On asymptotic expansions of Stirling numbers of the first and second kinds

A. N. Timashev

Diskretnaya Matematika, Tome 10 (1998) no. 3, pp. 148-159.

Résumé

We consider the problem of asymptotic estimation of Stirling numbers $s(n,N)$ of the first kind and Stirling numbers $\sigma(n,N)$ of the second kind under the condition that $n,N\to\infty$ so that $$ 1\alpha_0\le \alpha=\frac{n}{N}\le \alpha_1\infty, $$ where $\alpha_0$, $\alpha_1$ are some constants. Under this condition, by making use of the saddle point method, we demonstrate that the coefficients of the negative powers of the form $N^{-m}$, $m=1,2,\dots$, in asymptotic expansions of the numbers $s(n,N)$ and $\sigma(n,N)$ in powers of $N^{-1}$ are determined from the representation in the form of a power series of a certain function that depends on the solution of a given non-linear differential equation of the first order with a given initial condition. These results allow us to show that these coefficients obey some linear recurrence relations in the complex plane. As corollaries, we give explicit formulas for the coefficient of $N^{-1}$.