These two papers answer your question in the introductory paragraphs:
- Norbert Wiener's "The historical background of harmonic analysis"
and
The characteristic-function approach still abounds in generating series related to combinatorics in the umbral calculus / Sheffer sequences / finite operator calculus of Rota et al., where one might define the umbral variables as moments of distributions, defined by characteristic functions, and, of course, in quantum field theory and statistical mechanics with their diverse partition functions and cumulant-moment expansion theorems and associated enumerative diagrammatics, incuding Feynman diagrams, (cf. OEIS A036040 and A127671). E.g., the Laplace transform version gives \begin{gather*} (b_\cdot)^n = b_n = (-1)^n\left.\frac{d^n}{dt^n}\langle\exp(-tx)\rangle\right|_{t=0} \\ = (-1)^n\left.\frac{d^n}{dt^n} \int_0^\infty \exp(-tx)\operatorname{pdf}(x) dx\right|_{t=0} \\ = \int_0^\infty x^n\operatorname{pdf}(x) dx = \langle x^n\rangle, \end{gather*} where the characteristic function for the probability distribution function $\operatorname{pdf}(x)$ is
$$\langle\exp(-tx)\rangle = \int_0^\infty \exp(-tx)\operatorname{pdf}(x) dx .$$
There is an analogous Fourier transform characteristic function
$$\langle\exp(ixt)\rangle.$$
The gaussian distribution and the central limit theorem are key historical focal points in this appoach to probability theory, which is rife with enumerative combinatorics.
More recently, free probability theory employs the Cauchy transform to define characteristic functions for the generating functions of free moments, related to noncrossing partitions, parking functions, random matrices and the Wigner semicircular distribution—the counterpart to the gaussian distribution (cf. "A Simple Introduction to Free Probability Theory and Its Application to Random Matrices" by Xiang-Gen Xia, and A134264.)
