A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions

TL;DR

This paper explores integral formulas and transformations that convert between ordinary and exponential generating functions, providing new variants and generalizations for these conversions using complex analysis techniques.

Contribution

It introduces new integral representations and generalizations for converting between OGFs and EGFs, including variants involving Hankel contours and Fourier series.

Findings

Derived integral formulas for OGF-to-EGF conversion using Hankel contours.

Proved variants of the transformations involving reciprocal gamma functions.

Suggested several generalizations and provided new examples.

Abstract

The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively provide a mechanism for converting between a sequence's ordinary and exponential generating function (OGF and EGF, respectively) and vice versa. The Laplace transform provides an integral formula for the EGF-to-OGF transformation, where the reverse OGF-to-EGF operation requires more careful integration techniques. We prove two variants of the OGF-to-EGF transformation integrals from the Hankel loop contour for the reciprocal gamma function and from Fourier series expansions of integral representations for the Hadamard product of two generating functions, respectively. We also suggest several generalizations of these integral formulas and provide new examples along the way.