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

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 Γ ( 𝑎 + 𝑖 𝑡 ) and Γ ( 𝑎 + 𝑖 𝑡 ) Γ ( 𝑏 − 𝑖 𝑡 ) are evaluated in terms of Stirling numbers.

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