During my work i've encountered generating functions in the form $$U(z) = \sum a_ne^{-nz} \quad.$$ Now, i've just begun to study generating functions a little bit more in depth but up until now i've only encountered ordinary generating functions and exponential generating functions. I've also been told not to "think too much into it" as if it's just a transformation $$ U(e^{-z}) = \sum a_n (e^{-z})^k = \sum a_ne^{-kz}\quad.$$ I don't really understand the purpose of this transformation (most often a generating function is straight up defined in such manner). And how to deal with them when working with the standard generating function tools I learned so far (i've been reading generatingfunctionology).
How is that form called (if it has a proper name)?
For example: I have the generating function $U(z)$ defined as above that i plug into a PDE and end up obtaining an implicit solution for $U(z)$ in the form of $z = \phi(u(z))$. I've been trying to use the Lagrange expansion to extract the coefficients of $U$ but up until now i've encountered only references to OGF or EGF. I'm mainly concerned about the analytical aspects as i know that from a formal power series point of view I only care about the coefficients and not about the convergence.
How can I manipulate $U(z)$ when in this form? Can i bring it in a standard form?