Suppose I have a sequence $c_n$, and I make the formal power series $\phi(t) = \sum_{n=0}^{\infty}c_nt^n$. I can take the (inverse) fourier transform of $\phi(t)$
$f_X(x) = \frac{1}{2\pi}\int_\mathbb{R}{e^{itx}\phi(t)dt}$.
I could then compute the integral to find $f_X(x)$ in terms of $c_n$, but doing so is proving to be beyond me. What I'm ultimately interested in is how the $d_m$ in the expansion $f_X(x) = \sum_{m=0}^{\infty}d_mt^m$ are related to the $c_n$ in the expansion of $\phi(t)$.