Asymptotic analysis of the Nörlund and Stirling polynomials (Q2853264)

The Nörlund (actually Nørlund) polynomials \(b_n^{\langle\alpha\rangle}\), \(\alpha\in\mathbb R\), are defined through the exponential generating function \((t/(e^t-1))^\alpha=\sum_nb_n^{\langle\alpha\rangle}z^n/n!\) (thus \(b_n^\alpha=B_n^\alpha(0)\), where \(B_n^\alpha(0)\) are also known as the generalized Bernoulli polynomial), while the Stirling polynomials \(\sigma_n(x)\) are understood as defined trough \((ze^z/(e^z-1))^x=x\sum_n\sigma_n(x)z^n\), that is \(b_n^{\langle\alpha\rangle}=\alpha n!(-1)^n \sigma_n(\alpha)\). The author presents a complete asymptotic analysis of \(b_n^{\langle\alpha\rangle}/n!\) and \(\sigma_n(\alpha)\) for any positive \(\alpha\in\mathbb R\) as \(n\to\infty\). The case when \(\alpha\) is an integer is simpler, while the analysis when \(\alpha\) is non-integral requires more sophisticated tools based on ideas of \textit{P. Flajolet} and \textit{R. Sedgewick} [Analytic combinatorics, Cambridge Univ. Press (2009; Zbl 1165.05001), p. 381--384].