Infinite series, probably related to Fourier transform?

[fluidistic]

Homework Statement

A function f(x) has the following series expansion: $f(x)=\sum _{n=0}^\infty \frac{c_n x^n}{n!}$.
Write down the function $g(y)=\sum _{n=0}^\infty c_n y^n$ under a closed form in function of f(x).

Homework Equations

Not sure at all.

The Attempt at a Solution

Totally stuck.
I've tried to write the first few terms of both series and it basically looks like I'm asked to write a sort of exponential function in terms of a polynomial of infinite degree or something similar to this.
Really, I think I need a tip because I'm totally stuck. That problem appears after the end of a chapter on integral transforms in Mathews and Walker's book on mathematical methods in physics.

 

[haruspex]

The integral transform hint suggests investigating $\int_0^\infty f(x)e^{-xt}dx$