Poisson–Mellin–Newton cycle

Definition

Let $(f_n)_{n \ge 0}$ be a sequence of complex numbers. The Poisson–Mellin–Newton cycle of $(f_n)$ is the following chain of three objects.

1. The Poisson generating function $g(t) = e^{-t} \sum_{n \ge 0} f_n \frac{t^n}{n!}$, for $t > 0$ where the series converges.
2. The Newton coefficients $b_m = \sum_{k=0}^{m} \binom{m}{k} (-1)^k f_k$, which satisfy $g(t) = \sum_{m \ge 0} b_m \frac{(-t)^m}{m!}$ and $f_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k b_k$.
3. The Mellin transform $\varphi(s) = \int_0^\infty g(t)\, t^{s-1}\, dt$, where it converges.

Assume that $b_m = b(m)$ for a function $b$ that is holomorphic and of moderate growth in a half-plane. Then the cycle closes by the relations $$\varphi(s) = \Gamma(s)\, b(-s), \qquad f_n = \frac{(-1)^n}{2\pi i} \oint_{C} b(s)\, \frac{n!}{s (s - 1) \cdots (s - n)}\, ds,$$ where $C$ is a positively oriented contour that encircles the points $0, 1, \dots, n$ and no singularity of $b$.