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A uniform asymptotic expansion of the single variable Bell polynomials. (English) Zbl 1025.41020
Bell polynomials \(P_n(z)\) of a single variable are defined by the generating function \[ \exp(z(e^u-1))=1+\sum_{n=1}^\infty P_n(z) u^n/n! \] A representation in terms of Stirling numbers \(S(n,k)\) of the second kind reads \(P_n(z)=\sum_{k=0}^n S(n,k)z^k\). This paper discusses in detail the asymptotic behaviour of \(P_n(z)\) for large values of \(n\). The expansions are in terms of Airy-type expansions and hold uniformly with respect to \(z\) in certain intervals of the negative axis. The surprising result is that the Cauchy-type integral that follows from the generating function has to be written, for \(z\) in certain intervals on the negative axis, as a finite sum of integrals that each are expanded in terms of Airy-type expansions. Furthermore, one expansion is derived, also in terms of the Airy functions, completed with error bounds, holding uniformly for \(z\in (-\infty,-\delta]\) for positive \(\delta\).

41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
33E20 Other functions defined by series and integrals
Full Text: DOI
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