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Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals. (English) Zbl 1237.11011

The author provides a short elementary review of the exponential polynomials (also called single-variable Bell polynomials) \(\phi_n(x)\) defined combinatorially by \(\phi_n(x)=\sum_{k=0}^n S(n,k)x^k\) where \(S(n,k)\) denotes the number of partitions of \(\{1,2,\dots,n\}\) into \(k\) nonempty sets, or formally by \((x{d\over dx})^ne^x= \phi_ n(x)e^ x\). Some new properties are included, and several analysis-related applications are mentioned. One application is given in detail, namely certain Fourier integrals involving \(\Gamma (a+it)\) and \(\Gamma (a+it)\Gamma (b-it)\) are evaluated in terms of Stirling numbers.

MSC:

11B73 Bell and Stirling numbers
33B15 Gamma, beta and polygamma functions
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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