \mapsto \, \tilde{f}(w) = \sum_n \frac{f^{(n)} \, w^{n+1}}{n! (n+1)!}.$$ The derivative of the output function looks a bit nicer, $$\tilde{f}'(w) = \sum_n \frac{f^{(n)} \, w^n}{(n!)^2}.$$ We've computed several special cases (for instance, it converts exponentials to Bessel functions), but in practice we need to evaluate it for more general functions, such as exponentials divided by polynomials, where the resulting series seems hard to sum. However, it's simple enough that we suspect it already has a name, or at least some known properties that would be helpful. Is this transform known in the mathematical literature, and is there anything known that could help with evaluating it on functions such as $e^{iz}/(z-z_0)(z-z_1)$?

Answer 1

$f^\tilde'(w)$ is the Borel transform of the Maclaurin series of $f$.