Asymptotic Expansions for the Stirling Numbers of the First Kind

@article{Kwang1995AsymptoticEF,
  title={Asymptotic Expansions for the Stirling Numbers of the First Kind},
  author={Hsien-Kuei Kwang},
  journal={J. Comb. Theory, Ser. A},
  year={1995},
  volume={71},
  pages={343-351}
}
Let s(n,m) denote the (unsigned) Stirling numbers of the first kind: s(n,m) := [w] (w(w + 1) · · · (w + n− 1)) (1 ≤ m ≤ n, n ≥ 1). Many different asymptotic expressions for s(n,m), as n→∞, have been proposed in the literature due to their wide applications, cf. Temme [8] for a brief survey of known results together with a uniform asymptotic expansion valid for all m, 1 ≤ m ≤ n. Recently, Wilf [10] provided a considerably more explicit formula when m = O(1), m ≥ 1. His main result is, for m = O… CONTINUE READING