Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals

Abstract

This article is a short elementary review of the exponential polynomials (also called single-variable Bell polynomials) from the point of view of analysis. Some new properties are included, and several analysis-related applications are mentioned. At the end of the paper one application is described in details—certain Fourier integrals involving $\Gamma \left(a+it\right)$ and $\Gamma \left(a+it\right)\Gamma \left(b-it\right)$ are evaluated in terms of Stirling numbers.